4.24

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
4.24.E2	${\displaystyle{\displaystyle\operatorname{arccos}z=(2(1-z))^{1/2}\*\left(1+\sum_{n=1}^{\infty}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}(2n+1)n!}(1-z)^{n}\right)}}$ \acos@@{z} = (2(1-z))^{1/2}\*\left(1+\sum_{n=1}^{\infty}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}(2n+1)n!}(1-z)^{n}\right)	${\displaystyle{\displaystyle|1-z|\leq 2}}$	arccos(z) = (2*(1 - z))^(1/2)*(1 + sum((1 * 3 * 5*(2*n - 1))/((2)^(2*n)*(2*n + 1)*factorial(n))*(1 - z)^(n), n = 1..infinity))	ArcCos[z] == (2*(1 - z))^(1/2)*(1 + Sum[Divide[1 * 3 * 5*(2*n - 1),(2)^(2*n)*(2*n + 1)*(n)!]*(1 - z)^(n), {n, 1, Infinity}, GenerateConditions->None])	Failure	Failure	Failed [7 / 7] Result: .3065228369+.5552108774*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: -3.012742443+4.300365362*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: .3498215011-1.819822265*I Test Values: {z = 1/2-1/2*I*3^(1/2)} Result: -5.876013992-3.037981862*I Test Values: {z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [7 / 7] Result: Complex[0.3065228364484756, 0.5552108781095243] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-3.0127424460165777, 4.300365361528893] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.24.E6	${\displaystyle{\displaystyle x^{2}-y^{2}=-\tfrac{1}{2}}}$ x^{2}-y^{2} = -\tfrac{1}{2}		(x)^(2)- (y)^(2) = -(1)/(2)	(x)^(2)- (y)^(2) == -Divide[1,2]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.24.E7	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsin}z=(1-z^{2})^{-1/2}}}$ \deriv{}{z}\asin@@{z} = (1-z^{2})^{-1/2}		diff(arcsin(z), z) = (1 - (z)^(2))^(- 1/2)	D[ArcSin[z], z] == (1 - (z)^(2))^(- 1/2)	Successful	Successful	-	Successful [Tested: 7]
4.24.E8	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccos}z=-(1-z^{2})^{-1/2}}}$ \deriv{}{z}\acos@@{z} = -(1-z^{2})^{-1/2}		diff(arccos(z), z) = -(1 - (z)^(2))^(- 1/2)	D[ArcCos[z], z] == -(1 - (z)^(2))^(- 1/2)	Successful	Successful	-	Successful [Tested: 7]
4.24.E9	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arctan}z=\frac{1}{1+z^{2}}}}$ \deriv{}{z}\atan@@{z} = \frac{1}{1+z^{2}}		diff(arctan(z), z) = (1)/(1 + (z)^(2))	D[ArcTan[z], z] == Divide[1,1 + (z)^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.24.E10	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsc}z=-\frac{1}{z(z^{2}-1)^{1/2}}}}$ \deriv{}{z}\acsc@@{z} = -\frac{1}{z(z^{2}-1)^{1/2}}		diff(arccsc(z), z) = -(1)/(z*((z)^(2)- 1)^(1/2))	D[ArcCsc[z], z] == -Divide[1,z*((z)^(2)- 1)^(1/2)]	Failure	Failure	Failed [2 / 7] Result: 1.074569932-1.074569932*I Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2} Result: -.6696152420e-9+2.000000000*I Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}	Successful [Tested: 1]
4.24.E10	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsc}z=+\frac{1}{z(z^{2}-1)^{1/2}}}}$ \deriv{}{z}\acsc@@{z} = +\frac{1}{z(z^{2}-1)^{1/2}}		diff(arccsc(z), z) = +(1)/(z*((z)^(2)- 1)^(1/2))	D[ArcCsc[z], z] == +Divide[1,z*((z)^(2)- 1)^(1/2)]	Failure	Failure	Failed [5 / 7] Result: -.6696152420e-9+2.000000000*I Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2} Result: 1.074569932-1.074569932*I Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2} Result: -1.192569588 Test Values: {z = 1.5, z = 1/2} Result: 4.618802153*I Test Values: {z = .5, z = 1/2} ... skip entries to safe data	Failed [1 / 1] Result: Complex[0.0, 4.618802153517007] Test Values: {Rule[z, Rational[1, 2]]}
4.24.E11	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsec}z=+\frac{1}{z(z^{2}-1)^{1/2}}}}$ \deriv{}{z}\asec@@{z} = +\frac{1}{z(z^{2}-1)^{1/2}}		diff(arcsec(z), z) = +(1)/(z*((z)^(2)- 1)^(1/2))	D[ArcSec[z], z] == +Divide[1,z*((z)^(2)- 1)^(1/2)]	Failure	Failure	Failed [2 / 7] Result: -1.074569932+1.074569932*I Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2} Result: .6696152420e-9-2.000000000*I Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}	Successful [Tested: 1]
