Results of Elementary Functions I

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
4.2.E1	${\displaystyle{\displaystyle\operatorname{Ln}z=\int_{1}^{z}\frac{\mathrm{d}t}{% t}}}$ `\Ln@@{z} = \int_{1}^{z}\frac{\diff{t}}{t}`	${\displaystyle{\displaystyle z\neq 0}}$	ln(z) = int((1)/(t), t = 1..z)	Log[z] == Integrate[Divide[1,t], {t, 1, z}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.2.E2	${\displaystyle{\displaystyle\ln z=\int_{1}^{z}\frac{\mathrm{d}t}{t}}}$ `\ln@@{z} = \int_{1}^{z}\frac{\diff{t}}{t}`		ln(z) = int((1)/(t), t = 1..z)	Log[z] == Integrate[Divide[1,t], {t, 1, z}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.2.E3	${\displaystyle{\displaystyle\ln z=\ln\left\|z\right\|+\mathrm{i}\operatorname{ph% }z}}$ `\ln@@{z} = \ln@@{\abs{z}}+\iunit\phase@@{z}`	${\displaystyle{\displaystyle-\pi<\operatorname{ph}z,\operatorname{ph}z<\pi}}$	ln(z) = ln(abs(z))+ I*argument(z)	Log[z] == Log[Abs[z]]+ I*Arg[z]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.2.E4	${\displaystyle{\displaystyle z=x}}$ `z = x`	${\displaystyle{\displaystyle-\infty<x,x<0}}$	(x + y*I) = x	(x + y*I) == x	Skipped - no semantic math	Skipped - no semantic math	-	-
4.2.E5	${\displaystyle{\displaystyle\ln z=\ln\left\|z\right\|+\mathrm{i}\operatorname{ph% }z}}$ `\ln@@{z} = \ln@@{\abs{z}}+\iunit\phase@@{z}`	${\displaystyle{\displaystyle-\pi<\operatorname{ph}z,\operatorname{ph}z\leq\pi}}$	ln(z) = ln(abs(z))+ I*argument(z)	Log[z] == Log[Abs[z]]+ I*Arg[z]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.2.E6	${\displaystyle{\displaystyle\operatorname{Ln}z=\ln z+2k\pi\mathrm{i}}}$ `\Ln@@{z} = \ln@@{z}+2k\pi\iunit`		ln(z) = ln(z)+ 2kPi*I	Log[z] == Log[z]+ 2kPi*I	Failure	Failure	Failed [21 / 21] Result: -6.283185308I Test Values: {z = 1/23^(1/2)+1/2I, k = 1} Result: -12.56637062I Test Values: {z = 1/23^(1/2)+1/2I, k = 2} Result: -18.84955592I Test Values: {z = 1/23^(1/2)+1/2I, k = 3} Result: -6.283185308I Test Values: {z = -1/2+1/2I3^(1/2), k = 1} ... skip entries to safe data	Failed [21 / 21] Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0, -12.566370614359172] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.2.E7	${\displaystyle{\displaystyle\ln\left(x+\mathrm{i}0\right)=\ln\|x\|+i\pi}}$ `\ln@{x+\iunit 0} = \ln@@{\|x\|}+ i\pi`	${\displaystyle{\displaystyle-\infty<x,x<0}}$	ln(x + I0) = ln(abs(x))+ IPi	Log[x + I0] == Log[Abs[x]]+ IPi	Failure	Successful	Error	Skip - symbolical successful subtest
4.2.E7	${\displaystyle{\displaystyle\ln\left(x-\mathrm{i}0\right)=\ln\|x\|-i\pi}}$ `\ln@{x-\iunit 0} = \ln@@{\|x\|}- i\pi`	${\displaystyle{\displaystyle-\infty<x,x<0}}$	ln(x - I0) = ln(abs(x))- IPi	Log[x - I0] == Log[Abs[x]]- IPi	Failure	Failure	Error	Skip - No test values generated
4.2.E8	${\displaystyle{\displaystyle\operatorname{log}_{a}z=\ifrac{\ln z}{\ln a}}}$ `\genlog{a}@@{z} = \ifrac{\ln@@{z}}{\ln@@{a}}`		log[a](z) = (ln(z))/(ln(a))	Log[a,z] == Divide[Log[z],Log[a]]	Successful	Successful	-	Successful [Tested: 42]
4.2.E9	${\displaystyle{\displaystyle\operatorname{log}_{a}z=\frac{\operatorname{log}_{% b}z}{\operatorname{log}_{b}a}}}$ `\genlog{a}@@{z} = \frac{\genlog{b}@@{z}}{\genlog{b}@@{a}}`		log[a](z) = (log[b](z))/(log[b](a))	Log[a,z] == Divide[Log[b,z],Log[b,a]]	Successful	Successful	-	Successful [Tested: 252]
4.2.E10	${\displaystyle{\displaystyle\operatorname{log}_{a}b=\frac{1}{\operatorname{log% }_{b}a}}}$ `\genlog{a}@@{b} = \frac{1}{\genlog{b}@@{a}}`		log[a](b) = (1)/(log[b](a))	Log[a,b] == Divide[1,Log[b,a]]	Successful	Successful	-	Successful [Tested: 36]
4.2.E11	${\displaystyle{\displaystyle e=2.71828\ 18284\ 59045\ 23536\dots}}$ `e = 2.71828\ 18284\ 59045\ 23536\dots`		exp(1) = 2.71828182845904523536	E == 2.71828182845904523536	Successful	Successful	-	Successful [Tested: 1]
4.2.E12	${\displaystyle{\displaystyle\ln e=1}}$ `\ln@@{e} = 1`		ln(exp(1)) = 1	Log[E] == 1	Successful	Successful	-	Successful [Tested: 1]
4.2.E13	${\displaystyle{\displaystyle\int_{1}^{e}\frac{\mathrm{d}t}{t}=1}}$ `\int_{1}^{e}\frac{\diff{t}}{t} = 1`		int((1)/(t), t = 1..exp(1)) = 1	Integrate[Divide[1,t], {t, 1, E}, GenerateConditions->None] == 1	Successful	Successful	-	Successful [Tested: 1]
4.2.E14	${\displaystyle{\displaystyle\operatorname{log}_{e}z=\ln z}}$ `\genlog{e}@@{z} = \ln@@{z}`		log[exp(1)](z) = ln(z)	Log[E,z] == Log[z]	Successful	Successful	-	Successful [Tested: 7]
4.2.E15	${\displaystyle{\displaystyle\operatorname{log}_{10}z=\ifrac{(\ln z)}{(\ln 10)}}}$ `\genlog{10}@@{z} = \ifrac{(\ln@@{z})}{(\ln@@{10})}`		log[10](z) = (ln(z))/(ln(10))	Log[10,z] == Divide[Log[z],Log[10]]	Successful	Successful	-	Successful [Tested: 7]
4.2.E15	${\displaystyle{\displaystyle\ifrac{(\ln z)}{(\ln 10)}=(\operatorname{log}_{10}% e)\ln z}}$ `\ifrac{(\ln@@{z})}{(\ln@@{10})} = (\genlog{10}@@{e})\ln@@{z}`		(ln(z))/(ln(10)) = (log[10](exp(1)))*ln(z)	Divide[Log[z],Log[10]] == (Log[10,E])*Log[z]	Successful	Successful	-	Successful [Tested: 7]
4.2.E16	${\displaystyle{\displaystyle\ln z=(\ln 10)\operatorname{log}_{10}z}}$ `\ln@@{z} = (\ln@@{10})\genlog{10}@@{z}`		ln(z) = (ln(10))*log[10](z)	Log[z] == (Log[10])*Log[10,z]	Successful	Successful	-	Successful [Tested: 7]
4.2.E17	${\displaystyle{\displaystyle\operatorname{log}_{10}e=0.43429\ 44819\ 03251\ 82% 765\dots}}$ `\genlog{10}@@{e} = 0.43429\ 44819\ 03251\ 82765\dots`		log[10](exp(1)) = 0.43429448190325182765	Log[10,E] == 0.43429448190325182765	Failure	Successful	Successful [Tested: 0]	Successful [Tested: 1]
4.2.E18	${\displaystyle{\displaystyle\ln 10=2.30258\ 50929\ 94045\ 68401\dots}}$ `\ln@@{10} = 2.30258\ 50929\ 94045\ 68401\dots`		ln(10) = 2.30258509299404568401	Log[10] == 2.30258509299404568401	Successful	Successful	-	Successful [Tested: 1]
4.2.E20	${\displaystyle{\displaystyle\exp\left(z+2\pi i\right)=\exp z}}$ `\exp@{z+2\pi i} = \exp@@{z}`		exp(z + 2PiI) = exp(z)	Exp[z + 2PiI] == Exp[z]	Successful	Successful	-	Successful [Tested: 7]
4.2.E21	${\displaystyle{\displaystyle\exp\left(-z\right)=1/\exp\left(z\right)}}$ `\exp@{-z} = 1/\exp@{z}`		exp(- z) = 1/exp(z)	Exp[- z] == 1/Exp[z]	Successful	Successful	-	Successful [Tested: 7]
4.2.E22	${\displaystyle{\displaystyle\|\exp z\|=\exp\left(\Re z\right)}}$ `\|\exp@@{z}\| = \exp@{\realpart@@{z}}`		abs(exp(z)) = exp(Re(z))	Abs[Exp[z]] == Exp[Re[z]]	Successful	Successful	-	Successful [Tested: 7]
4.2.E23	${\displaystyle{\displaystyle\operatorname{ph}\left(\exp z\right)=\Im z+2k\pi}}$ `\phase@{\exp@@{z}} = \imagpart@@{z}+2k\pi`		argument(exp(z)) = Im(z)+ 2kPi	Arg[Exp[z]] == Im[z]+ 2kPi	Failure	Failure	Failed [21 / 21] Result: -6.283185308 Test Values: {z = 1/23^(1/2)+1/2I, k = 1, k = 3} Result: -12.56637062 Test Values: {z = 1/23^(1/2)+1/2I, k = 2, k = 3} Result: -18.84955592 Test Values: {z = 1/23^(1/2)+1/2I, k = 3, k = 3} Result: -6.283185308 Test Values: {z = -1/2+1/2I3^(1/2), k = 1, k = 3} ... skip entries to safe data	Failed [7 / 7] Result: -18.84955592153876 Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: -18.84955592153876 Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.2.E24	${\displaystyle{\displaystyle\exp z=e^{x}\cos y+ie^{x}\sin y}}$ `\exp@@{z} = e^{x}\cos@@{y}+ie^{x}\sin@@{y}`		exp(x + yI) = exp(x)cos(y)+ Iexp(x)sin(y)	Exp[x + yI] == Exp[x]Cos[y]+ IExp[x]Sin[y]	Successful	Successful	-	Successful [Tested: 18]
4.2.E26	${\displaystyle{\displaystyle z^{a}=\exp\left(a\operatorname{Ln}z\right)}}$ `z^{a} = \exp@{a\Ln@@{z}}`	${\displaystyle{\displaystyle z\neq 0}}$	(z)^(a) = exp(a*ln(z))	(z)^(a) == Exp[a*Log[z]]	Successful	Successful	-	Successful [Tested: 42]
4.2.E28	${\displaystyle{\displaystyle z^{a}=\exp\left(a\ln z\right)}}$ `z^{a} = \exp@{a\ln@@{z}}`		(z)^(a) = exp(a*ln(z))	(z)^(a) == Exp[a*Log[z]]	Successful	Successful	-	Successful [Tested: 42]
4.2.E29	${\displaystyle{\displaystyle\|z^{a}\|=\|z\|^{\Re a}\exp\left(-(\Im a)\operatorname% {ph}z\right)}}$ `\|z^{a}\| = \|z\|^{\realpart@@{a}}\exp@{-(\imagpart@@{a})\phase@@{z}}`		abs((z)^(a)) = (abs(z))^(Re(a))* exp(-(Im(a))*argument(z))	Abs[(z)^(a)] == (Abs[z])^(Re[a])* Exp[-(Im[a])*Arg[z]]	Failure	Failure	Successful [Tested: 42]	Successful [Tested: 42]
4.2.E30	${\displaystyle{\displaystyle\operatorname{ph}\left(z^{a}\right)=(\Re a)% \operatorname{ph}z+(\Im a)\ln\|z\|}}$ `\phase@{z^{a}} = (\realpart@@{a})\phase@@{z}+(\imagpart@@{a})\ln@@{\|z\|}`		argument((z)^(a)) = (Re(a))argument(z)+(Im(a))ln(abs(z))	Arg[(z)^(a)] == (Re[a])Arg[z]+(Im[a])Log[Abs[z]]	Failure	Failure	Failed [6 / 42] Result: -6.283185308 Test Values: {a = -1.5, z = -1/23^(1/2)-1/2I} Result: 6.283185308 Test Values: {a = 1.5, z = -1/23^(1/2)-1/2I} Result: 6.283185307 Test Values: {a = -2, z = -1/2+1/2I3^(1/2)} Result: -6.283185309 Test Values: {a = -2, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [6 / 42] Result: -6.283185307179586 Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} Result: 6.283185307179586 Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
4.2#Ex1	${\displaystyle{\displaystyle\|z^{a}\|=\|z\|^{a}}}$ `\|z^{a}\| = \|z\|^{a}`		abs((z)^(a)) = (abs(z))^(a)	Abs[(z)^(a)] == (Abs[z])^(a)	Skipped - no semantic math	Skipped - no semantic math	-	-
4.2#Ex2	${\displaystyle{\displaystyle\operatorname{ph}\left(z^{a}\right)=a\operatorname% {ph}z}}$ `\phase@{z^{a}} = a\phase@@{z}`		argument((z)^(a)) = a*argument(z)	Arg[(z)^(a)] == a*Arg[z]	Failure	Failure	Failed [6 / 42] Result: -6.283185308 Test Values: {a = -1.5, z = -1/23^(1/2)-1/2I} Result: 6.283185308 Test Values: {a = 1.5, z = -1/23^(1/2)-1/2I} Result: 6.283185307 Test Values: {a = -2, z = -1/2+1/2I3^(1/2)} Result: -6.283185309 Test Values: {a = -2, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [6 / 42] Result: -6.283185307179586 Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} Result: 6.283185307179586 Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
4.2.E32	${\displaystyle{\displaystyle e^{z}=\exp z}}$ `e^{z} = \exp@@{z}`		exp(z) = exp(z)	Exp[z] == Exp[z]	Successful	Successful	-	Successful [Tested: 7]
4.2.E33	${\displaystyle{\displaystyle e^{z}=(\exp z)\exp\left(2kz\pi\mathrm{i}\right)}}$ `e^{z} = (\exp@@{z})\exp@{2kz\pi\iunit}`		exp(z) = (exp(z))exp(2kzPi*I)	Exp[z] == (Exp[z])Exp[2kzPi*I]	Failure	Failure	Failed [16 / 21] Result: 1.989606315+1.174241786I Test Values: {z = 1/23^(1/2)+1/2I, k = 1, k = 3} Result: 2.084725711+1.143917762I Test Values: {z = 1/23^(1/2)+1/2I, k = 2, k = 3} Result: 2.086486474+1.139979111I Test Values: {z = 1/23^(1/2)+1/2I, k = 3, k = 3} Result: .3946493584+.4640329579I Test Values: {z = -1/2+1/2I3^(1/2), k = 1, k = 3} ... skip entries to safe data	Failed [6 / 7] Result: Complex[2.0864864733305994, 1.139979110702337] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.3929465878104918, 0.4620308216689905] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.2.E36	${\displaystyle{\displaystyle-\pi\leq\Im\left(\frac{1}{a}\operatorname{Ln}w% \right)}}$ `-\pi \leq \imagpart@@{\left(\frac{1}{a}\Ln@@{w}\right)}`		- Pi <= Im((1)/(a)*ln(w))	- Pi <= Im[Divide[1,a]*Log[w]]	Failure	Failure	Failed [5 / 60] Result: -3.141592654 <= -4.188790204 Test Values: {a = -.5, w = -1/2+1/2I3^(1/2)} Result: -3.141592654 <= -6.283185308 Test Values: {a = -.5, w = -1.5} Result: -3.141592654 <= -6.283185308 Test Values: {a = -.5, w = -.5} Result: -3.141592654 <= -6.283185308 Test Values: {a = -.5, w = -2} ... skip entries to safe data	Failed [5 / 60] Result: False Test Values: {Rule[a, -0.5], Rule[w, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: False Test Values: {Rule[a, -0.5], Rule[w, -1.5]} ... skip entries to safe data
4.2.E36	${\displaystyle{\displaystyle\Im\left(\frac{1}{a}\operatorname{Ln}w\right)\leq% \pi}}$ `\imagpart@@{\left(\frac{1}{a}\Ln@@{w}\right)} \leq \pi`		Im((1)/(a)*ln(w)) <= Pi	Im[Divide[1,a]*Log[w]] <= Pi	Failure	Failure	Failed [5 / 60] Result: 5.235987758 <= 3.141592654 Test Values: {a = -.5, w = -1/23^(1/2)-1/2I} Result: 4.188790204 <= 3.141592654 Test Values: {a = .5, w = -1/2+1/2I3^(1/2)} Result: 6.283185308 <= 3.141592654 Test Values: {a = .5, w = -1.5} Result: 6.283185308 <= 3.141592654 Test Values: {a = .5, w = -.5} ... skip entries to safe data	Failed [5 / 60] Result: False Test Values: {Rule[a, -0.5], Rule[w, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} Result: False Test Values: {Rule[a, 0.5], Rule[w, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.4.E1	${\displaystyle{\displaystyle\ln 1=0}}$ `\ln@@{1} = 0`		ln(1) = 0	Log[1] == 0	Successful	Successful	-	Successful [Tested: 1]
4.4.E2	${\displaystyle{\displaystyle\ln\left(-1+\mathrm{i}0\right)=+\pi\mathrm{i}}}$ `\ln@{-1+\iunit 0} = +\pi\iunit`		ln(- 1 + I0) = + PiI	Log[- 1 + I0] == + PiI	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 1]
4.4.E2	${\displaystyle{\displaystyle\ln\left(-1-\mathrm{i}0\right)=-\pi\mathrm{i}}}$ `\ln@{-1-\iunit 0} = -\pi\iunit`		ln(- 1 - I0) = - PiI	Log[- 1 - I0] == - PiI	Failure	Failure	Failed [1 / 1] Result: 6.283185308*I Test Values: {}	Failed [1 / 1] Result: Complex[0.0, 6.283185307179586] Test Values: {}
4.4.E3	${\displaystyle{\displaystyle\ln\left(+\mathrm{i}\right)=+\tfrac{1}{2}\pi% \mathrm{i}}}$ `\ln@{+\iunit} = +\tfrac{1}{2}\pi\iunit`		ln(+ I) = +(1)/(2)PiI	Log[+ I] == +Divide[1,2]PiI	Successful	Successful	-	Successful [Tested: 1]
4.4.E3	${\displaystyle{\displaystyle\ln\left(-\mathrm{i}\right)=-\tfrac{1}{2}\pi% \mathrm{i}}}$ `\ln@{-\iunit} = -\tfrac{1}{2}\pi\iunit`		ln(- I) = -(1)/(2)PiI	Log[- I] == -Divide[1,2]PiI	Successful	Successful	-	Successful [Tested: 1]
4.4.E4	${\displaystyle{\displaystyle e^{0}=1}}$ `e^{0} = 1`		exp(0) = 1	Exp[0] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
4.4.E5	${\displaystyle{\displaystyle e^{+\pi\mathrm{i}}=-1}}$ `e^{+\pi\iunit} = -1`		exp(+ Pi*I) = - 1	Exp[+ Pi*I] == - 1	Successful	Successful	-	Successful [Tested: 1]
4.4.E5	${\displaystyle{\displaystyle e^{-\pi\mathrm{i}}=-1}}$ `e^{-\pi\iunit} = -1`		exp(- Pi*I) = - 1	Exp[- Pi*I] == - 1	Successful	Successful	-	Successful [Tested: 1]
4.4.E6	${\displaystyle{\displaystyle e^{+\pi\mathrm{i}/2}=+\mathrm{i}}}$ `e^{+\pi\iunit/2} = +\iunit`		exp(+ Pi*I/2) = + I	Exp[+ Pi*I/2] == + I	Successful	Successful	-	Successful [Tested: 1]
4.4.E6	${\displaystyle{\displaystyle e^{-\pi\mathrm{i}/2}=-\mathrm{i}}}$ `e^{-\pi\iunit/2} = -\iunit`		exp(- Pi*I/2) = - I	Exp[- Pi*I/2] == - I	Successful	Successful	-	Successful [Tested: 1]
4.4.E7	${\displaystyle{\displaystyle e^{2\pi k\mathrm{i}}=1}}$ `e^{2\pi k\iunit} = 1`		exp(2Pik*I) = 1	Exp[2Pik*I] == 1	Successful	Successful	-	Successful [Tested: 1]
4.4.E8	${\displaystyle{\displaystyle e^{+\pi\mathrm{i}/3}=\frac{1}{2}+\mathrm{i}\frac{% \sqrt{3}}{2}}}$ `e^{+\pi\iunit/3} = \frac{1}{2}+\iunit\frac{\sqrt{3}}{2}`		exp(+ PiI/3) = (1)/(2)+ I(sqrt(3))/(2)	Exp[+ PiI/3] == Divide[1,2]+ IDivide[Sqrt[3],2]	Successful	Successful	-	Successful [Tested: 1]
4.4.E8	${\displaystyle{\displaystyle e^{-\pi\mathrm{i}/3}=\frac{1}{2}-\mathrm{i}\frac{% \sqrt{3}}{2}}}$ `e^{-\pi\iunit/3} = \frac{1}{2}-\iunit\frac{\sqrt{3}}{2}`		exp(- PiI/3) = (1)/(2)- I(sqrt(3))/(2)	Exp[- PiI/3] == Divide[1,2]- IDivide[Sqrt[3],2]	Successful	Successful	-	Successful [Tested: 1]
4.4.E9	${\displaystyle{\displaystyle e^{+2\pi\mathrm{i}/3}=-\frac{1}{2}+\mathrm{i}% \frac{\sqrt{3}}{2}}}$ `e^{+ 2\pi\iunit/3} = -\frac{1}{2}+\iunit\frac{\sqrt{3}}{2}`		exp(+ 2PiI/3) = -(1)/(2)+ I*(sqrt(3))/(2)	Exp[+ 2PiI/3] == -Divide[1,2]+ I*Divide[Sqrt[3],2]	Successful	Successful	-	Successful [Tested: 1]
4.4.E9	${\displaystyle{\displaystyle e^{-2\pi\mathrm{i}/3}=-\frac{1}{2}-\mathrm{i}% \frac{\sqrt{3}}{2}}}$ `e^{- 2\pi\iunit/3} = -\frac{1}{2}-\iunit\frac{\sqrt{3}}{2}`		exp(- 2PiI/3) = -(1)/(2)- I*(sqrt(3))/(2)	Exp[- 2PiI/3] == -Divide[1,2]- I*Divide[Sqrt[3],2]	Successful	Successful	-	Successful [Tested: 1]
4.4.E10	${\displaystyle{\displaystyle e^{+\pi\mathrm{i}/4}=\frac{1}{\sqrt{2}}+\mathrm{i% }\frac{1}{\sqrt{2}}}}$ `e^{+\pi\iunit/4} = \frac{1}{\sqrt{2}}+\iunit\frac{1}{\sqrt{2}}`		exp(+ PiI/4) = (1)/(sqrt(2))+ I(1)/(sqrt(2))	Exp[+ PiI/4] == Divide[1,Sqrt[2]]+ IDivide[1,Sqrt[2]]	Successful	Successful	-	Successful [Tested: 1]
4.4.E10	${\displaystyle{\displaystyle e^{-\pi\mathrm{i}/4}=\frac{1}{\sqrt{2}}-\mathrm{i% }\frac{1}{\sqrt{2}}}}$ `e^{-\pi\iunit/4} = \frac{1}{\sqrt{2}}-\iunit\frac{1}{\sqrt{2}}`		exp(- PiI/4) = (1)/(sqrt(2))- I(1)/(sqrt(2))	Exp[- PiI/4] == Divide[1,Sqrt[2]]- IDivide[1,Sqrt[2]]	Successful	Successful	-	Successful [Tested: 1]
4.4.E11	${\displaystyle{\displaystyle e^{+3\pi\mathrm{i}/4}=-\frac{1}{\sqrt{2}}+\mathrm% {i}\frac{1}{\sqrt{2}}}}$ `e^{+ 3\pi\iunit/4} = -\frac{1}{\sqrt{2}}+\iunit\frac{1}{\sqrt{2}}`		exp(+ 3PiI/4) = -(1)/(sqrt(2))+ I*(1)/(sqrt(2))	Exp[+ 3PiI/4] == -Divide[1,Sqrt[2]]+ I*Divide[1,Sqrt[2]]	Successful	Successful	-	Successful [Tested: 1]
4.4.E11	${\displaystyle{\displaystyle e^{-3\pi\mathrm{i}/4}=-\frac{1}{\sqrt{2}}-\mathrm% {i}\frac{1}{\sqrt{2}}}}$ `e^{- 3\pi\iunit/4} = -\frac{1}{\sqrt{2}}-\iunit\frac{1}{\sqrt{2}}`		exp(- 3PiI/4) = -(1)/(sqrt(2))- I*(1)/(sqrt(2))	Exp[- 3PiI/4] == -Divide[1,Sqrt[2]]- I*Divide[1,Sqrt[2]]	Successful	Successful	-	Successful [Tested: 1]
4.4.E12	${\displaystyle{\displaystyle{\mathrm{i}^{+\mathrm{i}}}=e^{-\pi/2}}}$ `\iunit^{+\iunit} = e^{-\pi/2}`		(I)^(+ I) = exp(- Pi/2)	(I)^(+ I) == Exp[- Pi/2]	Successful	Successful	-	Successful [Tested: 1]
4.4.E12	${\displaystyle{\displaystyle{\mathrm{i}^{-\mathrm{i}}}=e^{+\pi/2}}}$ `\iunit^{-\iunit} = e^{+\pi/2}`		(I)^(- I) = exp(+ Pi/2)	(I)^(- I) == Exp[+ Pi/2]	Successful	Successful	-	Successful [Tested: 1]
4.4.E13	${\displaystyle{\displaystyle\lim_{x\to\infty}x^{-a}\ln x=0}}$ `\lim_{x\to\infty}x^{-a}\ln@@{x} = 0`	${\displaystyle{\displaystyle\Re a>0}}$	limit((x)^(- a)* ln(x), x = infinity) = 0	Limit[(x)^(- a)* Log[x], x -> Infinity, GenerateConditions->None] == 0	Successful	Successful	-	Successful [Tested: 3]
4.4.E14	${\displaystyle{\displaystyle\lim_{x\to 0}x^{a}\ln x=0}}$ `\lim_{x\to 0}x^{a}\ln@@{x} = 0`	${\displaystyle{\displaystyle\Re a>0}}$	limit((x)^(a)* ln(x), x = 0) = 0	Limit[(x)^(a)* Log[x], x -> 0, GenerateConditions->None] == 0	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
4.4.E15	${\displaystyle{\displaystyle\lim_{x\to\infty}x^{a}e^{-x}=0}}$ `\lim_{x\to\infty}x^{a}e^{-x} = 0`		limit((x)^(a)* exp(- x), x = infinity) = 0	Limit[(x)^(a)* Exp[- x], x -> Infinity, GenerateConditions->None] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
4.4.E16	${\displaystyle{\displaystyle\lim_{z\to\infty}z^{a}e^{-z}=0}}$ `\lim_{z\to\infty}z^{a}e^{-z} = 0`	${\displaystyle{\displaystyle\|\operatorname{ph}z\|\leq\tfrac{1}{2}\pi-\delta,% \tfrac{1}{2}\pi-\delta<\tfrac{1}{2}\pi}}$	limit((z)^(a)* exp(- z), z = infinity) = 0	Limit[(z)^(a)* Exp[- z], z -> Infinity, GenerateConditions->None] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
4.4.E17	${\displaystyle{\displaystyle\lim_{n\to\infty}\left(1+\frac{z}{n}\right)^{n}=e^% {z}}}$ `\lim_{n\to\infty}\left(1+\frac{z}{n}\right)^{n} = e^{z}`	${\displaystyle{\displaystyle z=}}$	limit((1 +(z)/(n))^(n), n = infinity) = exp(z)	Limit[(1 +Divide[z,n])^(n), n -> Infinity, GenerateConditions->None] == Exp[z]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.4.E18	${\displaystyle{\displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}=e}}$ `\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n} = e`		limit((1 +(1)/(n))^(n), n = infinity) = exp(1)	Limit[(1 +Divide[1,n])^(n), n -> Infinity, GenerateConditions->None] == E	Skipped - no semantic math	Skipped - no semantic math	-	-
4.4.E19	${\displaystyle{\displaystyle\lim_{n\to\infty}\left(\left(\sum^{n}_{k=1}\frac{1% }{k}\right)-\ln n\right)=\gamma}}$ `\lim_{n\to\infty}\left(\left(\sum^{n}_{k=1}\frac{1}{k}\right)-\ln@@{n}\right) = \EulerConstant`		limit((sum((1)/(k), k = 1..n))- ln(n), n = infinity) = gamma	Limit[(Sum[Divide[1,k], {k, 1, n}, GenerateConditions->None])- Log[n], n -> Infinity, GenerateConditions->None] == EulerGamma	Successful	Successful	-	Successful [Tested: 1]
4.4.E19	${\displaystyle{\displaystyle\gamma=0.57721\ 56649\ 01532\ 86060\dots}}$ `\EulerConstant = 0.57721\ 56649\ 01532\ 86060\dots`		gamma = 0.57721566490153286060	EulerGamma == 0.57721566490153286060	Successful	Successful	-	Successful [Tested: 1]
4.5.E1	${\displaystyle{\displaystyle\frac{x}{1+x}<\ln\left(1+x\right)}}$ `\frac{x}{1+x} < \ln@{1+x}`	${\displaystyle{\displaystyle x>-1,x\neq 0}}$	(x)/(1 + x) < ln(1 + x)	Divide[x,1 + x] < Log[1 + x]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.5.E1	${\displaystyle{\displaystyle\ln\left(1+x\right)<x}}$ `\ln@{1+x} < x`	${\displaystyle{\displaystyle x>-1,x\neq 0}}$	ln(1 + x) < x	Log[1 + x] < x	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.5.E2	${\displaystyle{\displaystyle x<-\ln\left(1-x\right)}}$ `x < -\ln@{1-x}`	${\displaystyle{\displaystyle x<1,x\neq 0}}$	x < - ln(1 - x)	x < - Log[1 - x]	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 1]
4.5.E2	${\displaystyle{\displaystyle-\ln\left(1-x\right)<\frac{x}{1-x}}}$ `-\ln@{1-x} < \frac{x}{1-x}`	${\displaystyle{\displaystyle x<1,x\neq 0}}$	- ln(1 - x) < (x)/(1 - x)	- Log[1 - x] < Divide[x,1 - x]	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 1]
4.5.E3	${\displaystyle{\displaystyle\|\ln\left(1-x\right)\|<\tfrac{3}{2}x}}$ `\|\ln@{1-x}\| < \tfrac{3}{2}x`	${\displaystyle{\displaystyle 0<x,x\leq 0.5828\dots}}$	abs(ln(1 - x)) < (3)/(2)*x	Abs[Log[1 - x]] < Divide[3,2]*x	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 1]
4.5.E4	${\displaystyle{\displaystyle\ln x\leq x-1}}$ `\ln@@{x} \leq x-1`	${\displaystyle{\displaystyle x>0}}$	ln(x) <= x - 1	Log[x] <= x - 1	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.5.E5	${\displaystyle{\displaystyle\ln x\leq a(x^{1/a}-1)}}$ `\ln@@{x} \leq a(x^{1/a}-1)`	${\displaystyle{\displaystyle a>0,x>0}}$	ln(x) <= a*((x)^(1/a)- 1)	Log[x] <= a*((x)^(1/a)- 1)	Error	Failure	-	Successful [Tested: 9]
4.5.E6	${\displaystyle{\displaystyle\|\ln\left(1+z\right)\|\leq-\ln\left(1-\|z\|\right)}}$ `\|\ln@{1+z}\| \leq -\ln@{1-\|z\|}`	${\displaystyle{\displaystyle\|z\|<1}}$	abs(ln(1 + z)) <= - ln(1 -abs(z))	Abs[Log[1 + z]] <= - Log[1 -Abs[z]]	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 1]
4.5.E7	${\displaystyle{\displaystyle e^{-x/(1-x)}<1-x}}$ `e^{-x/(1-x)} < 1-x`	${\displaystyle{\displaystyle x<1}}$	exp(- x/(1 - x)) < 1 - x	Exp[- x/(1 - x)] < 1 - x	Skipped - no semantic math	Failure	-	Successful [Tested: 1]
4.5.E7	${\displaystyle{\displaystyle 1-x<e^{-x}}}$ `1-x < e^{-x}`	${\displaystyle{\displaystyle x<1}}$	1 - x < exp(- x)	1 - x < Exp[- x]	Error	Failure	-	Successful [Tested: 1]
4.5.E8	${\displaystyle{\displaystyle 1+x<e^{x}}}$ `1+x < e^{x}`	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	1 + x < exp(x)	1 + x < Exp[x]	Skipped - no semantic math	Failure	-	Successful [Tested: 3]
4.5.E9	${\displaystyle{\displaystyle e^{x}<\frac{1}{1-x}}}$ `e^{x} < \frac{1}{1-x}`	${\displaystyle{\displaystyle x<1}}$	exp(x) < (1)/(1 - x)	Exp[x] < Divide[1,1 - x]	Skipped - no semantic math	Failure	-	Successful [Tested: 1]
4.5.E10	${\displaystyle{\displaystyle\frac{x}{1+x}<1-e^{-x}}}$ `\frac{x}{1+x} < 1-e^{-x}`	${\displaystyle{\displaystyle x>-1}}$	(x)/(1 + x) < 1 - exp(- x)	Divide[x,1 + x] < 1 - Exp[- x]	Skipped - no semantic math	Failure	-	Successful [Tested: 3]
4.5.E10	${\displaystyle{\displaystyle 1-e^{-x}<x}}$ `1-e^{-x} < x`	${\displaystyle{\displaystyle x>-1}}$	1 - exp(- x) < x	1 - Exp[- x] < x	Error	Failure	-	Successful [Tested: 3]
4.5.E11	${\displaystyle{\displaystyle x<e^{x}-1}}$ `x < e^{x}-1`	${\displaystyle{\displaystyle x<1}}$	x < exp(x)- 1	x < Exp[x]- 1	Skipped - no semantic math	Failure	-	Successful [Tested: 1]
4.5.E11	${\displaystyle{\displaystyle e^{x}-1<\frac{x}{1-x}}}$ `e^{x}-1 < \frac{x}{1-x}`	${\displaystyle{\displaystyle x<1}}$	exp(x)- 1 < (x)/(1 - x)	Exp[x]- 1 < Divide[x,1 - x]	Error	Failure	-	Successful [Tested: 1]
4.5.E12	${\displaystyle{\displaystyle e^{x/(1+x)}<1+x}}$ `e^{x/(1+x)} < 1+x`	${\displaystyle{\displaystyle x>-1}}$	exp(x/(1 + x)) < 1 + x	Exp[x/(1 + x)] < 1 + x	Skipped - no semantic math	Failure	-	Successful [Tested: 3]
4.5.E13	${\displaystyle{\displaystyle e^{xy/(x+y)}<\left(1+\frac{x}{y}\right)^{y}}}$ `e^{xy/(x+y)} < \left(1+\frac{x}{y}\right)^{y}`	${\displaystyle{\displaystyle x>0,y>0}}$	exp(x*y/(x + y)) < (1 +(x)/(y))^(y)	Exp[x*y/(x + y)] < (1 +Divide[x,y])^(y)	Skipped - no semantic math	Failure	-	Successful [Tested: 9]
4.5.E13	${\displaystyle{\displaystyle\left(1+\frac{x}{y}\right)^{y}<e^{x}}}$ `\left(1+\frac{x}{y}\right)^{y} < e^{x}`	${\displaystyle{\displaystyle x>0,y>0}}$	(1 +(x)/(y))^(y) < exp(x)	(1 +Divide[x,y])^(y) < Exp[x]	Error	Failure	-	Successful [Tested: 9]
4.5.E14	${\displaystyle{\displaystyle e^{-x}<1-\tfrac{1}{2}x}}$ `e^{-x} < 1-\tfrac{1}{2}x`	${\displaystyle{\displaystyle 0<x,x\leq 1.5936\dots}}$	exp(- x) < 1 -(1)/(2)*x	Exp[- x] < 1 -Divide[1,2]*x	Skipped - no semantic math	Failure	-	Successful [Tested: 2]
4.5.E15	${\displaystyle{\displaystyle\tfrac{1}{4}\|z\|<\|e^{z}-1\|}}$ `\tfrac{1}{4}\|z\| < \|e^{z}-1\|`	${\displaystyle{\displaystyle 0<\|z\|,\|z\|<1}}$	(1)/(4)*abs(z) < abs(exp(z)- 1)	Divide[1,4]*Abs[z] < Abs[Exp[z]- 1]	Skipped - no semantic math	Failure	-	Successful [Tested: 1]
4.5.E15	${\displaystyle{\displaystyle\|e^{z}-1\|<\tfrac{7}{4}\|z\|}}$ `\|e^{z}-1\| < \tfrac{7}{4}\|z\|`	${\displaystyle{\displaystyle 0<\|z\|,\|z\|<1}}$	abs(exp(z)- 1) < (7)/(4)*abs(z)	Abs[Exp[z]- 1] < Divide[7,4]*Abs[z]	Error	Failure	-	Successful [Tested: 1]
4.5.E16	${\displaystyle{\displaystyle\|e^{z}-1\|\leq e^{\|z\|}-1}}$ `\|e^{z}-1\| \leq e^{\|z\|}-1`		abs(exp(z)- 1) <= exp(abs(z))- 1	Abs[Exp[z]- 1] <= Exp[Abs[z]]- 1	Skipped - no semantic math	Failure	-	Successful [Tested: 1]
4.5.E16	${\displaystyle{\displaystyle e^{\|z\|}-1\leq\|z\|e^{\|z\|}}}$ `e^{\|z\|}-1 \leq \|z\|e^{\|z\|}`		exp(abs(z))- 1 <= abs(z)*exp(abs(z))	Exp[Abs[z]]- 1 <= Abs[z]*Exp[Abs[z]]	Error	Failure	-	Successful [Tested: 1]
4.7.E1	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\ln z=\frac{1}{z}}}$ `\deriv{}{z}\ln@@{z} = \frac{1}{z}`		diff(ln(z), z) = (1)/(z)	D[Log[z], z] == Divide[1,z]	Successful	Successful	-	Successful [Tested: 7]
4.7.E2	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{Ln}z=% \frac{1}{z}}}$ `\deriv{}{z}\Ln@@{z} = \frac{1}{z}`		diff(ln(z), z) = (1)/(z)	D[Log[z], z] == Divide[1,z]	Successful	Successful	-	Successful [Tested: 7]
4.7.E3	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\ln z=(-% 1)^{n-1}(n-1)!z^{-n}}}$ `\deriv[n]{}{z}\ln@@{z} = (-1)^{n-1}(n-1)!z^{-n}`		diff(ln(z), [z$(n)]) = (- 1)^(n - 1)factorial(n - 1)(z)^(- n)	D[Log[z], {z, n}] == (- 1)^(n - 1)(n - 1)!(z)^(- n)	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
4.7.E4	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}% \operatorname{Ln}z=(-1)^{n-1}(n-1)!z^{-n}}}$ `\deriv[n]{}{z}\Ln@@{z} = (-1)^{n-1}(n-1)!z^{-n}`		diff(ln(z), [z$(n)]) = (- 1)^(n - 1)factorial(n - 1)(z)^(- n)	D[Log[z], {z, n}] == (- 1)^(n - 1)(n - 1)!(z)^(- n)	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
4.7.E7	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}e^{z}=e^{z}}}$ `\deriv{}{z}e^{z} = e^{z}`		diff(exp(z), z) = exp(z)	D[Exp[z], z] == Exp[z]	Successful	Successful	-	Successful [Tested: 7]
4.7.E8	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}e^{az}=ae^{az}}}$ `\deriv{}{z}e^{az} = ae^{az}`		diff(exp(az), z) = aexp(a*z)	D[Exp[az], z] == aExp[a*z]	Successful	Successful	-	Successful [Tested: 42]
4.7.E9	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}a^{z}=a^{z}\ln a}}$ `\deriv{}{z}a^{z} = a^{z}\ln@@{a}`	${\displaystyle{\displaystyle a\neq 0}}$	diff((a)^(z), z) = (a)^(z)* ln(a)	D[(a)^(z), z] == (a)^(z)* Log[a]	Successful	Successful	-	Successful [Tested: 42]
4.7.E10	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}z^{a}=az^{a-1}}}$ `\deriv{}{z}z^{a} = az^{a-1}`		diff((z)^(a), z) = a*(z)^(a - 1)	D[(z)^(a), z] == a*(z)^(a - 1)	Successful	Successful	-	Successful [Tested: 42]
4.7.E14	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=aw}}$ `\deriv[2]{w}{z} = aw`	${\displaystyle{\displaystyle a\neq 0}}$	diff(w, [z$(2)]) = a*w	D[w, {z, 2}] == a*w	Failure	Failure	Failed [300 / 300] Result: 1.299038106+.7500000000I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I} Result: 1.299038106+.7500000000I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} Result: 1.299038106+.7500000000I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = 1/2-1/2I3^(1/2)} Result: 1.299038106+.7500000000I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.299038105676658, 0.7499999999999999] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.299038105676658, 0.7499999999999999] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.7.E15	${\displaystyle{\displaystyle w=Ae^{\sqrt{a}z}+Be^{-\sqrt{a}z}}}$ `w = Ae^{\sqrt{a}z}+Be^{-\sqrt{a}z}`		w = Aexp(sqrt(a)z)+ Bexp(-sqrt(a)z)	w == AExp[Sqrt[a]z]+ BExp[-Sqrt[a]z]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.8.E1	${\displaystyle{\displaystyle\operatorname{Ln}\left(z_{1}z_{2}\right)=% \operatorname{Ln}z_{1}+\operatorname{Ln}z_{2}}}$ `\Ln@{z_{1}z_{2}} = \Ln@@{z_{1}}+\Ln@@{z_{2}}`		ln(z[1]*z[2]) = ln(z[1])+ ln(z[2])	Log[Subscript[z, 1]*Subscript[z, 2]] == Log[Subscript[z, 1]]+ Log[Subscript[z, 2]]	Failure	Failure	Failed [25 / 100] Result: 0.-6.283185308I Test Values: {z[1] = 1/23^(1/2)+1/2I, z[2] = -1.5} Result: 0.-6.283185308I Test Values: {z[1] = 1/23^(1/2)+1/2I, z[2] = -.5} Result: -.1e-9-6.283185308I Test Values: {z[1] = 1/23^(1/2)+1/2I, z[2] = -2} Result: .133199999e-10-6.283185307I Test Values: {z[1] = -1/2+1/2I3^(1/2), z[2] = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [25 / 100] Result: Complex[0.0, -6.283185307179587] Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], -1.5]} Result: Complex[0.0, -6.283185307179587] Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], -0.5]} ... skip entries to safe data
4.8.E2	${\displaystyle{\displaystyle\ln\left(z_{1}z_{2}\right)=\ln z_{1}+\ln z_{2}}}$ `\ln@{z_{1}z_{2}} = \ln@@{z_{1}}+\ln@@{z_{2}}`	${\displaystyle{\displaystyle-\pi\leq\operatorname{ph}z_{1}+\operatorname{ph}z_% {2},\operatorname{ph}z_{1}+\operatorname{ph}z_{2}\leq\pi}}$	ln(z[1]*z[2]) = ln(z[1])+ ln(z[2])	Log[Subscript[z, 1]*Subscript[z, 2]] == Log[Subscript[z, 1]]+ Log[Subscript[z, 2]]	Failure	Failure	Successful [Tested: 59]	Successful [Tested: 75]
4.8.E3	${\displaystyle{\displaystyle\operatorname{Ln}\frac{z_{1}}{z_{2}}=\operatorname% {Ln}z_{1}-\operatorname{Ln}z_{2}}}$ `\Ln@@{\frac{z_{1}}{z_{2}}} = \Ln@@{z_{1}}-\Ln@@{z_{2}}`		ln((z[1])/(z[2])) = ln(z[1])- ln(z[2])	Log[Divide[Subscript[z, 1],Subscript[z, 2]]] == Log[Subscript[z, 1]]- Log[Subscript[z, 2]]	Failure	Failure	Failed [25 / 100] Result: 0.-6.283185307I Test Values: {z[1] = -1/2+1/2I3^(1/2), z[2] = -1/23^(1/2)-1/2I} Result: 0.+6.283185307I Test Values: {z[1] = 1/2-1/2I3^(1/2), z[2] = -1/2+1/2I3^(1/2)} Result: .1e-9+6.283185307I Test Values: {z[1] = 1/2-1/2I3^(1/2), z[2] = -1.5} Result: -.1e-9+6.283185307I Test Values: {z[1] = 1/2-1/2I3^(1/2), z[2] = -.5} ... skip entries to safe data	Failed [25 / 100] Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.8.E4	${\displaystyle{\displaystyle\ln\frac{z_{1}}{z_{2}}=\ln z_{1}-\ln z_{2}}}$ `\ln@@{\frac{z_{1}}{z_{2}}} = \ln@@{z_{1}}-\ln@@{z_{2}}`	${\displaystyle{\displaystyle-\pi\leq\operatorname{ph}z_{1}-\operatorname{ph}z_% {2},\operatorname{ph}z_{1}-\operatorname{ph}z_{2}\leq\pi}}$	ln((z[1])/(z[2])) = ln(z[1])- ln(z[2])	Log[Divide[Subscript[z, 1],Subscript[z, 2]]] == Log[Subscript[z, 1]]- Log[Subscript[z, 2]]	Failure	Failure	Failed [3 / 70] Result: 0.+6.283185307I Test Values: {z[1] = 1/2-1/2I3^(1/2), z[2] = -1/2+1/2I3^(1/2)} Result: 0.+6.283185308I Test Values: {z[1] = -1/23^(1/2)-1/2I, z[2] = 1/23^(1/2)+1/2I} Result: 6.283185308*I Test Values: {z[1] = 2, z[2] = -2}	Failed [11 / 86] Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.8.E5	${\displaystyle{\displaystyle\operatorname{Ln}\left(z^{n}\right)=n\operatorname% {Ln}z}}$ `\Ln@{z^{n}} = n\Ln@@{z}`		ln((z)^(n)) = n*ln(z)	Log[(z)^(n)] == n*Log[z]	Failure	Failure	Failed [5 / 21] Result: .133199999e-10-6.283185307I Test Values: {z = -1/2+1/2I3^(1/2), n = 2, n = 3} Result: .4399599996e-9-6.283185306I Test Values: {z = -1/2+1/2I3^(1/2), n = 3, n = 3} Result: .4399599996e-9+6.283185306I Test Values: {z = 1/2-1/2I3^(1/2), n = 3, n = 3} Result: .133199999e-10+6.283185307I Test Values: {z = -1/23^(1/2)-1/2I, n = 2, n = 3} ... skip entries to safe data	Failed [3 / 7] Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.8.E6	${\displaystyle{\displaystyle\ln\left(z^{n}\right)=n\ln z}}$ `\ln@{z^{n}} = n\ln@@{z}`	${\displaystyle{\displaystyle-\pi\leq n\operatorname{ph}z,n\operatorname{ph}z% \leq\pi}}$	ln((z)^(n)) = n*ln(z)	Log[(z)^(n)] == n*Log[z]	Failure	Failure	Failed [1 / 17] Result: .4399599996e-9+6.283185306I Test Values: {z = 1/2-1/2I*3^(1/2), n = 3, n = 3}	Failed [3 / 7] Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.8.E7	${\displaystyle{\displaystyle\ln\frac{1}{z}=-\ln z}}$ `\ln@@{\frac{1}{z}} = -\ln@@{z}`	${\displaystyle{\displaystyle\|\operatorname{ph}z\|\leq\pi}}$	ln((1)/(z)) = - ln(z)	Log[Divide[1,z]] == - Log[z]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.8.E8	${\displaystyle{\displaystyle\operatorname{Ln}\left(\exp z\right)=z+2k\pi% \mathrm{i}}}$ `\Ln@{\exp@@{z}} = z+2k\pi\iunit`		ln(exp(z)) = z + 2kPi*I	Log[Exp[z]] == z + 2kPi*I	Failure	Failure	Failed [21 / 21] Result: -.1e-9-6.283185308I Test Values: {z = 1/23^(1/2)+1/2I, k = 1, k = 3} Result: -.1e-9-12.56637062I Test Values: {z = 1/23^(1/2)+1/2I, k = 2, k = 3} Result: -.1e-9-18.84955592I Test Values: {z = 1/23^(1/2)+1/2I, k = 3, k = 3} Result: 0.-6.283185308I Test Values: {z = -1/2+1/2I3^(1/2), k = 1, k = 3} ... skip entries to safe data	Failed [7 / 7] Result: Complex[0.0, -18.84955592153876] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0, -18.84955592153876] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.8.E9	${\displaystyle{\displaystyle\ln\left(\exp z\right)=z}}$ `\ln@{\exp@@{z}} = z`	${\displaystyle{\displaystyle-\pi\leq\Im z,\Im z\leq\pi}}$	ln(exp(z)) = z	Log[Exp[z]] == z	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.8.E10	${\displaystyle{\displaystyle\exp\left(\ln z\right)=\exp\left(\operatorname{Ln}% z\right)}}$ `\exp@{\ln@@{z}} = \exp@{\Ln@@{z}}`		exp(ln(z)) = exp(ln(z))	Exp[Log[z]] == Exp[Log[z]]	Successful	Successful	-	Successful [Tested: 7]
4.8.E10	${\displaystyle{\displaystyle\exp\left(\operatorname{Ln}z\right)=z}}$ `\exp@{\Ln@@{z}} = z`		exp(ln(z)) = z	Exp[Log[z]] == z	Successful	Successful	-	Successful [Tested: 7]
4.8.E11	${\displaystyle{\displaystyle\operatorname{Ln}\left(a^{z}\right)=z\operatorname% {Ln}a+2k\pi\mathrm{i}}}$ `\Ln@{a^{z}} = z\Ln@@{a}+2k\pi\iunit`		ln((a)^(z)) = zln(a)+ 2kPiI	Log[(a)^(z)] == zLog[a]+ 2kPiI	Failure	Failure	Failed [126 / 126] Result: 0.-6.283185308I Test Values: {a = -1.5, z = 1/23^(1/2)+1/2I, k = 1, k = 3} Result: 0.-12.56637062I Test Values: {a = -1.5, z = 1/23^(1/2)+1/2I, k = 2, k = 3} Result: 0.-18.84955592I Test Values: {a = -1.5, z = 1/23^(1/2)+1/2I, k = 3, k = 3} Result: 0.-6.283185308I Test Values: {a = -1.5, z = -1/2+1/2I3^(1/2), k = 1, k = 3} ... skip entries to safe data	Failed [42 / 42] Result: Complex[0.0, -18.84955592153876] Test Values: {Rule[a, -1.5], Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0, -18.84955592153876] Test Values: {Rule[a, -1.5], Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.8.E12	${\displaystyle{\displaystyle\ln\left(a^{z}\right)=z\ln a+2k\pi\mathrm{i}}}$ `\ln@{a^{z}} = z\ln@@{a}+2k\pi\iunit`		ln((a)^(z)) = zln(a)+ 2kPiI	Log[(a)^(z)] == zLog[a]+ 2kPiI	Failure	Failure	Failed [126 / 126] Result: 0.-6.283185308I Test Values: {a = -1.5, z = 1/23^(1/2)+1/2I, k = 1} Result: 0.-12.56637062I Test Values: {a = -1.5, z = 1/23^(1/2)+1/2I, k = 2} Result: 0.-18.84955592I Test Values: {a = -1.5, z = 1/23^(1/2)+1/2I, k = 3} Result: 0.-6.283185308I Test Values: {a = -1.5, z = -1/2+1/2I3^(1/2), k = 1} ... skip entries to safe data	Failed [126 / 126] Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0, -12.566370614359172] Test Values: {Rule[a, -1.5], Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.8.E13	${\displaystyle{\displaystyle\ln\left(a^{x}\right)=x\ln a}}$ `\ln@{a^{x}} = x\ln@@{a}`	${\displaystyle{\displaystyle a>0}}$	ln((a)^(x)) = x*ln(a)	Log[(a)^(x)] == x*Log[a]	Successful	Failure	-	Successful [Tested: 9]
4.8.E14	${\displaystyle{\displaystyle a^{z_{1}}a^{z_{2}}=a^{z_{1}+z_{2}}}}$ `a^{z_{1}}a^{z_{2}} = a^{z_{1}+z_{2}}`	${\displaystyle{\displaystyle a\neq 0}}$	(a)^(z[1])* (a)^(z[2]) = (a)^(z[1]+ z[2])	(a)^(Subscript[z, 1])* (a)^(Subscript[z, 2]) == (a)^(Subscript[z, 1]+ Subscript[z, 2])	Skipped - no semantic math	Skipped - no semantic math	-	-
4.8.E15	${\displaystyle{\displaystyle a^{z}b^{z}=(ab)^{z}}}$ `a^{z}b^{z} = (ab)^{z}`	${\displaystyle{\displaystyle-\pi\leq\operatorname{ph}a+\operatorname{ph}b,% \operatorname{ph}a+\operatorname{ph}b\leq\pi}}$	(a)^(z)* (b)^(z) = (a*b)^(z)	(a)^(z)* (b)^(z) == (a*b)^(z)	Skipped - no semantic math	Skipped - no semantic math	-	-
4.8.E16	${\displaystyle{\displaystyle e^{z_{1}}e^{z_{2}}=e^{z_{1}+z_{2}}}}$ `e^{z_{1}}e^{z_{2}} = e^{z_{1}+z_{2}}`		exp(z[1])*exp(z[2]) = exp(z[1]+ z[2])	Exp[Subscript[z, 1]]*Exp[Subscript[z, 2]] == Exp[Subscript[z, 1]+ Subscript[z, 2]]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.8.E17	${\displaystyle{\displaystyle(e^{z_{1}})^{z_{2}}=e^{z_{1}z_{2}}}}$ `(e^{z_{1}})^{z_{2}} = e^{z_{1}z_{2}}`	${\displaystyle{\displaystyle-\pi\leq\Im z_{1},\Im z_{1}\leq\pi}}$	(exp(z[1]))^(z[2]) = exp(z[1]*z[2])	(Exp[Subscript[z, 1]])^(Subscript[z, 2]) == Exp[Subscript[z, 1]*Subscript[z, 2]]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.10.E1	${\displaystyle{\displaystyle\int\frac{\mathrm{d}z}{z}=\ln z}}$ `\int\frac{\diff{z}}{z} = \ln@@{z}`		int((1)/(z), z) = ln(z)	Integrate[Divide[1,z], z, GenerateConditions->None] == Log[z]	Successful	Successful	-	Successful [Tested: 7]
4.10.E2	${\displaystyle{\displaystyle\int\ln z\mathrm{d}z=z\ln z-z}}$ `\int\ln@@{z}\diff{z} = z\ln@@{z}-z`		int(ln(z), z) = z*ln(z)- z	Integrate[Log[z], z, GenerateConditions->None] == z*Log[z]- z	Successful	Successful	-	Successful [Tested: 7]
4.10.E3	${\displaystyle{\displaystyle\int z^{n}\ln z\mathrm{d}z=\frac{z^{n+1}}{n+1}\ln z% -\frac{z^{n+1}}{(n+1)^{2}}}}$ `\int z^{n}\ln@@{z}\diff{z} = \frac{z^{n+1}}{n+1}\ln@@{z}-\frac{z^{n+1}}{(n+1)^{2}}`	${\displaystyle{\displaystyle n\neq-1}}$	int((z)^(n)* ln(z), z) = ((z)^(n + 1))/(n + 1)*ln(z)-((z)^(n + 1))/((n + 1)^(2))	Integrate[(z)^(n)* Log[z], z, GenerateConditions->None] == Divide[(z)^(n + 1),n + 1]*Log[z]-Divide[(z)^(n + 1),(n + 1)^(2)]	Successful	Successful	-	Successful [Tested: 21]
4.10.E4	${\displaystyle{\displaystyle\int\frac{\mathrm{d}z}{z\ln z}=\ln\left(\ln z% \right)}}$ `\int\frac{\diff{z}}{z\ln@@{z}} = \ln@{\ln@@{z}}`		int((1)/(z*ln(z)), z) = ln(ln(z))	Integrate[Divide[1,z*Log[z]], z, GenerateConditions->None] == Log[Log[z]]	Successful	Successful	-	Successful [Tested: 7]
4.10.E5	${\displaystyle{\displaystyle\int_{0}^{1}\frac{\ln t}{1-t}\mathrm{d}t=-\frac{% \pi^{2}}{6}}}$ `\int_{0}^{1}\frac{\ln@@{t}}{1-t}\diff{t} = -\frac{\pi^{2}}{6}`		int((ln(t))/(1 - t), t = 0..1) = -((Pi)^(2))/(6)	Integrate[Divide[Log[t],1 - t], {t, 0, 1}, GenerateConditions->None] == -Divide[(Pi)^(2),6]	Successful	Successful	-	Successful [Tested: 1]
4.10.E6	${\displaystyle{\displaystyle\int_{0}^{1}\frac{\ln t}{1+t}\mathrm{d}t=-\frac{% \pi^{2}}{12}}}$ `\int_{0}^{1}\frac{\ln@@{t}}{1+t}\diff{t} = -\frac{\pi^{2}}{12}`		int((ln(t))/(1 + t), t = 0..1) = -((Pi)^(2))/(12)	Integrate[Divide[Log[t],1 + t], {t, 0, 1}, GenerateConditions->None] == -Divide[(Pi)^(2),12]	Successful	Successful	-	Successful [Tested: 1]
4.10.E8	${\displaystyle{\displaystyle\int e^{az}\mathrm{d}z=\frac{e^{az}}{a}}}$ `\int e^{az}\diff{z} = \frac{e^{az}}{a}`		int(exp(az), z) = (exp(az))/(a)	Integrate[Exp[az], z, GenerateConditions->None] == Divide[Exp[az],a]	Successful	Successful	-	Successful [Tested: 42]
4.10.E9	${\displaystyle{\displaystyle\int\frac{\mathrm{d}z}{e^{az}+b}=\frac{1}{ab}(az-% \ln\left(e^{az}+b\right))}}$ `\int\frac{\diff{z}}{e^{az}+b} = \frac{1}{ab}(az-\ln@{e^{az}+b})`		int((1)/(exp(az)+ b), z) = (1)/(ab)(az - ln(exp(a*z)+ b))	Integrate[Divide[1,Exp[az]+ b], z, GenerateConditions->None] == Divide[1,ab](az - Log[Exp[a*z]+ b])	Failure	Successful	Successful [Tested: 252]	Successful [Tested: 252]
4.10.E10	${\displaystyle{\displaystyle\int\frac{e^{az}-1}{e^{az}+1}\mathrm{d}z=\frac{2}{% a}\ln\left(e^{az/2}+e^{-az/2}\right)}}$ `\int\frac{e^{az}-1}{e^{az}+1}\diff{z} = \frac{2}{a}\ln@{e^{az/2}+e^{-az/2}}`		int((exp(az)- 1)/(exp(az)+ 1), z) = (2)/(a)ln(exp(az/2)+ exp(- a*z/2))	Integrate[Divide[Exp[az]- 1,Exp[az]+ 1], z, GenerateConditions->None] == Divide[2,a]Log[Exp[az/2]+ Exp[- a*z/2]]	Failure	Failure	Successful [Tested: 42]	Successful [Tested: 42]
4.10.E11	${\displaystyle{\displaystyle\int_{-\infty}^{\infty}e^{-cx^{2}}\mathrm{d}x=% \sqrt{\frac{\pi}{c}}}}$ `\int_{-\infty}^{\infty}e^{-cx^{2}}\diff{x} = \sqrt{\frac{\pi}{c}}`	${\displaystyle{\displaystyle\Re c>0}}$	int(exp(- c*(x)^(2)), x = - infinity..infinity) = sqrt((Pi)/(c))	Integrate[Exp[- c*(x)^(2)], {x, - Infinity, Infinity}, GenerateConditions->None] == Sqrt[Divide[Pi,c]]	Successful	Successful	-	Successful [Tested: 3]
4.10.E12	${\displaystyle{\displaystyle\int_{0}^{\ln 2}\frac{xe^{x}}{e^{x}-1}\mathrm{d}x=% \frac{\pi^{2}}{12}}}$ `\int_{0}^{\ln@@{2}}\frac{xe^{x}}{e^{x}-1}\diff{x} = \frac{\pi^{2}}{12}`		int((x*exp(x))/(exp(x)- 1), x = 0..ln(2)) = ((Pi)^(2))/(12)	Integrate[Divide[x*Exp[x],Exp[x]- 1], {x, 0, Log[2]}, GenerateConditions->None] == Divide[(Pi)^(2),12]	Successful	Successful	-	Successful [Tested: 1]
4.10.E13	${\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\mathrm{d}x}{e^{x}+1}=\ln 2}}$ `\int_{0}^{\infty}\frac{\diff{x}}{e^{x}+1} = \ln@@{2}`		int((1)/(exp(x)+ 1), x = 0..infinity) = ln(2)	Integrate[Divide[1,Exp[x]+ 1], {x, 0, Infinity}, GenerateConditions->None] == Log[2]	Successful	Successful	-	Successful [Tested: 1]
4.12.E1	${\displaystyle{\displaystyle\phi(x+1)=e^{\phi(x)}}}$ `\phi(x+1) = e^{\phi(x)}`	${\displaystyle{\displaystyle-1<x,x<\infty}}$	phi(x + 1) = exp(phi(x))	\[Phi][x + 1] == Exp[\[Phi][x]]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.12.E2	${\displaystyle{\displaystyle\phi(0)=0}}$ `\phi(0) = 0`		phi(0) = 0	\[Phi][0] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
4.12.E3	${\displaystyle{\displaystyle\psi(e^{x})=1+\psi(x)}}$ `\psi(e^{x}) = 1+\psi(x)`	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	psi(exp(x)) = 1 + psi(x)	\[Psi][Exp[x]] == 1 + \[Psi][x]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.12.E4	${\displaystyle{\displaystyle\psi(0)=0}}$ `\psi(0) = 0`		psi(0) = 0	\[Psi][0] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
4.12.E5	${\displaystyle{\displaystyle\phi(x)=\psi(x)}}$ `\phi(x) = \psi(x)`	${\displaystyle{\displaystyle 0\leq x,x\leq 1}}$	phi(x) = psi(x)	\[Phi][x] == \[Psi][x]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.12.E6	${\displaystyle{\displaystyle\phi(x)=\ln\left(x+1\right)}}$ `\phi(x) = \ln@{x+1}`	${\displaystyle{\displaystyle-1<x,x<0}}$	phi(x) = ln(x + 1)	\[Phi][x] == Log[x + 1]	Failure	Failure	Error	Skip - No test values generated
4.12.E8	${\displaystyle{\displaystyle\psi(x)=e^{x}-1}}$ `\psi(x) = e^{x}-1`	${\displaystyle{\displaystyle-\infty<x,x<0}}$	psi(x) = exp(x)- 1	\[Psi][x] == Exp[x]- 1	Skipped - no semantic math	Skipped - no semantic math	-	-
4.13.E1	${\displaystyle{\displaystyle We^{W}=x}}$ `We^{W} = x`		W*exp(W) = x	W*Exp[W] == x	Failure	Failure	Failed [30 / 30] Result: -.263026030+2.030302705I Test Values: {W = 1/23^(1/2)+1/2I, x = 1.5} Result: .736973970+2.030302705I Test Values: {W = 1/23^(1/2)+1/2I, x = .5} Result: -.763026030+2.030302705I Test Values: {W = 1/23^(1/2)+1/2I, x = 2} Result: -2.096603674+.1092863076I Test Values: {W = -1/2+1/2I3^(1/2), x = 1.5} ... skip entries to safe data	Failed [30 / 30] Result: Complex[-0.2630260306572938, 2.0303027048207967] Test Values: {Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]} Result: Complex[0.7369739693427062, 2.0303027048207967] Test Values: {Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]} ... skip entries to safe data
4.13#Ex1	${\displaystyle{\displaystyle\mathrm{Wp}\left(-1/e\right)=\mathrm{Wm}\left(-1/e% \right)}}$ `\LambertWp@{-1/e} = \LambertWm@{-1/e}`		LambertW(0, - 1/exp(1)) = LambertW(-1, - 1/exp(1))	ProductLog[0, - 1/E] == ProductLog[-1, - 1/E]	Successful	Successful	-	Successful [Tested: 1]
4.13#Ex1	${\displaystyle{\displaystyle\mathrm{Wm}\left(-1/e\right)=-1}}$ `\LambertWm@{-1/e} = -1`		LambertW(-1, - 1/exp(1)) = - 1	ProductLog[-1, - 1/E] == - 1	Successful	Successful	-	Successful [Tested: 1]
4.13#Ex2	${\displaystyle{\displaystyle\mathrm{Wp}\left(0\right)=0}}$ `\LambertWp@{0} = 0`		LambertW(0, 0) = 0	ProductLog[0, 0] == 0	Successful	Successful	-	Successful [Tested: 1]
4.13#Ex3	${\displaystyle{\displaystyle\mathrm{Wp}\left(e\right)=1}}$ `\LambertWp@{e} = 1`		LambertW(0, exp(1)) = 1	ProductLog[0, E] == 1	Successful	Successful	-	Successful [Tested: 1]
4.13#Ex4	${\displaystyle{\displaystyle U+\ln U=x}}$ `U+\ln@@{U} = x`		U + ln(U) = x	U + Log[U] == x	Failure	Failure	Failed [30 / 30] Result: -.6339745958+1.023598776I Test Values: {U = 1/23^(1/2)+1/2I, x = 1.5} Result: .3660254042+1.023598776I Test Values: {U = 1/23^(1/2)+1/2I, x = .5} Result: -1.133974596+1.023598776I Test Values: {U = 1/23^(1/2)+1/2I, x = 2} Result: -2.000000000+2.960420506I Test Values: {U = -1/2+1/2I3^(1/2), x = 1.5} ... skip entries to safe data	Failed [30 / 30] Result: Complex[-0.6339745962155613, 1.0235987755982987] Test Values: {Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]} Result: Complex[0.3660254037844387, 1.0235987755982987] Test Values: {Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]} ... skip entries to safe data
4.13#Ex5	${\displaystyle{\displaystyle U=U(x)}}$ `U = U(x)`		U = U*(x)	U == U*(x)	Failure	Failure	Failed [30 / 30] Result: -.4330127020-.2500000000I Test Values: {U = 1/23^(1/2)+1/2I, x = 1.5} Result: .4330127020+.2500000000I Test Values: {U = 1/23^(1/2)+1/2I, x = .5} Result: -.8660254040-.5000000000I Test Values: {U = 1/23^(1/2)+1/2I, x = 2} Result: .2500000000-.4330127020I Test Values: {U = -1/2+1/2I3^(1/2), x = 1.5} ... skip entries to safe data	Failed [30 / 30] Result: Complex[-0.4330127018922193, -0.24999999999999994] Test Values: {Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]} Result: Complex[0.43301270189221935, 0.24999999999999997] Test Values: {Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]} ... skip entries to safe data
4.13#Ex5	${\displaystyle{\displaystyle U(x)=W\left(e^{x}\right)}}$ `U(x) = \LambertW@{e^{x}}`		U(x) = LambertW(exp(x))	U[x] == ProductLog[Exp[x]]	Failure	Failure	Failed [30 / 30] Result: .34078386e-1+.7500000000I Test Values: {U = 1/23^(1/2)+1/2I, x = 1.5} Result: -.3332359062+.2500000000I Test Values: {U = 1/23^(1/2)+1/2I, x = .5} Result: .174905209+1.I Test Values: {U = 1/23^(1/2)+1/2I, x = 2} Result: -2.014959720+1.299038106I Test Values: {U = -1/2+1/2I3^(1/2), x = 1.5} ... skip entries to safe data	Failed [30 / 30] Result: Complex[0.0340783855511575, 0.7499999999999999] Test Values: {Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]} Result: Complex[-0.333235906269531, 0.24999999999999997] Test Values: {Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]} ... skip entries to safe data
4.13.E5	${\displaystyle{\displaystyle\mathrm{Wp}\left(x\right)=\sum_{n=1}^{\infty}(-1)^% {n-1}\frac{n^{n-2}}{(n-1)!}x^{n}}}$ `\LambertWp@{x} = \sum_{n=1}^{\infty}(-1)^{n-1}\frac{n^{n-2}}{(n-1)!}x^{n}`	${\displaystyle{\displaystyle\|x\|<\dfrac{1}{e}}}$	LambertW(0, x) = sum((- 1)^(n - 1)((n)^(n - 2))/(factorial(n - 1))(x)^(n), n = 1..infinity)	ProductLog[0, x] == Sum[(- 1)^(n - 1)Divide[(n)^(n - 2),(n - 1)!](x)^(n), {n, 1, Infinity}, GenerateConditions->None]	Failure	Successful	Error	Successful [Tested: 0]
4.13.E6	${\displaystyle{\displaystyle W\left(-e^{-1-(t^{2}/2)}\right)=\sum_{n=0}^{% \infty}(-1)^{n-1}c_{n}t^{n}}}$ `\LambertW@{-e^{-1-(t^{2}/2)}} = \sum_{n=0}^{\infty}(-1)^{n-1}c_{n}t^{n}`	${\displaystyle{\displaystyle\|t\|<2\sqrt{\pi}}}$	LambertW(- exp(- 1 -((t)^(2)/2))) = sum((- 1)^(n - 1)* c[n]*(t)^(n), n = 0..infinity)	ProductLog[- Exp[- 1 -((t)^(2)/2)]] == Sum[(- 1)^(n - 1)* Subscript[c, n]*(t)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [60 / 60] Result: Float(infinity)+Float(infinity)I Test Values: {t = -1.5, c[n] = 1/23^(1/2)+1/2I} Result: Float(infinity)+Float(infinity)I Test Values: {t = -1.5, c[n] = -1/2+1/2I3^(1/2)} Result: Float(infinity)+Float(infinity)I Test Values: {t = -1.5, c[n] = 1/2-1/2I3^(1/2)} Result: Float(infinity)+Float(infinity)I Test Values: {t = -1.5, c[n] = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [60 / 60] Result: Plus[-0.13696418431579768, Times[-1.0, NSum[Times[Power[-1.5, n], Power[-1, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[Subscript[c, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[-0.13696418431579768, Times[-1.0, NSum[Times[Power[-1.5, n], Power[-1, Plus[-1, n]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[Subscript[c, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.13.E7	${\displaystyle{\displaystyle c_{0}=1,c_{1}}}$ `c_{0} = 1,c_{1}`		c[0] = 1; c[1]	Subscript[c, 0] == 1 Subscript[c, 1]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.13.E8	${\displaystyle{\displaystyle c_{n}=\frac{1}{n+1}\left(c_{n-1}-\sum_{k=2}^{n-1}% kc_{k}c_{n+1-k}\right)}}$ `c_{n} = \frac{1}{n+1}\left(c_{n-1}-\sum_{k=2}^{n-1}kc_{k}c_{n+1-k}\right)`	${\displaystyle{\displaystyle n\geq 2}}$	c[n] = (1)/(n + 1)(c[n - 1]- sum(kc[k]*c[n + 1 - k], k = 2..n - 1))	Subscript[c, n] == Divide[1,n + 1](Subscript[c, n - 1]- Sum[kSubscript[c, k]*Subscript[c, n + 1 - k], {k, 2, n - 1}, GenerateConditions->None])	Skipped - no semantic math	Skipped - no semantic math	-	-
4.13.E9	${\displaystyle{\displaystyle 1\cdot 3\cdot 5\cdots(2n+1)c_{2n+1}=g_{n}}}$ `1\cdot 3\cdot 5\cdots(2n+1)c_{2n+1} = g_{n}`		1 * 3 * 5(2n + 1)c[2n + 1] = g[n]	1 * 3 * 5(2n + 1)Subscript[c, 2n + 1] == Subscript[g, n]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.14.E1	${\displaystyle{\displaystyle\sin z=\frac{e^{\mathrm{i}z}-e^{-\mathrm{i}z}}{2% \mathrm{i}}}}$ `\sin@@{z} = \frac{e^{\iunit z}-e^{-\iunit z}}{2\iunit}`		sin(z) = (exp(Iz)- exp(- Iz))/(2*I)	Sin[z] == Divide[Exp[Iz]- Exp[- Iz],2*I]	Successful	Successful	-	Successful [Tested: 7]
4.14.E2	${\displaystyle{\displaystyle\cos z=\frac{e^{\mathrm{i}z}+e^{-\mathrm{i}z}}{2}}}$ `\cos@@{z} = \frac{e^{\iunit z}+e^{-\iunit z}}{2}`		cos(z) = (exp(Iz)+ exp(- Iz))/(2)	Cos[z] == Divide[Exp[Iz]+ Exp[- Iz],2]	Successful	Successful	-	Successful [Tested: 7]
4.14.E3	${\displaystyle{\displaystyle\cos z+i\sin z=e^{+iz}}}$ `\cos@@{z}+ i\sin@@{z} = e^{+ iz}`		cos(z)+ Isin(z) = exp(+ Iz)	Cos[z]+ ISin[z] == Exp[+ Iz]	Successful	Successful	-	Successful [Tested: 7]
4.14.E3	${\displaystyle{\displaystyle\cos z-i\sin z=e^{-iz}}}$ `\cos@@{z}- i\sin@@{z} = e^{- iz}`		cos(z)- Isin(z) = exp(- Iz)	Cos[z]- ISin[z] == Exp[- Iz]	Successful	Successful	-	Successful [Tested: 7]
4.14.E4	${\displaystyle{\displaystyle\tan z=\frac{\sin z}{\cos z}}}$ `\tan@@{z} = \frac{\sin@@{z}}{\cos@@{z}}`		tan(z) = (sin(z))/(cos(z))	Tan[z] == Divide[Sin[z],Cos[z]]	Successful	Successful	-	Successful [Tested: 7]
4.14.E5	${\displaystyle{\displaystyle\csc z=\frac{1}{\sin z}}}$ `\csc@@{z} = \frac{1}{\sin@@{z}}`		csc(z) = (1)/(sin(z))	Csc[z] == Divide[1,Sin[z]]	Successful	Successful	-	Successful [Tested: 7]
4.14.E6	${\displaystyle{\displaystyle\sec z=\frac{1}{\cos z}}}$ `\sec@@{z} = \frac{1}{\cos@@{z}}`		sec(z) = (1)/(cos(z))	Sec[z] == Divide[1,Cos[z]]	Successful	Successful	-	Successful [Tested: 7]
4.14.E7	${\displaystyle{\displaystyle\cot z=\frac{\cos z}{\sin z}}}$ `\cot@@{z} = \frac{\cos@@{z}}{\sin@@{z}}`		cot(z) = (cos(z))/(sin(z))	Cot[z] == Divide[Cos[z],Sin[z]]	Successful	Successful	-	Successful [Tested: 7]
4.14.E7	${\displaystyle{\displaystyle\frac{\cos z}{\sin z}=\frac{1}{\tan z}}}$ `\frac{\cos@@{z}}{\sin@@{z}} = \frac{1}{\tan@@{z}}`		(cos(z))/(sin(z)) = (1)/(tan(z))	Divide[Cos[z],Sin[z]] == Divide[1,Tan[z]]	Successful	Successful	-	Successful [Tested: 7]
4.14.E8	${\displaystyle{\displaystyle\sin\left(z+2k\pi\right)=\sin z}}$ `\sin@{z+2k\pi} = \sin@@{z}`		sin(z + 2kPi) = sin(z)	Sin[z + 2kPi] == Sin[z]	Successful	Failure	-	Successful [Tested: 21]
4.14.E9	${\displaystyle{\displaystyle\cos\left(z+2k\pi\right)=\cos z}}$ `\cos@{z+2k\pi} = \cos@@{z}`		cos(z + 2kPi) = cos(z)	Cos[z + 2kPi] == Cos[z]	Successful	Failure	-	Successful [Tested: 21]
4.14.E10	${\displaystyle{\displaystyle\tan\left(z+k\pi\right)=\tan z}}$ `\tan@{z+k\pi} = \tan@@{z}`		tan(z + k*Pi) = tan(z)	Tan[z + k*Pi] == Tan[z]	Successful	Failure	-	Successful [Tested: 21]
4.15.E1	${\displaystyle{\displaystyle\cos\left(x+iy\right)=\sin\left(x+\tfrac{1}{2}\pi+% iy\right)}}$ `\cos@{x+iy} = \sin@{x+\tfrac{1}{2}\pi+iy}`		cos(x + Iy) = sin(x +(1)/(2)Pi + I*y)	Cos[x + Iy] == Sin[x +Divide[1,2]Pi + I*y]	Successful	Successful	-	Successful [Tested: 18]
4.15.E2	${\displaystyle{\displaystyle\cot\left(x+iy\right)=-\tan\left(x+\tfrac{1}{2}\pi% +iy\right)}}$ `\cot@{x+iy} = -\tan@{x+\tfrac{1}{2}\pi+iy}`		cot(x + Iy) = - tan(x +(1)/(2)Pi + I*y)	Cot[x + Iy] == - Tan[x +Divide[1,2]Pi + I*y]	Successful	Successful	-	Successful [Tested: 18]
4.15.E3	${\displaystyle{\displaystyle\sec\left(x+iy\right)=\csc\left(x+\tfrac{1}{2}\pi+% iy\right)}}$ `\sec@{x+iy} = \csc@{x+\tfrac{1}{2}\pi+iy}`		sec(x + Iy) = csc(x +(1)/(2)Pi + I*y)	Sec[x + Iy] == Csc[x +Divide[1,2]Pi + I*y]	Successful	Successful	-	Successful [Tested: 18]
4.17.E1	${\displaystyle{\displaystyle\lim_{z\to 0}\frac{\sin z}{z}=1}}$ `\lim_{z\to 0}\frac{\sin@@{z}}{z} = 1`		limit((sin(z))/(z), z = 0) = 1	Limit[Divide[Sin[z],z], z -> 0, GenerateConditions->None] == 1	Successful	Successful	-	Successful [Tested: 1]
4.17.E2	${\displaystyle{\displaystyle\lim_{z\to 0}\frac{\tan z}{z}=1}}$ `\lim_{z\to 0}\frac{\tan@@{z}}{z} = 1`		limit((tan(z))/(z), z = 0) = 1	Limit[Divide[Tan[z],z], z -> 0, GenerateConditions->None] == 1	Successful	Successful	-	Successful [Tested: 1]
4.17.E3	${\displaystyle{\displaystyle\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}}}$ `\lim_{z\to 0}\frac{1-\cos@@{z}}{z^{2}} = \frac{1}{2}`		limit((1 - cos(z))/((z)^(2)), z = 0) = (1)/(2)	Limit[Divide[1 - Cos[z],(z)^(2)], z -> 0, GenerateConditions->None] == Divide[1,2]	Successful	Successful	-	Successful [Tested: 1]
4.18.E1	${\displaystyle{\displaystyle\frac{2x}{\pi}\leq\sin x}}$ `\frac{2x}{\pi} \leq \sin@@{x}`	${\displaystyle{\displaystyle 0\leq x,x\leq\frac{1}{2}\pi}}$	(2*x)/(Pi) <= sin(x)	Divide[2*x,Pi] <= Sin[x]	Failure	Failure	Successful [Tested: 2]	Successful [Tested: 2]
4.18.E1	${\displaystyle{\displaystyle\sin x\leq x}}$ `\sin@@{x} \leq x`	${\displaystyle{\displaystyle 0\leq x,x\leq\frac{1}{2}\pi}}$	sin(x) <= x	Sin[x] <= x	Failure	Failure	Successful [Tested: 2]	Successful [Tested: 2]
4.18.E2	${\displaystyle{\displaystyle x\leq\tan x}}$ `x \leq \tan@@{x}`	${\displaystyle{\displaystyle 0\leq x,x<\frac{1}{2}\pi}}$	x <= tan(x)	x <= Tan[x]	Failure	Failure	Successful [Tested: 2]	Successful [Tested: 2]
4.18.E3	${\displaystyle{\displaystyle\cos x\leq\frac{\sin x}{x}}}$ `\cos@@{x} \leq \frac{\sin@@{x}}{x}`	${\displaystyle{\displaystyle 0\leq x,x\leq\pi}}$	cos(x) <= (sin(x))/(x)	Cos[x] <= Divide[Sin[x],x]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.18.E3	${\displaystyle{\displaystyle\frac{\sin x}{x}\leq 1}}$ `\frac{\sin@@{x}}{x} \leq 1`	${\displaystyle{\displaystyle 0\leq x,x\leq\pi}}$	(sin(x))/(x) <= 1	Divide[Sin[x],x] <= 1	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.18.E4	${\displaystyle{\displaystyle\pi<\frac{\sin\left(\pi x\right)}{x(1-x)}}}$ `\pi < \frac{\sin@{\pi x}}{x(1-x)}`	${\displaystyle{\displaystyle 0<x,x<1}}$	Pi < (sin(Pix))/(x(1 - x))	Pi < Divide[Sin[Pix],x(1 - x)]	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 1]
4.18.E4	${\displaystyle{\displaystyle\frac{\sin\left(\pi x\right)}{x(1-x)}\leq 4}}$ `\frac{\sin@{\pi x}}{x(1-x)} \leq 4`	${\displaystyle{\displaystyle 0<x,x<1}}$	(sin(Pix))/(x(1 - x)) <= 4	Divide[Sin[Pix],x(1 - x)] <= 4	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 1]
4.18.E5	${\displaystyle{\displaystyle\|\sinh y\|\leq\|\sin z\|\leq\cosh y}}$ `\|\sinh@@{y}\| \leq \|\sin@@{z}\|\leq\cosh@@{y}`		abs(sinh(y)) <= abs(sin(x + y*I)) <= cosh(y)	Abs[Sinh[y]] <= Abs[Sin[x + y*I]] <= Cosh[y]	Failure	Failure	Error	Successful [Tested: 18]
4.18.E6	${\displaystyle{\displaystyle\|\sinh y\|\leq\|\cos z\|\leq\cosh y}}$ `\|\sinh@@{y}\| \leq \|\cos@@{z}\|\leq\cosh@@{y}`		abs(sinh(y)) <= abs(cos(x + y*I)) <= cosh(y)	Abs[Sinh[y]] <= Abs[Cos[x + y*I]] <= Cosh[y]	Failure	Failure	Error	Successful [Tested: 18]
4.18.E7	${\displaystyle{\displaystyle\|\csc z\|\leq\operatorname{csch}\|y\|}}$ `\|\csc@@{z}\| \leq \csch@@{\|y\|}`		abs(csc(x + y*I)) <= csch(abs(y))	Abs[Csc[x + y*I]] <= Csch[Abs[y]]	Failure	Failure	Successful [Tested: 18]	Successful [Tested: 18]
4.18.E8	${\displaystyle{\displaystyle\|\cos z\|\leq\cosh\|z\|}}$ `\|\cos@@{z}\| \leq \cosh@@{\|z\|}`		abs(cos(z)) <= cosh(abs(z))	Abs[Cos[z]] <= Cosh[Abs[z]]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.18.E9	${\displaystyle{\displaystyle\|\sin z\|\leq\sinh\|z\|}}$ `\|\sin@@{z}\| \leq \sinh@@{\|z\|}`		abs(sin(z)) <= sinh(abs(z))	Abs[Sin[z]] <= Sinh[Abs[z]]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.18#Ex1	${\displaystyle{\displaystyle\|\cos z\|<2}}$ `\|\cos@@{z}\| < 2`		abs(cos(z)) < 2	Abs[Cos[z]] < 2	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.18#Ex2	${\displaystyle{\displaystyle\|\sin z\|\leq\tfrac{6}{5}\|z\|}}$ `\|\sin@@{z}\| \leq \tfrac{6}{5}\|z\|`	${\displaystyle{\displaystyle\|z\|<1}}$	abs(sin(z)) <= (6)/(5)*abs(z)	Abs[Sin[z]] <= Divide[6,5]*Abs[z]	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 1]
4.19.E7	${\displaystyle{\displaystyle\ln\left(\frac{\sin z}{z}\right)=\sum_{n=1}^{% \infty}\frac{(-1)^{n}2^{2n-1}B_{2n}}{n(2n)!}z^{2n}}}$ `\ln@{\frac{\sin@@{z}}{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}\BernoullinumberB{2n}}{n(2n)!}z^{2n}`	${\displaystyle{\displaystyle\|z\|<\pi}}$	ln((sin(z))/(z)) = sum(((- 1)^(n)* (2)^(2n - 1) bernoulli(2n))/(nfactorial(2n))(z)^(2*n), n = 1..infinity)	Log[Divide[Sin[z],z]] == Sum[Divide[(- 1)^(n)* (2)^(2n - 1) BernoulliB[2n],n(2n)!](z)^(2*n), {n, 1, Infinity}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.19.E8	${\displaystyle{\displaystyle\ln\left(\cos z\right)=\sum_{n=1}^{\infty}\frac{(-% 1)^{n}2^{2n-1}(2^{2n}-1)B_{2n}}{n(2n)!}z^{2n}}}$ `\ln@{\cos@@{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}(2^{2n}-1)\BernoullinumberB{2n}}{n(2n)!}z^{2n}`	${\displaystyle{\displaystyle\|z\|<\frac{1}{2}\pi}}$	ln(cos(z)) = sum(((- 1)^(n)* (2)^(2n - 1)((2)^(2n)- 1)bernoulli(2n))/(nfactorial(2n))(z)^(2*n), n = 1..infinity)	Log[Cos[z]] == Sum[Divide[(- 1)^(n)* (2)^(2n - 1)((2)^(2n)- 1)BernoulliB[2n],n(2n)!](z)^(2*n), {n, 1, Infinity}, GenerateConditions->None]	Failure	Failure	Manual Skip!	Successful [Tested: 6]
4.19.E9	${\displaystyle{\displaystyle\ln\left(\frac{\tan z}{z}\right)=\sum_{n=1}^{% \infty}\frac{(-1)^{n-1}2^{2n}(2^{2n-1}-1)B_{2n}}{n(2n)!}z^{2n}}}$ `\ln@{\frac{\tan@@{z}}{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{2n}(2^{2n-1}-1)\BernoullinumberB{2n}}{n(2n)!}z^{2n}`	${\displaystyle{\displaystyle\|z\|<\frac{1}{2}\pi}}$	ln((tan(z))/(z)) = sum(((- 1)^(n - 1)* (2)^(2n)((2)^(2n - 1)- 1)bernoulli(2n))/(nfactorial(2n))(z)^(2*n), n = 1..infinity)	Log[Divide[Tan[z],z]] == Sum[Divide[(- 1)^(n - 1)* (2)^(2n)((2)^(2n - 1)- 1)BernoulliB[2n],n(2n)!](z)^(2*n), {n, 1, Infinity}, GenerateConditions->None]	Failure	Failure	Manual Skip!	Successful [Tested: 6]
4.20.E1	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sin z=\cos z}}$ `\deriv{}{z}\sin@@{z} = \cos@@{z}`		diff(sin(z), z) = cos(z)	D[Sin[z], z] == Cos[z]	Successful	Successful	-	Successful [Tested: 7]
4.20.E2	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cos z=-\sin z}}$ `\deriv{}{z}\cos@@{z} = -\sin@@{z}`		diff(cos(z), z) = - sin(z)	D[Cos[z], z] == - Sin[z]	Successful	Successful	-	Successful [Tested: 7]
4.20.E3	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\tan z={\sec^{2}}z}}$ `\deriv{}{z}\tan@@{z} = \sec^{2}@@{z}`		diff(tan(z), z) = (sec(z))^(2)	D[Tan[z], z] == (Sec[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.20.E4	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\csc z=-\csc z\cot z}}$ `\deriv{}{z}\csc@@{z} = -\csc@@{z}\cot@@{z}`		diff(csc(z), z) = - csc(z)*cot(z)	D[Csc[z], z] == - Csc[z]*Cot[z]	Successful	Successful	-	Successful [Tested: 7]
4.20.E5	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sec z=\sec z\tan z}}$ `\deriv{}{z}\sec@@{z} = \sec@@{z}\tan@@{z}`		diff(sec(z), z) = sec(z)*tan(z)	D[Sec[z], z] == Sec[z]*Tan[z]	Successful	Successful	-	Successful [Tested: 7]
4.20.E6	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cot z=-{\csc^{2}}z}}$ `\deriv{}{z}\cot@@{z} = -\csc^{2}@@{z}`		diff(cot(z), z) = - (csc(z))^(2)	D[Cot[z], z] == - (Csc[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.20.E7	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\sin z=% \sin\left(z+\tfrac{1}{2}n\pi\right)}}$ `\deriv[n]{}{z}\sin@@{z} = \sin@{z+\tfrac{1}{2}n\pi}`		diff(sin(z), [z$(n)]) = sin(z +(1)/(2)nPi)	D[Sin[z], {z, n}] == Sin[z +Divide[1,2]nPi]	Successful	Successful	-	Successful [Tested: 21]
4.20.E8	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\cos z=% \cos\left(z+\tfrac{1}{2}n\pi\right)}}$ `\deriv[n]{}{z}\cos@@{z} = \cos@{z+\tfrac{1}{2}n\pi}`		diff(cos(z), [z$(n)]) = cos(z +(1)/(2)nPi)	D[Cos[z], {z, n}] == Cos[z +Divide[1,2]nPi]	Successful	Successful	-	Successful [Tested: 21]
4.20.E9	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+a^{2}w% =0}}$ `\deriv[2]{w}{z}+a^{2}w = 0`		diff(w, [z$(2)])+ (a)^(2)* w = 0	D[w, {z, 2}]+ (a)^(2)* w == 0	Failure	Failure	Failed [300 / 300] Result: 1.948557159+1.125000000I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I} Result: 1.948557159+1.125000000I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} Result: 1.948557159+1.125000000I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = 1/2-1/2I3^(1/2)} Result: 1.948557159+1.125000000I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.948557158514987, 1.1249999999999998] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.948557158514987, 1.1249999999999998] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.20.E10	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}+a% ^{2}w^{2}=1}}$ `\left(\deriv{w}{z}\right)^{2}+a^{2}w^{2} = 1`		(diff(w, z))^(2)+ (a)^(2)* (w)^(2) = 1	(D[w, z])^(2)+ (a)^(2)* (w)^(2) == 1	Failure	Failure	Failed [272 / 300] Result: .125000001+1.948557159I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I} Result: .125000001+1.948557159I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} Result: .125000001+1.948557159I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = 1/2-1/2I3^(1/2)} Result: .125000001+1.948557159I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [272 / 300] Result: Complex[0.12500000000000022, 1.9485571585149868] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.12500000000000022, 1.9485571585149868] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.20.E11	${\displaystyle{\displaystyle\frac{\mathrm{d}w}{\mathrm{d}z}-a^{2}w^{2}=1}}$ `\deriv{w}{z}-a^{2}w^{2} = 1`		diff(w, z)- (a)^(2)* (w)^(2) = 1	D[w, z]- (a)^(2)* (w)^(2) == 1	Failure	Failure	Failed [300 / 300] Result: -2.125000001-1.948557159I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I} Result: -2.125000001-1.948557159I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} Result: -2.125000001-1.948557159I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = 1/2-1/2I3^(1/2)} Result: -2.125000001-1.948557159I Test Values: {a = -1.5, w = 1/23^(1/2)+1/2I, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-2.125, -1.9485571585149868] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-2.125, -1.9485571585149868] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.20.E12	${\displaystyle{\displaystyle w=A\cos\left(az\right)+B\sin\left(az\right)}}$ `w = A\cos@{az}+B\sin@{az}`		w = Acos(az)+ Bsin(az)	w == ACos[az]+ BSin[az]	Failure	Failure	Failed [300 / 300] Result: 1.138704571+1.826991634I Test Values: {A = 1/23^(1/2)+1/2I, B = 1/23^(1/2)+1/2I, a = -1.5, w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I} Result: -1.586785764-.8180862806I Test Values: {A = 1/23^(1/2)+1/2I, B = 1/23^(1/2)+1/2I, a = -1.5, w = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} Result: 1.979513822-1.625744019I Test Values: {A = 1/23^(1/2)+1/2I, B = 1/23^(1/2)+1/2I, a = -1.5, w = 1/23^(1/2)+1/2I, z = 1/2-1/2I3^(1/2)} Result: -.8007246334+.1975056737I Test Values: {A = 1/23^(1/2)+1/2I, B = 1/23^(1/2)+1/2I, a = -1.5, w = 1/23^(1/2)+1/2I, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.138704570618858, 1.8269916342928783] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.5867857625486925, -0.8180862808059206] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.20.E13	${\displaystyle{\displaystyle w=(1/a)\sin\left(az+c\right)}}$ `w = (1/a)\sin@{az+c}`		w = (1/a)sin(az + c)	w == (1/a)Sin[az + c]	Failure	Failure	Failed [300 / 300] Result: .5761075690+1.016359912I Test Values: {a = -1.5, c = -1.5, w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I} Result: -.288669860e-1-.3275339707I Test Values: {a = -1.5, c = -1.5, w = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} Result: -.1554713530-.2104590960I Test Values: {a = -1.5, c = -1.5, w = 1/23^(1/2)+1/2I, z = 1/2-1/2I3^(1/2)} Result: .6937358929+1.037178419I Test Values: {a = -1.5, c = -1.5, w = 1/23^(1/2)+1/2I, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.5761075684969701, 1.0163599120046827] Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.028866985825810376, -0.3275339701177746] Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.20.E14	${\displaystyle{\displaystyle w=(1/a)\tan\left(az+c\right)}}$ `w = (1/a)\tan@{az+c}`		w = (1/a)tan(az + c)	w == (1/a)Tan[az + c]	Failure	Failure	Failed [300 / 300] Result: 1.000937702+.460093509e-1I Test Values: {a = -1.5, c = -1.5, w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I} Result: .7686167751-.1524919258I Test Values: {a = -1.5, c = -1.5, w = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} Result: .9655903492+1.180557377I Test Values: {a = -1.5, c = -1.5, w = 1/23^(1/2)+1/2I, z = 1/2-1/2I3^(1/2)} Result: .7863384613+.9337431086I Test Values: {a = -1.5, c = -1.5, w = 1/23^(1/2)+1/2I, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.0009377022129278, 0.04600935086169866] Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.7686167748870922, -0.1524919257161706] Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.21.E1	${\displaystyle{\displaystyle\sin u+\cos u=\sqrt{2}\sin\left(u+\tfrac{1}{4}\pi% \right)}}$ `\sin@@{u}+\cos@@{u} = \sqrt{2}\sin@{u+\tfrac{1}{4}\pi}`		sin(u)+ cos(u) = sqrt(2)sin(u +(1)/(4)Pi)	Sin[u]+ Cos[u] == Sqrt[2]Sin[u +Divide[1,4]Pi]	Successful	Successful	-	Successful [Tested: 10]
4.21.E1	${\displaystyle{\displaystyle\sin u-\cos u=\sqrt{2}\sin\left(u-\tfrac{1}{4}\pi% \right)}}$ `\sin@@{u}-\cos@@{u} = \sqrt{2}\sin@{u-\tfrac{1}{4}\pi}`		sin(u)- cos(u) = sqrt(2)sin(u -(1)/(4)Pi)	Sin[u]- Cos[u] == Sqrt[2]Sin[u -Divide[1,4]Pi]	Successful	Successful	-	Successful [Tested: 10]
4.21.E1	${\displaystyle{\displaystyle\sqrt{2}\sin\left(u+\tfrac{1}{4}\pi\right)=+\sqrt{% 2}\cos\left(u-\tfrac{1}{4}\pi\right)}}$ `\sqrt{2}\sin@{u+\tfrac{1}{4}\pi} = +\sqrt{2}\cos@{u-\tfrac{1}{4}\pi}`		sqrt(2)sin(u +(1)/(4)Pi) = +sqrt(2)cos(u -(1)/(4)Pi)	Sqrt[2]Sin[u +Divide[1,4]Pi] == +Sqrt[2]Cos[u -Divide[1,4]Pi]	Successful	Successful	-	Successful [Tested: 10]
4.21.E1	${\displaystyle{\displaystyle\sqrt{2}\sin\left(u-\tfrac{1}{4}\pi\right)=-\sqrt{% 2}\cos\left(u+\tfrac{1}{4}\pi\right)}}$ `\sqrt{2}\sin@{u-\tfrac{1}{4}\pi} = -\sqrt{2}\cos@{u+\tfrac{1}{4}\pi}`		sqrt(2)sin(u -(1)/(4)Pi) = -sqrt(2)cos(u +(1)/(4)Pi)	Sqrt[2]Sin[u -Divide[1,4]Pi] == -Sqrt[2]Cos[u +Divide[1,4]Pi]	Successful	Successful	-	Successful [Tested: 10]
4.21.E2	${\displaystyle{\displaystyle\sin\left(u+v\right)=\sin u\cos v+\cos u\sin v}}$ `\sin@{u+ v} = \sin@@{u}\cos@@{v}+\cos@@{u}\sin@@{v}`		sin(u + v) = sin(u)cos(v)+ cos(u)sin(v)	Sin[u + v] == Sin[u]Cos[v]+ Cos[u]Sin[v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E2	${\displaystyle{\displaystyle\sin\left(u-v\right)=\sin u\cos v-\cos u\sin v}}$ `\sin@{u- v} = \sin@@{u}\cos@@{v}-\cos@@{u}\sin@@{v}`		sin(u - v) = sin(u)cos(v)- cos(u)sin(v)	Sin[u - v] == Sin[u]Cos[v]- Cos[u]Sin[v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E3	${\displaystyle{\displaystyle\cos\left(u+v\right)=\cos u\cos v-\sin u\sin v}}$ `\cos@{u+ v} = \cos@@{u}\cos@@{v}-\sin@@{u}\sin@@{v}`		cos(u + v) = cos(u)cos(v)- sin(u)sin(v)	Cos[u + v] == Cos[u]Cos[v]- Sin[u]Sin[v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E3	${\displaystyle{\displaystyle\cos\left(u-v\right)=\cos u\cos v+\sin u\sin v}}$ `\cos@{u- v} = \cos@@{u}\cos@@{v}+\sin@@{u}\sin@@{v}`		cos(u - v) = cos(u)cos(v)+ sin(u)sin(v)	Cos[u - v] == Cos[u]Cos[v]+ Sin[u]Sin[v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E4	${\displaystyle{\displaystyle\tan\left(u+v\right)=\frac{\tan u+\tan v}{1-\tan u% \tan v}}}$ `\tan@{u+ v} = \frac{\tan@@{u}+\tan@@{v}}{1-\tan@@{u}\tan@@{v}}`		tan(u + v) = (tan(u)+ tan(v))/(1 - tan(u)*tan(v))	Tan[u + v] == Divide[Tan[u]+ Tan[v],1 - Tan[u]*Tan[v]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E4	${\displaystyle{\displaystyle\tan\left(u-v\right)=\frac{\tan u-\tan v}{1+\tan u% \tan v}}}$ `\tan@{u- v} = \frac{\tan@@{u}-\tan@@{v}}{1+\tan@@{u}\tan@@{v}}`		tan(u - v) = (tan(u)- tan(v))/(1 + tan(u)*tan(v))	Tan[u - v] == Divide[Tan[u]- Tan[v],1 + Tan[u]*Tan[v]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E5	${\displaystyle{\displaystyle\cot\left(u+v\right)=\frac{+\cot u\cot v-1}{\cot u% +\cot v}}}$ `\cot@{u+ v} = \frac{+\cot@@{u}\cot@@{v}-1}{\cot@@{u}+\cot@@{v}}`		cot(u + v) = (+ cot(u)*cot(v)- 1)/(cot(u)+ cot(v))	Cot[u + v] == Divide[+ Cot[u]*Cot[v]- 1,Cot[u]+ Cot[v]]	Successful	Successful	-	Failed [10 / 100] Result: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} Result: Complex[1.9674787081851645^15, 2.0439439417914815^15] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.21.E5	${\displaystyle{\displaystyle\cot\left(u-v\right)=\frac{-\cot u\cot v-1}{\cot u% -\cot v}}}$ `\cot@{u- v} = \frac{-\cot@@{u}\cot@@{v}-1}{\cot@@{u}-\cot@@{v}}`		cot(u - v) = (- cot(u)*cot(v)- 1)/(cot(u)- cot(v))	Cot[u - v] == Divide[- Cot[u]*Cot[v]- 1,Cot[u]- Cot[v]]	Successful	Successful	-	Failed [10 / 100] Result: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.21.E6	${\displaystyle{\displaystyle\sin u+\sin v=2\sin\left(\frac{u+v}{2}\right)\cos% \left(\frac{u-v}{2}\right)}}$ `\sin@@{u}+\sin@@{v} = 2\sin@{\frac{u+v}{2}}\cos@{\frac{u-v}{2}}`		sin(u)+ sin(v) = 2sin((u + v)/(2))cos((u - v)/(2))	Sin[u]+ Sin[v] == 2Sin[Divide[u + v,2]]Cos[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E7	${\displaystyle{\displaystyle\sin u-\sin v=2\cos\left(\frac{u+v}{2}\right)\sin% \left(\frac{u-v}{2}\right)}}$ `\sin@@{u}-\sin@@{v} = 2\cos@{\frac{u+v}{2}}\sin@{\frac{u-v}{2}}`		sin(u)- sin(v) = 2cos((u + v)/(2))sin((u - v)/(2))	Sin[u]- Sin[v] == 2Cos[Divide[u + v,2]]Sin[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E8	${\displaystyle{\displaystyle\cos u+\cos v=2\cos\left(\frac{u+v}{2}\right)\cos% \left(\frac{u-v}{2}\right)}}$ `\cos@@{u}+\cos@@{v} = 2\cos@{\frac{u+v}{2}}\cos@{\frac{u-v}{2}}`		cos(u)+ cos(v) = 2cos((u + v)/(2))cos((u - v)/(2))	Cos[u]+ Cos[v] == 2Cos[Divide[u + v,2]]Cos[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E9	${\displaystyle{\displaystyle\cos u-\cos v=-2\sin\left(\frac{u+v}{2}\right)\sin% \left(\frac{u-v}{2}\right)}}$ `\cos@@{u}-\cos@@{v} = -2\sin@{\frac{u+v}{2}}\sin@{\frac{u-v}{2}}`		cos(u)- cos(v) = - 2sin((u + v)/(2))sin((u - v)/(2))	Cos[u]- Cos[v] == - 2Sin[Divide[u + v,2]]Sin[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E10	${\displaystyle{\displaystyle\tan u+\tan v=\frac{\sin\left(u+v\right)}{\cos u% \cos v}}}$ `\tan@@{u}+\tan@@{v} = \frac{\sin@{u+ v}}{\cos@@{u}\cos@@{v}}`		tan(u)+ tan(v) = (sin(u + v))/(cos(u)*cos(v))	Tan[u]+ Tan[v] == Divide[Sin[u + v],Cos[u]*Cos[v]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E10	${\displaystyle{\displaystyle\tan u-\tan v=\frac{\sin\left(u-v\right)}{\cos u% \cos v}}}$ `\tan@@{u}-\tan@@{v} = \frac{\sin@{u- v}}{\cos@@{u}\cos@@{v}}`		tan(u)- tan(v) = (sin(u - v))/(cos(u)*cos(v))	Tan[u]- Tan[v] == Divide[Sin[u - v],Cos[u]*Cos[v]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E11	${\displaystyle{\displaystyle\cot u+\cot v=\frac{\sin\left(v+u\right)}{\sin u% \sin v}}}$ `\cot@@{u}+\cot@@{v} = \frac{\sin@{v+ u}}{\sin@@{u}\sin@@{v}}`		cot(u)+ cot(v) = (sin(v + u))/(sin(u)*sin(v))	Cot[u]+ Cot[v] == Divide[Sin[v + u],Sin[u]*Sin[v]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E11	${\displaystyle{\displaystyle\cot u-\cot v=\frac{\sin\left(v-u\right)}{\sin u% \sin v}}}$ `\cot@@{u}-\cot@@{v} = \frac{\sin@{v- u}}{\sin@@{u}\sin@@{v}}`		cot(u)- cot(v) = (sin(v - u))/(sin(u)*sin(v))	Cot[u]- Cot[v] == Divide[Sin[v - u],Sin[u]*Sin[v]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E12	${\displaystyle{\displaystyle{\sin^{2}}z+{\cos^{2}}z=1}}$ `\sin^{2}@@{z}+\cos^{2}@@{z} = 1`		(sin(z))^(2)+ (cos(z))^(2) = 1	(Sin[z])^(2)+ (Cos[z])^(2) == 1	Successful	Successful	-	Successful [Tested: 7]
4.21.E13	${\displaystyle{\displaystyle{\sec^{2}}z=1+{\tan^{2}}z}}$ `\sec^{2}@@{z} = 1+\tan^{2}@@{z}`		(sec(z))^(2) = 1 + (tan(z))^(2)	(Sec[z])^(2) == 1 + (Tan[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.21.E14	${\displaystyle{\displaystyle{\csc^{2}}z=1+{\cot^{2}}z}}$ `\csc^{2}@@{z} = 1+\cot^{2}@@{z}`		(csc(z))^(2) = 1 + (cot(z))^(2)	(Csc[z])^(2) == 1 + (Cot[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.21.E15	${\displaystyle{\displaystyle 2\sin u\sin v=\cos\left(u-v\right)-\cos\left(u+v% \right)}}$ `2\sin@@{u}\sin@@{v} = \cos@{u-v}-\cos@{u+v}`		2sin(u)sin(v) = cos(u - v)- cos(u + v)	2Sin[u]Sin[v] == Cos[u - v]- Cos[u + v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E16	${\displaystyle{\displaystyle 2\cos u\cos v=\cos\left(u-v\right)+\cos\left(u+v% \right)}}$ `2\cos@@{u}\cos@@{v} = \cos@{u-v}+\cos@{u+v}`		2cos(u)cos(v) = cos(u - v)+ cos(u + v)	2Cos[u]Cos[v] == Cos[u - v]+ Cos[u + v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E17	${\displaystyle{\displaystyle 2\sin u\cos v=\sin\left(u-v\right)+\sin\left(u+v% \right)}}$ `2\sin@@{u}\cos@@{v} = \sin@{u-v}+\sin@{u+v}`		2sin(u)cos(v) = sin(u - v)+ sin(u + v)	2Sin[u]Cos[v] == Sin[u - v]+ Sin[u + v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E18	${\displaystyle{\displaystyle{\sin^{2}}u-{\sin^{2}}v=\sin\left(u+v\right)\sin% \left(u-v\right)}}$ `\sin^{2}@@{u}-\sin^{2}@@{v} = \sin@{u+v}\sin@{u-v}`		(sin(u))^(2)- (sin(v))^(2) = sin(u + v)*sin(u - v)	(Sin[u])^(2)- (Sin[v])^(2) == Sin[u + v]*Sin[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E19	${\displaystyle{\displaystyle{\cos^{2}}u-{\cos^{2}}v=-\sin\left(u+v\right)\sin% \left(u-v\right)}}$ `\cos^{2}@@{u}-\cos^{2}@@{v} = -\sin@{u+v}\sin@{u-v}`		(cos(u))^(2)- (cos(v))^(2) = - sin(u + v)*sin(u - v)	(Cos[u])^(2)- (Cos[v])^(2) == - Sin[u + v]*Sin[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E20	${\displaystyle{\displaystyle{\cos^{2}}u-{\sin^{2}}v=\cos\left(u+v\right)\cos% \left(u-v\right)}}$ `\cos^{2}@@{u}-\sin^{2}@@{v} = \cos@{u+v}\cos@{u-v}`		(cos(u))^(2)- (sin(v))^(2) = cos(u + v)*cos(u - v)	(Cos[u])^(2)- (Sin[v])^(2) == Cos[u + v]*Cos[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E21	${\displaystyle{\displaystyle\sin\frac{z}{2}=+\left(\frac{1-\cos z}{2}\right)^{% 1/2}}}$ `\sin@@{\frac{z}{2}} = +\left(\frac{1-\cos@@{z}}{2}\right)^{1/2}`		sin((z)/(2)) = +((1 - cos(z))/(2))^(1/2)	Sin[Divide[z,2]] == +(Divide[1 - Cos[z],2])^(1/2)	Failure	Failure	Failed [2 / 7] Result: -.5419255224+.8655716642I Test Values: {z = -1/2+1/2I3^(1/2)} Result: -.8655770340-.4585952894I Test Values: {z = -1/23^(1/2)-1/2I}	Failed [2 / 7] Result: Complex[-0.541925522457336, 0.8655716640572733] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.8655770337160631, -0.4585952893468805] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
4.21.E21	${\displaystyle{\displaystyle\sin\frac{z}{2}=-\left(\frac{1-\cos z}{2}\right)^{% 1/2}}}$ `\sin@@{\frac{z}{2}} = -\left(\frac{1-\cos@@{z}}{2}\right)^{1/2}`		sin((z)/(2)) = -((1 - cos(z))/(2))^(1/2)	Sin[Divide[z,2]] == -(Divide[1 - Cos[z],2])^(1/2)	Failure	Failure	Failed [5 / 7] Result: .8655770340+.4585952894I Test Values: {z = 1/23^(1/2)+1/2I} Result: .5419255224-.8655716642I Test Values: {z = 1/2-1/2I3^(1/2)} Result: 1.363277520 Test Values: {z = 1.5} Result: .4948079184 Test Values: {z = .5} ... skip entries to safe data	Failed [5 / 7] Result: Complex[0.8655770337160631, 0.4585952893468805] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.5419255224573365, -0.8655716640572731] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.21.E22	${\displaystyle{\displaystyle\cos\frac{z}{2}=+\left(\frac{1+\cos z}{2}\right)^{% 1/2}}}$ `\cos@@{\frac{z}{2}} = +\left(\frac{1+\cos@@{z}}{2}\right)^{1/2}`		cos((z)/(2)) = +((1 + cos(z))/(2))^(1/2)	Cos[Divide[z,2]] == +(Divide[1 + Cos[z],2])^(1/2)	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.21.E22	${\displaystyle{\displaystyle\cos\frac{z}{2}=-\left(\frac{1+\cos z}{2}\right)^{% 1/2}}}$ `\cos@@{\frac{z}{2}} = -\left(\frac{1+\cos@@{z}}{2}\right)^{1/2}`		cos((z)/(2)) = -((1 + cos(z))/(2))^(1/2)	Cos[Divide[z,2]] == -(Divide[1 + Cos[z],2])^(1/2)	Failure	Failure	Failed [7 / 7] Result: 1.872439139-.2119959694I Test Values: {z = 1/23^(1/2)+1/2I} Result: 2.122352334+.2210167318I Test Values: {z = -1/2+1/2I3^(1/2)} Result: 2.122352334+.2210167318I Test Values: {z = 1/2-1/2I3^(1/2)} Result: 1.872439139-.2119959694I Test Values: {z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [7 / 7] Result: Complex[1.872439138961815, -0.2119959693051084] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[2.1223523339444896, 0.22101673165487346] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.21.E23	${\displaystyle{\displaystyle\tan\frac{z}{2}=+\left(\frac{1-\cos z}{1+\cos z}% \right)^{1/2}}}$ `\tan@@{\frac{z}{2}} = +\left(\frac{1-\cos@@{z}}{1+\cos@@{z}}\right)^{1/2}`		tan((z)/(2)) = +((1 - cos(z))/(1 + cos(z)))^(1/2)	Tan[Divide[z,2]] == +(Divide[1 - Cos[z],1 + Cos[z]])^(1/2)	Failure	Failure	Failed [2 / 7] Result: -.4211742148+.8595320616I Test Values: {z = -1/2+1/2I3^(1/2)} Result: -.8580864930-.5869891489I Test Values: {z = -1/23^(1/2)-1/2I}	Failed [2 / 7] Result: Complex[-0.4211742148849969, 0.8595320613685856] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.858086492859854, -0.5869891488727426] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
4.21.E23	${\displaystyle{\displaystyle\tan\frac{z}{2}=-\left(\frac{1-\cos z}{1+\cos z}% \right)^{1/2}}}$ `\tan@@{\frac{z}{2}} = -\left(\frac{1-\cos@@{z}}{1+\cos@@{z}}\right)^{1/2}`		tan((z)/(2)) = -((1 - cos(z))/(1 + cos(z)))^(1/2)	Tan[Divide[z,2]] == -(Divide[1 - Cos[z],1 + Cos[z]])^(1/2)	Failure	Failure	Failed [5 / 7] Result: .8580864930+.5869891489I Test Values: {z = 1/23^(1/2)+1/2I} Result: .4211742148-.8595320616I Test Values: {z = 1/2-1/2I3^(1/2)} Result: 1.863192920 Test Values: {z = 1.5} Result: .5106838424 Test Values: {z = .5} ... skip entries to safe data	Failed [5 / 7] Result: Complex[0.858086492859854, 0.5869891488727426] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.4211742148849973, -0.8595320613685857] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.21.E23	${\displaystyle{\displaystyle+\left(\frac{1-\cos z}{1+\cos z}\right)^{1/2}=% \frac{1-\cos z}{\sin z}}}$ `+\left(\frac{1-\cos@@{z}}{1+\cos@@{z}}\right)^{1/2} = \frac{1-\cos@@{z}}{\sin@@{z}}`		+((1 - cos(z))/(1 + cos(z)))^(1/2) = (1 - cos(z))/(sin(z))	+(Divide[1 - Cos[z],1 + Cos[z]])^(1/2) == Divide[1 - Cos[z],Sin[z]]	Failure	Failure	Failed [2 / 7] Result: .4211742148-.8595320615I Test Values: {z = -1/2+1/2I3^(1/2)} Result: .8580864930+.5869891489I Test Values: {z = -1/23^(1/2)-1/2I}	Failed [2 / 7] Result: Complex[0.42117421488499684, -0.8595320613685857] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.8580864928598539, 0.5869891488727426] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
4.21.E23	${\displaystyle{\displaystyle-\left(\frac{1-\cos z}{1+\cos z}\right)^{1/2}=% \frac{1-\cos z}{\sin z}}}$ `-\left(\frac{1-\cos@@{z}}{1+\cos@@{z}}\right)^{1/2} = \frac{1-\cos@@{z}}{\sin@@{z}}`		-((1 - cos(z))/(1 + cos(z)))^(1/2) = (1 - cos(z))/(sin(z))	-(Divide[1 - Cos[z],1 + Cos[z]])^(1/2) == Divide[1 - Cos[z],Sin[z]]	Failure	Failure	Failed [5 / 7] Result: -.8580864930-.5869891489I Test Values: {z = 1/23^(1/2)+1/2I} Result: -.4211742148+.8595320615I Test Values: {z = 1/2-1/2I3^(1/2)} Result: -1.863192920 Test Values: {z = 1.5} Result: -.5106838424 Test Values: {z = .5} ... skip entries to safe data	Failed [5 / 7] Result: Complex[-0.8580864928598539, -0.5869891488727426] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.4211742148849972, 0.8595320613685855] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.21.E23	${\displaystyle{\displaystyle\frac{1-\cos z}{\sin z}=\frac{\sin z}{1+\cos z}}}$ `\frac{1-\cos@@{z}}{\sin@@{z}} = \frac{\sin@@{z}}{1+\cos@@{z}}`		(1 - cos(z))/(sin(z)) = (sin(z))/(1 + cos(z))	Divide[1 - Cos[z],Sin[z]] == Divide[Sin[z],1 + Cos[z]]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 7]
4.21.E24	${\displaystyle{\displaystyle\sin\left(-z\right)=-\sin z}}$ `\sin@{-z} = -\sin@@{z}`		sin(- z) = - sin(z)	Sin[- z] == - Sin[z]	Successful	Successful	-	Successful [Tested: 7]
4.21.E25	${\displaystyle{\displaystyle\cos\left(-z\right)=\cos z}}$ `\cos@{-z} = \cos@@{z}`		cos(- z) = cos(z)	Cos[- z] == Cos[z]	Successful	Successful	-	Successful [Tested: 7]
4.21.E26	${\displaystyle{\displaystyle\tan\left(-z\right)=-\tan z}}$ `\tan@{-z} = -\tan@@{z}`		tan(- z) = - tan(z)	Tan[- z] == - Tan[z]	Successful	Successful	-	Successful [Tested: 7]
4.21.E27	${\displaystyle{\displaystyle\sin\left(2z\right)=2\sin z\cos z}}$ `\sin@{2z} = 2\sin@@{z}\cos@@{z}`		sin(2z) = 2sin(z)*cos(z)	Sin[2z] == 2Sin[z]*Cos[z]	Successful	Successful	-	Successful [Tested: 7]
4.21.E27	${\displaystyle{\displaystyle 2\sin z\cos z=\frac{2\tan z}{1+{\tan^{2}}z}}}$ `2\sin@@{z}\cos@@{z} = \frac{2\tan@@{z}}{1+\tan^{2}@@{z}}`		2sin(z)cos(z) = (2*tan(z))/(1 + (tan(z))^(2))	2Sin[z]Cos[z] == Divide[2*Tan[z],1 + (Tan[z])^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.21.E28	${\displaystyle{\displaystyle\cos\left(2z\right)=2{\cos^{2}}z-1}}$ `\cos@{2z} = 2\cos^{2}@@{z}-1`		cos(2z) = 2(cos(z))^(2)- 1	Cos[2z] == 2(Cos[z])^(2)- 1	Successful	Successful	-	Successful [Tested: 7]
4.21.E28	${\displaystyle{\displaystyle 2{\cos^{2}}z-1=1-2{\sin^{2}}z}}$ `2\cos^{2}@@{z}-1 = 1-2\sin^{2}@@{z}`		2(cos(z))^(2)- 1 = 1 - 2(sin(z))^(2)	2(Cos[z])^(2)- 1 == 1 - 2(Sin[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.21.E28	${\displaystyle{\displaystyle 1-2{\sin^{2}}z={\cos^{2}}z-{\sin^{2}}z}}$ `1-2\sin^{2}@@{z} = \cos^{2}@@{z}-\sin^{2}@@{z}`		1 - 2*(sin(z))^(2) = (cos(z))^(2)- (sin(z))^(2)	1 - 2*(Sin[z])^(2) == (Cos[z])^(2)- (Sin[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.21.E28	${\displaystyle{\displaystyle{\cos^{2}}z-{\sin^{2}}z=\frac{1-{\tan^{2}}z}{1+{% \tan^{2}}z}}}$ `\cos^{2}@@{z}-\sin^{2}@@{z} = \frac{1-\tan^{2}@@{z}}{1+\tan^{2}@@{z}}`		(cos(z))^(2)- (sin(z))^(2) = (1 - (tan(z))^(2))/(1 + (tan(z))^(2))	(Cos[z])^(2)- (Sin[z])^(2) == Divide[1 - (Tan[z])^(2),1 + (Tan[z])^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.21.E29	${\displaystyle{\displaystyle\tan\left(2z\right)=\frac{2\tan z}{1-{\tan^{2}}z}}}$ `\tan@{2z} = \frac{2\tan@@{z}}{1-\tan^{2}@@{z}}`		tan(2z) = (2tan(z))/(1 - (tan(z))^(2))	Tan[2z] == Divide[2Tan[z],1 - (Tan[z])^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.21.E29	${\displaystyle{\displaystyle\frac{2\tan z}{1-{\tan^{2}}z}=\frac{2\cot z}{{\cot% ^{2}}z-1}}}$ `\frac{2\tan@@{z}}{1-\tan^{2}@@{z}} = \frac{2\cot@@{z}}{\cot^{2}@@{z}-1}`		(2tan(z))/(1 - (tan(z))^(2)) = (2cot(z))/((cot(z))^(2)- 1)	Divide[2Tan[z],1 - (Tan[z])^(2)] == Divide[2Cot[z],(Cot[z])^(2)- 1]	Successful	Successful	-	Successful [Tested: 7]
4.21.E29	${\displaystyle{\displaystyle\frac{2\cot z}{{\cot^{2}}z-1}=\frac{2}{\cot z-\tan z% }}}$ `\frac{2\cot@@{z}}{\cot^{2}@@{z}-1} = \frac{2}{\cot@@{z}-\tan@@{z}}`		(2*cot(z))/((cot(z))^(2)- 1) = (2)/(cot(z)- tan(z))	Divide[2*Cot[z],(Cot[z])^(2)- 1] == Divide[2,Cot[z]- Tan[z]]	Successful	Successful	-	Successful [Tested: 7]
4.21.E30	${\displaystyle{\displaystyle\sin\left(3z\right)=3\sin z-4{\sin^{3}}z}}$ `\sin@{3z} = 3\sin@@{z}-4\sin^{3}@@{z}`		sin(3z) = 3sin(z)- 4*(sin(z))^(3)	Sin[3z] == 3Sin[z]- 4*(Sin[z])^(3)	Successful	Successful	-	Successful [Tested: 7]
4.21.E31	${\displaystyle{\displaystyle\cos\left(3z\right)=-3\cos z+4{\cos^{3}}z}}$ `\cos@{3z} = -3\cos@@{z}+4\cos^{3}@@{z}`		cos(3z) = - 3cos(z)+ 4*(cos(z))^(3)	Cos[3z] == - 3Cos[z]+ 4*(Cos[z])^(3)	Successful	Successful	-	Successful [Tested: 7]
4.21.E32	${\displaystyle{\displaystyle\sin\left(4z\right)=8{\cos^{3}}z\sin z-4\cos z\sin z}}$ `\sin@{4z} = 8\cos^{3}@@{z}\sin@@{z}-4\cos@@{z}\sin@@{z}`		sin(4z) = 8(cos(z))^(3)* sin(z)- 4cos(z)sin(z)	Sin[4z] == 8(Cos[z])^(3)* Sin[z]- 4Cos[z]Sin[z]	Successful	Successful	-	Successful [Tested: 7]
4.21.E33	${\displaystyle{\displaystyle\cos\left(4z\right)=8{\cos^{4}}z-8{\cos^{2}}z+1}}$ `\cos@{4z} = 8\cos^{4}@@{z}-8\cos^{2}@@{z}+1`		cos(4z) = 8(cos(z))^(4)- 8*(cos(z))^(2)+ 1	Cos[4z] == 8(Cos[z])^(4)- 8*(Cos[z])^(2)+ 1	Successful	Successful	-	Successful [Tested: 7]
4.21.E34	${\displaystyle{\displaystyle\cos\left(nz\right)+i\sin\left(nz\right)=(\cos z+i% \sin z)^{n}}}$ `\cos@{nz}+i\sin@{nz} = (\cos@@{z}+i\sin@@{z})^{n}`		cos(nz)+ Isin(nz) = (cos(z)+ Isin(z))^(n)	Cos[nz]+ ISin[nz] == (Cos[z]+ ISin[z])^(n)	Successful	Failure	-	Successful [Tested: 21]
4.21.E35	${\displaystyle{\displaystyle\sin\left(nz\right)=2^{n-1}\prod_{k=0}^{n-1}\sin% \left(z+\frac{k\pi}{n}\right)}}$ `\sin@{nz} = 2^{n-1}\prod_{k=0}^{n-1}\sin@{z+\frac{k\pi}{n}}`		sin(nz) = (2)^(n - 1) product(sin(z +(k*Pi)/(n)), k = 0..n - 1)	Sin[nz] == (2)^(n - 1) Product[Sin[z +Divide[k*Pi,n]], {k, 0, n - 1}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 21]	Successful [Tested: 7]
4.21#Ex1	${\displaystyle{\displaystyle\sin z=\frac{2t}{1+t^{2}}}}$ `\sin@@{z} = \frac{2t}{1+t^{2}}`		sin(z) = (2*t)/(1 + (t)^(2))	Sin[z] == Divide[2*t,1 + (t)^(2)]	Failure	Failure	Failed [42 / 42] Result: 1.782057258+.3375964631I Test Values: {t = -1.5, z = 1/23^(1/2)+1/2I} Result: .2523455641+.8586367171I Test Values: {t = -1.5, z = -1/2+1/2I3^(1/2)} Result: 1.593808282-.8586367171I Test Values: {t = -1.5, z = 1/2-1/2I3^(1/2)} Result: .640965885e-1-.3375964631I Test Values: {t = -1.5, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [42 / 42] Result: Complex[1.782057257377061, 0.33759646322287] Test Values: {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.25234556426971166, 0.8586367168171449] Test Values: {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.21#Ex2	${\displaystyle{\displaystyle\cos z=\frac{1-t^{2}}{1+t^{2}}}}$ `\cos@@{z} = \frac{1-t^{2}}{1+t^{2}}`		cos(z) = (1 - (t)^(2))/(1 + (t)^(2))	Cos[z] == Divide[1 - (t)^(2),1 + (t)^(2)]	Failure	Failure	Failed [42 / 42] Result: 1.115158404-.3969495503I Test Values: {t = -1.5, z = 1/23^(1/2)+1/2I} Result: 1.612380902+.4690753764I Test Values: {t = -1.5, z = -1/2+1/2I3^(1/2)} Result: 1.612380902+.4690753764I Test Values: {t = -1.5, z = 1/2-1/2I3^(1/2)} Result: 1.115158404-.3969495503I Test Values: {t = -1.5, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [42 / 42] Result: Complex[1.1151584036726099, -0.3969495502290325] Test Values: {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.612380901479495, 0.46907537626850365] Test Values: {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.21.E37	${\displaystyle{\displaystyle\sin z=\sin x\cosh y+\mathrm{i}\cos x\sinh y}}$ `\sin@@{z} = \sin@@{x}\cosh@@{y}+\iunit\cos@@{x}\sinh@@{y}`		sin(x + yI) = sin(x)cosh(y)+ Icos(x)sinh(y)	Sin[x + yI] == Sin[x]Cosh[y]+ ICos[x]Sinh[y]	Successful	Successful	-	Successful [Tested: 18]
4.21.E38	${\displaystyle{\displaystyle\cos z=\cos x\cosh y-\mathrm{i}\sin x\sinh y}}$ `\cos@@{z} = \cos@@{x}\cosh@@{y}-\iunit\sin@@{x}\sinh@@{y}`		cos(x + yI) = cos(x)cosh(y)- Isin(x)sinh(y)	Cos[x + yI] == Cos[x]Cosh[y]- ISin[x]Sinh[y]	Successful	Successful	-	Successful [Tested: 18]
4.21.E39	${\displaystyle{\displaystyle\tan z=\frac{\sin\left(2x\right)+\mathrm{i}\sinh% \left(2y\right)}{\cos\left(2x\right)+\cosh\left(2y\right)}}}$ `\tan@@{z} = \frac{\sin@{2x}+\iunit\sinh@{2y}}{\cos@{2x}+\cosh@{2y}}`		tan(x + yI) = (sin(2x)+ Isinh(2y))/(cos(2x)+ cosh(2y))	Tan[x + yI] == Divide[Sin[2x]+ ISinh[2y],Cos[2x]+ Cosh[2y]]	Successful	Successful	-	Successful [Tested: 18]
4.21.E40	${\displaystyle{\displaystyle\cot z=\frac{\sin\left(2x\right)-\mathrm{i}\sinh% \left(2y\right)}{\cosh\left(2y\right)-\cos\left(2x\right)}}}$ `\cot@@{z} = \frac{\sin@{2x}-\iunit\sinh@{2y}}{\cosh@{2y}-\cos@{2x}}`		cot(x + yI) = (sin(2x)- Isinh(2y))/(cosh(2y)- cos(2x))	Cot[x + yI] == Divide[Sin[2x]- ISinh[2y],Cosh[2y]- Cos[2x]]	Successful	Successful	-	Successful [Tested: 18]
4.21.E41	${\displaystyle{\displaystyle\|\sin z\|=({\sin^{2}}x+{\sinh^{2}}y)^{1/2}}}$ `\|\sin@@{z}\| = (\sin^{2}@@{x}+\sinh^{2}@@{y})^{1/2}`		abs(sin(x + y*I)) = ((sin(x))^(2)+ (sinh(y))^(2))^(1/2)	Abs[Sin[x + y*I]] == ((Sin[x])^(2)+ (Sinh[y])^(2))^(1/2)	Successful	Failure	-	Successful [Tested: 18]
4.21.E41	${\displaystyle{\displaystyle({\sin^{2}}x+{\sinh^{2}}y)^{1/2}=\left(\tfrac{1}{2% }\left(\cosh\left(2y\right)-\cos\left(2x\right)\right)\right)^{1/2}}}$ `(\sin^{2}@@{x}+\sinh^{2}@@{y})^{1/2} = \left(\tfrac{1}{2}\left(\cosh@{2y}-\cos@{2x}\right)\right)^{1/2}`		((sin(x))^(2)+ (sinh(y))^(2))^(1/2) = ((1)/(2)(cosh(2y)- cos(2*x)))^(1/2)	((Sin[x])^(2)+ (Sinh[y])^(2))^(1/2) == (Divide[1,2](Cosh[2y]- Cos[2*x]))^(1/2)	Successful	Successful	-	Successful [Tested: 18]
4.21.E42	${\displaystyle{\displaystyle\|\cos z\|=({\cos^{2}}x+{\sinh^{2}}y)^{1/2}}}$ `\|\cos@@{z}\| = (\cos^{2}@@{x}+\sinh^{2}@@{y})^{1/2}`		abs(cos(x + y*I)) = ((cos(x))^(2)+ (sinh(y))^(2))^(1/2)	Abs[Cos[x + y*I]] == ((Cos[x])^(2)+ (Sinh[y])^(2))^(1/2)	Successful	Failure	-	Successful [Tested: 18]
4.21.E42	${\displaystyle{\displaystyle({\cos^{2}}x+{\sinh^{2}}y)^{1/2}=\left(\tfrac{1}{2% }(\cosh\left(2y\right)+\cos\left(2x\right))\right)^{1/2}}}$ `(\cos^{2}@@{x}+\sinh^{2}@@{y})^{1/2} = \left(\tfrac{1}{2}(\cosh@{2y}+\cos@{2x})\right)^{1/2}`		((cos(x))^(2)+ (sinh(y))^(2))^(1/2) = ((1)/(2)(cosh(2y)+ cos(2*x)))^(1/2)	((Cos[x])^(2)+ (Sinh[y])^(2))^(1/2) == (Divide[1,2](Cosh[2y]+ Cos[2*x]))^(1/2)	Successful	Successful	-	Successful [Tested: 18]
4.21.E43	${\displaystyle{\displaystyle\|\tan z\|=\left(\frac{\cosh\left(2y\right)-\cos% \left(2x\right)}{\cosh\left(2y\right)+\cos\left(2x\right)}\right)^{1/2}}}$ `\|\tan@@{z}\| = \left(\frac{\cosh@{2y}-\cos@{2x}}{\cosh@{2y}+\cos@{2x}}\right)^{1/2}`		abs(tan(x + yI)) = ((cosh(2y)- cos(2x))/(cosh(2y)+ cos(2*x)))^(1/2)	Abs[Tan[x + yI]] == (Divide[Cosh[2y]- Cos[2x],Cosh[2y]+ Cos[2*x]])^(1/2)	Successful	Failure	-	Successful [Tested: 18]
4.22.E1	${\displaystyle{\displaystyle\sin z=z\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n% ^{2}\pi^{2}}\right)}}$ `\sin@@{z} = z\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n^{2}\pi^{2}}\right)`		sin(z) = zproduct(1 -((z)^(2))/((n)^(2) (Pi)^(2)), n = 1..infinity)	Sin[z] == zProduct[1 -Divide[(z)^(2),(n)^(2) (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
4.22.E2	${\displaystyle{\displaystyle\cos z=\prod_{n=1}^{\infty}\left(1-\frac{4z^{2}}{(% 2n-1)^{2}\pi^{2}}\right)}}$ `\cos@@{z} = \prod_{n=1}^{\infty}\left(1-\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}\right)`		cos(z) = product(1 -(4(z)^(2))/((2n - 1)^(2)* (Pi)^(2)), n = 1..infinity)	Cos[z] == Product[1 -Divide[4(z)^(2),(2n - 1)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
4.22.E3	${\displaystyle{\displaystyle\cot z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z% ^{2}-n^{2}\pi^{2}}}}$ `\cot@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}-n^{2}\pi^{2}}`		cot(z) = (1)/(z)+ 2zsum((1)/((z)^(2)- (n)^(2)* (Pi)^(2)), n = 1..infinity)	Cot[z] == Divide[1,z]+ 2zSum[Divide[1,(z)^(2)- (n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.22.E4	${\displaystyle{\displaystyle{\csc^{2}}z=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n% \pi)^{2}}}}$ `\csc^{2}@@{z} = \sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi)^{2}}`		(csc(z))^(2) = sum((1)/((z - n*Pi)^(2)), n = - infinity..infinity)	(Csc[z])^(2) == Sum[Divide[1,(z - n*Pi)^(2)], {n, - Infinity, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.22.E5	${\displaystyle{\displaystyle\csc z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)% ^{n}}{z^{2}-n^{2}\pi^{2}}}}$ `\csc@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}-n^{2}\pi^{2}}`		csc(z) = (1)/(z)+ 2zsum(((- 1)^(n))/((z)^(2)- (n)^(2)* (Pi)^(2)), n = 1..infinity)	Csc[z] == Divide[1,z]+ 2zSum[Divide[(- 1)^(n),(z)^(2)- (n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
4.23.E1	${\displaystyle{\displaystyle\operatorname{Arcsin}z=\int_{0}^{z}\frac{\mathrm{d% }t}{(1-t^{2})^{1/2}}}}$ `\Asin@@{z} = \int_{0}^{z}\frac{\diff{t}}{(1-t^{2})^{1/2}}`		Error	ArcSin[z] == Integrate[Divide[1,(1 - (t)^(2))^(1/2)], {t, 0, z}, GenerateConditions->None]	Missing Macro Error	Successful	-	Successful [Tested: 7]
4.23.E2	${\displaystyle{\displaystyle\operatorname{Arccos}z=\int_{z}^{1}\frac{\mathrm{d% }t}{(1-t^{2})^{1/2}}}}$ `\Acos@@{z} = \int_{z}^{1}\frac{\diff{t}}{(1-t^{2})^{1/2}}`		Error	ArcCos[z] == Integrate[Divide[1,(1 - (t)^(2))^(1/2)], {t, z, 1}, GenerateConditions->None]	Missing Macro Error	Successful	-	Successful [Tested: 7]
4.23.E3	${\displaystyle{\displaystyle\operatorname{Arctan}z=\int_{0}^{z}\frac{\mathrm{d% }t}{1+t^{2}}}}$ `\Atan@@{z} = \int_{0}^{z}\frac{\diff{t}}{1+t^{2}}`		Error	ArcTan[z] == Integrate[Divide[1,1 + (t)^(2)], {t, 0, z}, GenerateConditions->None]	Missing Macro Error	Successful	-	Successful [Tested: 1]
4.23.E4	${\displaystyle{\displaystyle\operatorname{Arccsc}z=\operatorname{Arcsin}\left(% 1/z\right)}}$ `\Acsc@@{z} = \Asin@{1/z}`		Error	ArcCsc[z] == ArcSin[1/z]	Missing Macro Error	Successful	-	Successful [Tested: 7]
4.23.E5	${\displaystyle{\displaystyle\operatorname{Arcsec}z=\operatorname{Arccos}\left(% 1/z\right)}}$ `\Asec@@{z} = \Acos@{1/z}`		Error	ArcSec[z] == ArcCos[1/z]	Missing Macro Error	Successful	-	Successful [Tested: 7]
4.23.E6	${\displaystyle{\displaystyle\operatorname{Arccot}z=\operatorname{Arctan}\left(% 1/z\right)}}$ `\Acot@@{z} = \Atan@{1/z}`		Error	ArcCot[z] == ArcTan[1/z]	Missing Macro Error	Successful	-	Successful [Tested: 7]
4.23.E7	${\displaystyle{\displaystyle\operatorname{arccsc}z=\operatorname{arcsin}\left(% 1/z\right)}}$ `\acsc@@{z} = \asin@{1/z}`		arccsc(z) = arcsin(1/z)	ArcCsc[z] == ArcSin[1/z]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.23.E8	${\displaystyle{\displaystyle\operatorname{arcsec}z=\operatorname{arccos}\left(% 1/z\right)}}$ `\asec@@{z} = \acos@{1/z}`		arcsec(z) = arccos(1/z)	ArcSec[z] == ArcCos[1/z]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.23.E9	${\displaystyle{\displaystyle\operatorname{arccot}z=\operatorname{arctan}\left(% 1/z\right)}}$ `\acot@@{z} = \atan@{1/z}`		arccot(z) = arctan(1/z)	ArcCot[z] == ArcTan[1/z]	Failure	Successful	Failed [2 / 7] Result: 3.141592654+0.I Test Values: {z = -1/2+1/2I3^(1/2), z = 1/2} Result: 3.141592654+0.I Test Values: {z = -1/23^(1/2)-1/2I, z = 1/2}	Successful [Tested: 1]
4.23.E10	${\displaystyle{\displaystyle\operatorname{arcsin}\left(-z\right)=-% \operatorname{arcsin}z}}$ `\asin@{-z} = -\asin@@{z}`		arcsin(- z) = - arcsin(z)	ArcSin[- z] == - ArcSin[z]	Successful	Successful	-	Successful [Tested: 7]
4.23.E11	${\displaystyle{\displaystyle\operatorname{arccos}\left(-z\right)=\pi-% \operatorname{arccos}z}}$ `\acos@{-z} = \pi-\acos@@{z}`		arccos(- z) = Pi - arccos(z)	ArcCos[- z] == Pi - ArcCos[z]	Successful	Successful	-	Successful [Tested: 7]
4.23.E12	${\displaystyle{\displaystyle\operatorname{arctan}\left(-z\right)=-% \operatorname{arctan}z}}$ `\atan@{-z} = -\atan@@{z}`		arctan(- z) = - arctan(z)	ArcTan[- z] == - ArcTan[z]	Successful	Successful	-	Successful [Tested: 1]
4.23.E13	${\displaystyle{\displaystyle\operatorname{arccsc}\left(-z\right)=-% \operatorname{arccsc}z}}$ `\acsc@{-z} = -\acsc@@{z}`		arccsc(- z) = - arccsc(z)	ArcCsc[- z] == - ArcCsc[z]	Successful	Successful	-	Successful [Tested: 7]
4.23.E14	${\displaystyle{\displaystyle\operatorname{arcsec}\left(-z\right)=\pi-% \operatorname{arcsec}z}}$ `\asec@{-z} = \pi-\asec@@{z}`		arcsec(- z) = Pi - arcsec(z)	ArcSec[- z] == Pi - ArcSec[z]	Successful	Successful	-	Successful [Tested: 7]
4.23.E15	${\displaystyle{\displaystyle\operatorname{arccot}\left(-z\right)=-% \operatorname{arccot}z}}$ `\acot@{-z} = -\acot@@{z}`		arccot(- z) = - arccot(z)	ArcCot[- z] == - ArcCot[z]	Failure	Successful	Skip - No test values generated	Successful [Tested: 1]
4.23.E16	${\displaystyle{\displaystyle\operatorname{arccos}z=\tfrac{1}{2}\pi-% \operatorname{arcsin}z}}$ `\acos@@{z} = \tfrac{1}{2}\pi-\asin@@{z}`		arccos(z) = (1)/(2)*Pi - arcsin(z)	ArcCos[z] == Divide[1,2]*Pi - ArcSin[z]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.23.E17	${\displaystyle{\displaystyle\operatorname{arcsec}z=\tfrac{1}{2}\pi-% \operatorname{arccsc}z}}$ `\asec@@{z} = \tfrac{1}{2}\pi-\acsc@@{z}`		arcsec(z) = (1)/(2)*Pi - arccsc(z)	ArcSec[z] == Divide[1,2]*Pi - ArcCsc[z]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.23.E18	${\displaystyle{\displaystyle\operatorname{arccot}z=+\tfrac{1}{2}\pi-% \operatorname{arctan}z}}$ `\acot@@{z} = +\tfrac{1}{2}\pi-\atan@@{z}`		arccot(z) = +(1)/(2)*Pi - arctan(z)	ArcCot[z] == +Divide[1,2]*Pi - ArcTan[z]	Successful	Failure	Skip - symbolical successful subtest	Successful [Tested: 1]
4.23.E18	${\displaystyle{\displaystyle\operatorname{arccot}z=-\tfrac{1}{2}\pi-% \operatorname{arctan}z}}$ `\acot@@{z} = -\tfrac{1}{2}\pi-\atan@@{z}`		arccot(z) = -(1)/(2)*Pi - arctan(z)	ArcCot[z] == -Divide[1,2]*Pi - ArcTan[z]	Failure	Failure	Failed [7 / 7] Result: 3.141592654+0.I Test Values: {z = 1/23^(1/2)+1/2I, z = 1/2} Result: 3.141592654+0.I Test Values: {z = -1/2+1/2I3^(1/2), z = 1/2} Result: 3.141592654+0.I Test Values: {z = 1/2-1/2I3^(1/2), z = 1/2} Result: 3.141592654+0.I Test Values: {z = -1/23^(1/2)-1/2I, z = 1/2} ... skip entries to safe data	Failed [1 / 1] Result: 3.141592653589793 Test Values: {Rule[z, Rational[1, 2]]}
4.23.E19	${\displaystyle{\displaystyle\operatorname{arcsin}z=-i\ln\left((1-z^{2})^{1/2}+% iz\right)}}$ `\asin@@{z} = -i\ln@{(1-z^{2})^{1/2}+iz}`		arcsin(z) = - Iln((1 - (z)^(2))^(1/2)+ Iz)	ArcSin[z] == - ILog[(1 - (z)^(2))^(1/2)+ Iz]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.23.E20	${\displaystyle{\displaystyle\operatorname{arcsin}x=\tfrac{1}{2}\pi+i\ln\left((% x^{2}-1)^{1/2}+x\right)}}$ `\asin@@{x} = \tfrac{1}{2}\pi+ i\ln@{(x^{2}-1)^{1/2}+x}`		arcsin(x) = (1)/(2)Pi + Iln(((x)^(2)- 1)^(1/2)+ x)	ArcSin[x] == Divide[1,2]Pi + ILog[((x)^(2)- 1)^(1/2)+ x]	Failure	Failure	Failed [2 / 3] Result: 0.-1.924847300I Test Values: {x = 1.5, x = 3/2} Result: 0.-2.633915794I Test Values: {x = 2, x = 3/2}	Failed [1 / 1] Result: Complex[0.0, -1.9248473002384139] Test Values: {Rule[x, Rational[3, 2]]}
4.23.E20	${\displaystyle{\displaystyle\operatorname{arcsin}x=\tfrac{1}{2}\pi-i\ln\left((% x^{2}-1)^{1/2}+x\right)}}$ `\asin@@{x} = \tfrac{1}{2}\pi- i\ln@{(x^{2}-1)^{1/2}+x}`		arcsin(x) = (1)/(2)Pi - Iln(((x)^(2)- 1)^(1/2)+ x)	ArcSin[x] == Divide[1,2]Pi - ILog[((x)^(2)- 1)^(1/2)+ x]	Failure	Failure	Failed [1 / 3] Result: -2.094395102+.1347500000e-10*I Test Values: {x = .5, x = 3/2}	Successful [Tested: 1]
4.23.E21	${\displaystyle{\displaystyle\operatorname{arcsin}x=-\tfrac{1}{2}\pi+i\ln\left(% (x^{2}-1)^{1/2}-x\right)}}$ `\asin@@{x} = -\tfrac{1}{2}\pi+ i\ln@{(x^{2}-1)^{1/2}-x}`		arcsin(x) = -(1)/(2)Pi + Iln(((x)^(2)- 1)^(1/2)- x)	ArcSin[x] == -Divide[1,2]Pi + ILog[((x)^(2)- 1)^(1/2)- x]	Failure	Failure	Failed [3 / 3] Result: 6.283185308+.7e-9I Test Values: {x = 1.5, x = -2} Result: 4.188790205-.1347500000e-10I Test Values: {x = .5, x = -2} Result: 6.283185308+.2e-8*I Test Values: {x = 2, x = -2}	Successful [Tested: 1]
4.23.E21	${\displaystyle{\displaystyle\operatorname{arcsin}x=-\tfrac{1}{2}\pi-i\ln\left(% (x^{2}-1)^{1/2}-x\right)}}$ `\asin@@{x} = -\tfrac{1}{2}\pi- i\ln@{(x^{2}-1)^{1/2}-x}`		arcsin(x) = -(1)/(2)Pi - Iln(((x)^(2)- 1)^(1/2)- x)	ArcSin[x] == -Divide[1,2]Pi - ILog[((x)^(2)- 1)^(1/2)- x]	Failure	Failure	Failed [2 / 3] Result: 0.-1.924847301I Test Values: {x = 1.5, x = -2} Result: 0.-2.633915796I Test Values: {x = 2, x = -2}	Failed [1 / 1] Result: Complex[0.0, 2.633915793849633] Test Values: {Rule[x, -2]}
4.23.E22	${\displaystyle{\displaystyle\operatorname{arccos}z=\tfrac{1}{2}\pi+i\ln\left((% 1-z^{2})^{1/2}+iz\right)}}$ `\acos@@{z} = \tfrac{1}{2}\pi+i\ln@{(1-z^{2})^{1/2}+iz}`		arccos(z) = (1)/(2)Pi + Iln((1 - (z)^(2))^(1/2)+ I*z)	ArcCos[z] == Divide[1,2]Pi + ILog[(1 - (z)^(2))^(1/2)+ I*z]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.23.E23	${\displaystyle{\displaystyle\operatorname{arccos}z=-2i\ln\left(\left(\frac{1+z% }{2}\right)^{1/2}+i\left(\frac{1-z}{2}\right)^{1/2}\right)}}$ `\acos@@{z} = -2i\ln@{\left(\frac{1+z}{2}\right)^{1/2}+i\left(\frac{1-z}{2}\right)^{1/2}}`		arccos(z) = - 2Iln(((1 + z)/(2))^(1/2)+ I*((1 - z)/(2))^(1/2))	ArcCos[z] == - 2ILog[(Divide[1 + z,2])^(1/2)+ I*(Divide[1 - z,2])^(1/2)]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.23.E24	${\displaystyle{\displaystyle\operatorname{arccos}x=-i\ln\left((x^{2}-1)^{1/2}+% x\right)}}$ `\acos@@{x} = - i\ln@{(x^{2}-1)^{1/2}+x}`		arccos(x) = - I*ln(((x)^(2)- 1)^(1/2)+ x)	ArcCos[x] == - I*Log[((x)^(2)- 1)^(1/2)+ x]	Failure	Failure	Failed [2 / 3] Result: 1.924847300I Test Values: {x = 1.5, x = 3/2} Result: 2.633915794I Test Values: {x = 2, x = 3/2}	Failed [1 / 1] Result: Complex[0.0, 1.9248473002384139] Test Values: {Rule[x, Rational[3, 2]]}
4.23.E24	${\displaystyle{\displaystyle\operatorname{arccos}x=+i\ln\left((x^{2}-1)^{1/2}+% x\right)}}$ `\acos@@{x} = + i\ln@{(x^{2}-1)^{1/2}+x}`		arccos(x) = + I*ln(((x)^(2)- 1)^(1/2)+ x)	ArcCos[x] == + I*Log[((x)^(2)- 1)^(1/2)+ x]	Failure	Failure	Failed [1 / 3] Result: 2.094395102-.1347500000e-10*I Test Values: {x = .5, x = 3/2}	Successful [Tested: 1]
4.23.E25	${\displaystyle{\displaystyle\operatorname{arccos}x=\pi-i\ln\left((x^{2}-1)^{1/% 2}-x\right)}}$ `\acos@@{x} = \pi- i\ln@{(x^{2}-1)^{1/2}-x}`		arccos(x) = Pi - I*ln(((x)^(2)- 1)^(1/2)- x)	ArcCos[x] == Pi - I*Log[((x)^(2)- 1)^(1/2)- x]	Failure	Failure	Failed [3 / 3] Result: -6.283185308-.7e-9I Test Values: {x = 1.5, x = -2} Result: -4.188790205+.1347500000e-10I Test Values: {x = .5, x = -2} Result: -6.283185308-.2e-8*I Test Values: {x = 2, x = -2}	Successful [Tested: 1]
4.23.E25	${\displaystyle{\displaystyle\operatorname{arccos}x=\pi+i\ln\left((x^{2}-1)^{1/% 2}-x\right)}}$ `\acos@@{x} = \pi+ i\ln@{(x^{2}-1)^{1/2}-x}`		arccos(x) = Pi + I*ln(((x)^(2)- 1)^(1/2)- x)	ArcCos[x] == Pi + I*Log[((x)^(2)- 1)^(1/2)- x]	Failure	Failure	Failed [2 / 3] Result: 0.+1.924847301I Test Values: {x = 1.5, x = -2} Result: 0.+2.633915796I Test Values: {x = 2, x = -2}	Failed [1 / 1] Result: Complex[0.0, -2.633915793849633] Test Values: {Rule[x, -2]}
4.23.E26	${\displaystyle{\displaystyle\operatorname{arctan}z=\frac{i}{2}\ln\left(\frac{i% +z}{i-z}\right)}}$ `\atan@@{z} = \frac{i}{2}\ln@{\frac{i+z}{i-z}}`		arctan(z) = (I)/(2)*ln((I + z)/(I - z))	ArcTan[z] == Divide[I,2]*Log[Divide[I + z,I - z]]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.23.E27	${\displaystyle{\displaystyle\operatorname{arctan}\left(iy\right)=+\frac{1}{2}% \pi+\frac{i}{2}\ln\left(\frac{y+1}{y-1}\right)}}$ `\atan@{iy} = +\frac{1}{2}\pi+\frac{i}{2}\ln@{\frac{y+1}{y-1}}`		arctan(Iy) = +(1)/(2)Pi +(I)/(2)*ln((y + 1)/(y - 1))	ArcTan[Iy] == +Divide[1,2]Pi +Divide[I,2]*Log[Divide[y + 1,y - 1]]	Failure	Failure	Failed [2 / 6] Result: -3.141592654-.2e-9I Test Values: {y = -1.5, y = -3/2} Result: -3.141592654+.2e-9I Test Values: {y = -2, y = -3/2}	Failed [1 / 1] Result: Complex[-3.141592653589793, -1.1102230246251565*^-16] Test Values: {Rule[y, Rational[-3, 2]]}
4.23.E27	${\displaystyle{\displaystyle\operatorname{arctan}\left(iy\right)=-\frac{1}{2}% \pi+\frac{i}{2}\ln\left(\frac{y+1}{y-1}\right)}}$ `\atan@{iy} = -\frac{1}{2}\pi+\frac{i}{2}\ln@{\frac{y+1}{y-1}}`		arctan(Iy) = -(1)/(2)Pi +(I)/(2)*ln((y + 1)/(y - 1))	ArcTan[Iy] == -Divide[1,2]Pi +Divide[I,2]*Log[Divide[y + 1,y - 1]]	Failure	Failure	Failed [4 / 6] Result: 3.141592654+.2e-9I Test Values: {y = 1.5, y = -3/2} Result: 3.141592654+.2e-9I Test Values: {y = -.5, y = -3/2} Result: 3.141592654-.2e-9I Test Values: {y = .5, y = -3/2} Result: 3.141592654-.2e-9I Test Values: {y = 2, y = -3/2}	Successful [Tested: 1]
4.23.E28	${\displaystyle{\displaystyle z=\sin w}}$ `z = \sin@@{w}`		z = sin(w)	z == Sin[w]	Failure	Failure	Failed [70 / 70] Result: .70450695e-2+.1624035369I Test Values: {w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I} Result: -1.358980334+.5284289409I Test Values: {w = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} Result: -.3589803345-1.203621867I Test Values: {w = 1/23^(1/2)+1/2I, z = 1/2-1/2I3^(1/2)} Result: -1.725005738-.8375964631I Test Values: {w = 1/23^(1/2)+1/2I, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [70 / 70] Result: Complex[0.007045069484300837, 0.16240353677712993] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.3589803343001376, 0.5284289405615687] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.23.E29	${\displaystyle{\displaystyle z=\cos w}}$ `z = \cos@@{w}`		z = cos(w)	z == Cos[w]	Failure	Failure	Failed [70 / 70] Result: .1354823851+.8969495503I Test Values: {w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I} Result: -1.230543019+1.262974954I Test Values: {w = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} Result: -.2305430189-.4690758537I Test Values: {w = 1/23^(1/2)+1/2I, z = 1/2-1/2I3^(1/2)} Result: -1.596568423-.1030504497I Test Values: {w = 1/23^(1/2)+1/2I, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [70 / 70] Result: Complex[0.13548238472721352, 0.8969495502290324] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.230543019057225, 1.2629749540134712] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.23.E30	${\displaystyle{\displaystyle z=\tan w}}$ `z = \tan@@{w}`		z = tan(w)	z == Tan[w]	Failure	Failure	Failed [70 / 70] Result: .1520945236-.3500402975I Test Values: {w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I} Result: -1.213930880+.159851065e-1I Test Values: {w = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} Result: -.2139308804-1.716065702I Test Values: {w = 1/23^(1/2)+1/2I, z = 1/2-1/2I3^(1/2)} Result: -1.579956284-1.350040298I Test Values: {w = 1/23^(1/2)+1/2I, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [70 / 70] Result: Complex[0.1520945235384168, -0.3500402971922752] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.2139308802460218, 0.015985106592163567] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.23.E31	${\displaystyle{\displaystyle w=\operatorname{Arcsin}z}}$ `w = \Asin@@{z}`		Error	w == ArcSin[z]	Missing Macro Error	Failure	-	Failed [70 / 70] Result: Complex[0.0806272403869902, -0.15847894846240845] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.2407598364931787, -0.3314429455293106] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.23.E31	${\displaystyle{\displaystyle\operatorname{Arcsin}z=(-1)^{k}\operatorname{% arcsin}z+k\pi}}$ `\Asin@@{z} = (-1)^{k}\asin@@{z}+k\pi`		Error	ArcSin[z] == (- 1)^(k)* ArcSin[z]+ k*Pi	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[-1.5707963267948961, 1.3169578969248168] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: -6.283185307179586 Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.23.E32	${\displaystyle{\displaystyle w=\operatorname{Arccos}z}}$ `w = \Acos@@{z}`		Error	w == ArcCos[z]	Missing Macro Error	Failure	-	Failed [70 / 70] Result: Complex[0.08062724038699065, 1.1584789484624083] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.0795053557191978, 1.3314429455293104] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.23.E32	${\displaystyle{\displaystyle\operatorname{Arccos}z=+\operatorname{arccos}z+2k% \pi}}$ `\Acos@@{z} = +\acos@@{z}+2k\pi`		Error	ArcCos[z] == + ArcCos[z]+ 2kPi	Missing Macro Error	Failure	-	Failed [21 / 21] Result: -6.283185307179586 Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: -12.566370614359172 Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.23.E32	${\displaystyle{\displaystyle\operatorname{Arccos}z=-\operatorname{arccos}z+2k% \pi}}$ `\Acos@@{z} = -\acos@@{z}+2k\pi`		Error	ArcCos[z] == - ArcCos[z]+ 2kPi	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[-4.71238898038469, -1.3169578969248168] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-10.995574287564276, -1.3169578969248168] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.23.E33	${\displaystyle{\displaystyle w=\operatorname{Arctan}z}}$ `w = \Atan@@{z}`		Error	w == ArcTan[z]	Missing Macro Error	Failure	-	Failed [10 / 10] Result: Complex[0.4023777947836326, 0.49999999999999994] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Rational[1, 2]]} Result: Complex[-0.9636476090008059, 0.8660254037844387] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[z, Rational[1, 2]]} ... skip entries to safe data
4.23.E33	${\displaystyle{\displaystyle\operatorname{Arctan}z=\operatorname{arctan}z+k\pi}}$ `\Atan@@{z} = \atan@@{z}+k\pi`		Error	ArcTan[z] == ArcTan[z]+ k*Pi	Missing Macro Error	Failure	-	Failed [3 / 3] Result: -3.141592653589793 Test Values: {Rule[k, 1], Rule[z, Rational[1, 2]]} Result: -6.283185307179586 Test Values: {Rule[k, 2], Rule[z, Rational[1, 2]]} ... skip entries to safe data
4.23.E34	${\displaystyle{\displaystyle\operatorname{arcsin}z=\operatorname{arcsin}\beta+% \mathrm{i}\operatorname{sign}\left(y\right)\ln\left(\alpha+(\alpha^{2}-1)^{1/2% }\right)}}$ `\asin@@{z} = \asin@@{\beta}+\iunit\sign@{y}\ln@{\alpha+(\alpha^{2}-1)^{1/2}}`		arcsin(x + yI) = arcsin((1)/(2)((x + 1)^(2)+ (y)^(2))^(1/2)-(1)/(2)((x - 1)^(2)+ (y)^(2))^(1/2))+ Isignum(y)ln(((1)/(2)((x + 1)^(2)+ (y)^(2))^(1/2)+(1)/(2)((x - 1)^(2)+ (y)^(2))^(1/2))+(((1)/(2)((x + 1)^(2)+ (y)^(2))^(1/2)+(1)/(2)*((x - 1)^(2)+ (y)^(2))^(1/2))^(2)- 1)^(1/2))	ArcSin[x + yI] == ArcSin[Divide[1,2]((x + 1)^(2)+ (y)^(2))^(1/2)-Divide[1,2]((x - 1)^(2)+ (y)^(2))^(1/2)]+ ISign[y]Log[(Divide[1,2]((x + 1)^(2)+ (y)^(2))^(1/2)+Divide[1,2]((x - 1)^(2)+ (y)^(2))^(1/2))+((Divide[1,2]((x + 1)^(2)+ (y)^(2))^(1/2)+Divide[1,2]*((x - 1)^(2)+ (y)^(2))^(1/2))^(2)- 1)^(1/2)]	Failure	Failure	Successful [Tested: 18]	Successful [Tested: 18]
4.23.E35	${\displaystyle{\displaystyle\operatorname{arccos}z=\operatorname{arccos}\beta-% \mathrm{i}\operatorname{sign}\left(y\right)\ln\left(\alpha+(\alpha^{2}-1)^{1/2% }\right)}}$ `\acos@@{z} = \acos@@{\beta}-\iunit\sign@{y}\ln@{\alpha+(\alpha^{2}-1)^{1/2}}`		arccos(x + yI) = arccos((1)/(2)((x + 1)^(2)+ (y)^(2))^(1/2)-(1)/(2)((x - 1)^(2)+ (y)^(2))^(1/2))- Isignum(y)ln(((1)/(2)((x + 1)^(2)+ (y)^(2))^(1/2)+(1)/(2)((x - 1)^(2)+ (y)^(2))^(1/2))+(((1)/(2)((x + 1)^(2)+ (y)^(2))^(1/2)+(1)/(2)*((x - 1)^(2)+ (y)^(2))^(1/2))^(2)- 1)^(1/2))	ArcCos[x + yI] == ArcCos[Divide[1,2]((x + 1)^(2)+ (y)^(2))^(1/2)-Divide[1,2]((x - 1)^(2)+ (y)^(2))^(1/2)]- ISign[y]Log[(Divide[1,2]((x + 1)^(2)+ (y)^(2))^(1/2)+Divide[1,2]((x - 1)^(2)+ (y)^(2))^(1/2))+((Divide[1,2]((x + 1)^(2)+ (y)^(2))^(1/2)+Divide[1,2]*((x - 1)^(2)+ (y)^(2))^(1/2))^(2)- 1)^(1/2)]	Failure	Failure	Successful [Tested: 18]	Successful [Tested: 18]
4.23.E36	${\displaystyle{\displaystyle\operatorname{arctan}z=\tfrac{1}{2}\operatorname{% arctan}\left(\frac{2x}{1-x^{2}-y^{2}}\right)+\tfrac{1}{4}i\ln\left(\frac{x^{2}% +(y+1)^{2}}{x^{2}+(y-1)^{2}}\right)}}$ `\atan@@{z} = \tfrac{1}{2}\atan@{\frac{2x}{1-x^{2}-y^{2}}}+\tfrac{1}{4}i\ln@{\frac{x^{2}+(y+1)^{2}}{x^{2}+(y-1)^{2}}}`		arctan(x + yI) = (1)/(2)arctan((2x)/(1 - (x)^(2)- (y)^(2)))+(1)/(4)I*ln(((x)^(2)+(y + 1)^(2))/((x)^(2)+(y - 1)^(2)))	ArcTan[x + yI] == Divide[1,2]ArcTan[Divide[2x,1 - (x)^(2)- (y)^(2)]]+Divide[1,4]I*Log[Divide[(x)^(2)+(y + 1)^(2),(x)^(2)+(y - 1)^(2)]]	Failure	Failure	Failed [16 / 18] Result: 1.570796327-.1e-9I Test Values: {x = 1.5, y = -1.5} Result: 1.570796327-.1e-9I Test Values: {x = 1.5, y = 1.5} Result: 1.570796327+0.I Test Values: {x = 1.5, y = -.5} Result: 1.570796327+0.I Test Values: {x = 1.5, y = .5} ... skip entries to safe data	Failed [16 / 18] Result: Complex[1.5707963267948968, 1.1102230246251565^-16] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[1.5707963267948968, -1.6653345369377348^-16] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
4.23.E39	${\displaystyle{\displaystyle\operatorname{gd}\left(x\right)=\int_{0}^{x}% \operatorname{sech}t\mathrm{d}t}}$ `\Gudermannian@{x} = \int_{0}^{x}\sech@@{t}\diff{t}`	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	arctan(sinh(x)) = int(sech(t), t = 0..x)	Gudermannian[x] == Integrate[Sech[t], {t, 0, x}, GenerateConditions->None]	Successful	Aborted	-	Successful [Tested: 3]
4.23.E40	${\displaystyle{\displaystyle\operatorname{gd}\left(x\right)=2\operatorname{% arctan}\left(e^{x}\right)-\tfrac{1}{2}\pi\\ }}$ `\Gudermannian@{x} = 2\atan@{e^{x}}-\tfrac{1}{2}\pi\\`	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	arctan(sinh(x)) = 2arctan(exp(x))-(1)/(2)Pi	Gudermannian[x] == 2ArcTan[Exp[x]]-Divide[1,2]Pi	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.23.E40	${\displaystyle{\displaystyle 2\operatorname{arctan}\left(e^{x}\right)-\tfrac{1% }{2}\pi\\ =\operatorname{arcsin}\left(\tanh x\right)}}$ `2\atan@{e^{x}}-\tfrac{1}{2}\pi\\ = \asin@{\tanh@@{x}}`	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	2arctan(exp(x))-(1)/(2)Pi = arcsin(tanh(x))	2ArcTan[Exp[x]]-Divide[1,2]Pi == ArcSin[Tanh[x]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.23.E40	${\displaystyle{\displaystyle\operatorname{arcsin}\left(\tanh x\right)=% \operatorname{arccsc}\left(\coth x\right)\\ }}$ `\asin@{\tanh@@{x}} = \acsc@{\coth@@{x}}\\`	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	arcsin(tanh(x)) = arccsc(coth(x))	ArcSin[Tanh[x]] == ArcCsc[Coth[x]]	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
4.23.E40	${\displaystyle{\displaystyle\operatorname{arccsc}\left(\coth x\right)\\ =\operatorname{arccos}\left(\operatorname{sech}x\right)}}$ `\acsc@{\coth@@{x}}\\ = \acos@{\sech@@{x}}`	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	arccsc(coth(x)) = arccos(sech(x))	ArcCsc[Coth[x]] == ArcCos[Sech[x]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.23.E40	${\displaystyle{\displaystyle\operatorname{arccos}\left(\operatorname{sech}x% \right)=\operatorname{arcsec}\left(\cosh x\right)\\ }}$ `\acos@{\sech@@{x}} = \asec@{\cosh@@{x}}\\`	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	arccos(sech(x)) = arcsec(cosh(x))	ArcCos[Sech[x]] == ArcSec[Cosh[x]]	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
4.23.E40	${\displaystyle{\displaystyle\operatorname{arcsec}\left(\cosh x\right)\\ =\operatorname{arctan}\left(\sinh x\right)}}$ `\asec@{\cosh@@{x}}\\ = \atan@{\sinh@@{x}}`	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	arcsec(cosh(x)) = arctan(sinh(x))	ArcSec[Cosh[x]] == ArcTan[Sinh[x]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.23.E40	${\displaystyle{\displaystyle\operatorname{arctan}\left(\sinh x\right)=% \operatorname{arccot}\left(\operatorname{csch}x\right)}}$ `\atan@{\sinh@@{x}} = \acot@{\csch@@{x}}`	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	arctan(sinh(x)) = arccot(csch(x))	ArcTan[Sinh[x]] == ArcCot[Csch[x]]	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
4.23.E41	${\displaystyle{\displaystyle{\operatorname{gd}^{-1}}\left(x\right)=\int_{0}^{x% }\sec t\mathrm{d}t}}$ `\aGudermannian@{x} = \int_{0}^{x}\sec@@{t}\diff{t}`	${\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}}$	arctanh(sin(x)) = int(sec(t), t = 0..x)	InverseGudermannian[x] == Integrate[Sec[t], {t, 0, x}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 2]	Successful [Tested: 2]
4.23.E42	${\displaystyle{\displaystyle{\operatorname{gd}^{-1}}\left(x\right)=\ln\tan% \left(\tfrac{1}{2}x+\tfrac{1}{4}\pi\right)}}$ `\aGudermannian@{x} = \ln@@{\tan@{\tfrac{1}{2}x+\tfrac{1}{4}\pi}}`	${\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}}$	arctanh(sin(x)) = ln(tan((1)/(2)x +(1)/(4)Pi))	InverseGudermannian[x] == Log[Tan[Divide[1,2]x +Divide[1,4]Pi]]	Failure	Successful	Successful [Tested: 2]	Successful [Tested: 2]
4.23.E42	${\displaystyle{\displaystyle\ln\tan\left(\tfrac{1}{2}x+\tfrac{1}{4}\pi\right)=% \ln\left(\sec x+\tan x\right)}}$ `\ln@@{\tan@{\tfrac{1}{2}x+\tfrac{1}{4}\pi}} = \ln@{\sec@@{x}+\tan@@{x}}`	${\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}}$	ln(tan((1)/(2)x +(1)/(4)Pi)) = ln(sec(x)+ tan(x))	Log[Tan[Divide[1,2]x +Divide[1,4]Pi]] == Log[Sec[x]+ Tan[x]]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 2]
4.23.E42	${\displaystyle{\displaystyle\ln\left(\sec x+\tan x\right)=\operatorname{% arcsinh}\left(\tan x\right)}}$ `\ln@{\sec@@{x}+\tan@@{x}} = \asinh@{\tan@@{x}}`	${\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}}$	ln(sec(x)+ tan(x)) = arcsinh(tan(x))	Log[Sec[x]+ Tan[x]] == ArcSinh[Tan[x]]	Failure	Failure	Successful [Tested: 2]	Failed [1 / 3] Result: Complex[3.046904887125347, 3.141592653589793] Test Values: {Rule[x, 2]}
4.23.E42	${\displaystyle{\displaystyle\operatorname{arcsinh}\left(\tan x\right)=% \operatorname{arccsch}\left(\cot x\right)}}$ `\asinh@{\tan@@{x}} = \acsch@{\cot@@{x}}`	${\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}}$	arcsinh(tan(x)) = arccsch(cot(x))	ArcSinh[Tan[x]] == ArcCsch[Cot[x]]	Failure	Successful	Successful [Tested: 2]	Successful [Tested: 2]
4.23.E42	${\displaystyle{\displaystyle\operatorname{arccsch}\left(\cot x\right)=% \operatorname{arccosh}\left(\sec x\right)}}$ `\acsch@{\cot@@{x}} = \acosh@{\sec@@{x}}`	${\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}}$	arccsch(cot(x)) = arccosh(sec(x))	ArcCsch[Cot[x]] == ArcCosh[Sec[x]]	Failure	Failure	Successful [Tested: 2]	Failed [1 / 3] Result: Complex[-3.046904887125347, -3.141592653589793] Test Values: {Rule[x, 2]}
4.23.E42	${\displaystyle{\displaystyle\operatorname{arccosh}\left(\sec x\right)=% \operatorname{arcsech}\left(\cos x\right)}}$ `\acosh@{\sec@@{x}} = \asech@{\cos@@{x}}`	${\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}}$	arccosh(sec(x)) = arcsech(cos(x))	ArcCosh[Sec[x]] == ArcSech[Cos[x]]	Failure	Successful	Successful [Tested: 2]	Successful [Tested: 2]
4.23.E42	${\displaystyle{\displaystyle\operatorname{arcsech}\left(\cos x\right)=% \operatorname{arctanh}\left(\sin x\right)}}$ `\asech@{\cos@@{x}} = \atanh@{\sin@@{x}}`	${\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}}$	arcsech(cos(x)) = arctanh(sin(x))	ArcSech[Cos[x]] == ArcTanh[Sin[x]]	Failure	Failure	Successful [Tested: 2]	Failed [1 / 3] Result: Complex[0.0, 3.141592653589793] Test Values: {Rule[x, 2]}
4.23.E42	${\displaystyle{\displaystyle\operatorname{arctanh}\left(\sin x\right)=% \operatorname{arccoth}\left(\csc x\right)}}$ `\atanh@{\sin@@{x}} = \acoth@{\csc@@{x}}`	${\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}}$	arctanh(sin(x)) = arccoth(csc(x))	ArcTanh[Sin[x]] == ArcCoth[Csc[x]]	Failure	Successful	Successful [Tested: 2]	Successful [Tested: 2]

Results of Elementary Functions I

Navigation menu

Search