4.24.E11	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsec}z=-\frac{1}{z(z^{2}-1)^{1/2}}}}$ \deriv{}{z}\asec@@{z} = -\frac{1}{z(z^{2}-1)^{1/2}}		diff(arcsec(z), z) = -(1)/(z*((z)^(2)- 1)^(1/2))	D[ArcSec[z], z] == -Divide[1,z*((z)^(2)- 1)^(1/2)]	Failure	Failure	Failed [5 / 7] Result: .6696152420e-9-2.000000000*I Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2} Result: -1.074569932+1.074569932*I Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2} Result: 1.192569588 Test Values: {z = 1.5, z = 1/2} Result: -4.618802153*I Test Values: {z = .5, z = 1/2} ... skip entries to safe data	Failed [1 / 1] Result: Complex[0.0, -4.618802153517007] Test Values: {Rule[z, Rational[1, 2]]}
4.24.E12	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccot}z=-\frac{1}{1+z^{2}}}}$ \deriv{}{z}\acot@@{z} = -\frac{1}{1+z^{2}}		diff(arccot(z), z) = -(1)/(1 + (z)^(2))	D[ArcCot[z], z] == -Divide[1,1 + (z)^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.24.E13	${\displaystyle{\displaystyle\operatorname{Arcsin}u+\operatorname{Arcsin}v=\operatorname{Arcsin}\left(u(1-v^{2})^{1/2}+v(1-u^{2})^{1/2}\right)}}$ \Asin@@{u}+\Asin@@{v} = \Asin@{u(1-v^{2})^{1/2}+ v(1-u^{2})^{1/2}}		Error	ArcSin[u]+ ArcSin[v] == ArcSin[u*(1 - (v)^(2))^(1/2)+ v*(1 - (u)^(2))^(1/2)]	Missing Macro Error	Failure	-	Failed [34 / 100] Result: Complex[4.440892098500626*^-16, 2.633915793849633] 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: Complex[1.5707963267948966, -0.6078894033135972] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, 1.5]} ... skip entries to safe data
4.24.E13	${\displaystyle{\displaystyle\operatorname{Arcsin}u-\operatorname{Arcsin}v=\operatorname{Arcsin}\left(u(1-v^{2})^{1/2}-v(1-u^{2})^{1/2}\right)}}$ \Asin@@{u}-\Asin@@{v} = \Asin@{u(1-v^{2})^{1/2}- v(1-u^{2})^{1/2}}		Error	ArcSin[u]- ArcSin[v] == ArcSin[u*(1 - (v)^(2))^(1/2)- v*(1 - (u)^(2))^(1/2)]	Missing Macro Error	Failure	-	Failed [34 / 100] Result: Complex[4.440892098500626*^-16, 2.633915793849633] 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[1.5707963267948966, -0.6078894033135972] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -1.5]} ... skip entries to safe data
4.24.E14	${\displaystyle{\displaystyle\operatorname{Arccos}u+\operatorname{Arccos}v=\operatorname{Arccos}\left(uv-((1-u^{2})(1-v^{2}))^{1/2}\right)}}$ \Acos@@{u}+\Acos@@{v} = \Acos@{uv-((1-u^{2})(1-v^{2}))^{1/2}}		Error	ArcCos[u]+ ArcCos[v] == ArcCos[u*v -((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)]	Missing Macro Error	Failure	-	Failed [63 / 100] Result: Complex[1.5707963267948963, -3.2418051971632305] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -1.5]} Result: Complex[1.5707963267948966, -3.9508736907744497] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -2]} ... skip entries to safe data
4.24.E14	${\displaystyle{\displaystyle\operatorname{Arccos}u-\operatorname{Arccos}v=\operatorname{Arccos}\left(uv+((1-u^{2})(1-v^{2}))^{1/2}\right)}}$ \Acos@@{u}-\Acos@@{v} = \Acos@{uv+((1-u^{2})(1-v^{2}))^{1/2}}		Error	ArcCos[u]- ArcCos[v] == ArcCos[u*v +((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)]	Missing Macro Error	Failure	-	Failed [63 / 100] Result: Complex[-2.3202651922123767, 0.3459279941338042] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.8213274613774166, -2.979843787983438] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.24.E15	${\displaystyle{\displaystyle\operatorname{Arctan}u+\operatorname{Arctan}v=\operatorname{Arctan}\left(\frac{u+v}{1-uv}\right)}}$ \Atan@@{u}+\Atan@@{v} = \Atan@{\frac{u+ v}{1- uv}}		Error	ArcTan[u]+ ArcTan[v] == ArcTan[Divide[u + v,1 - u*v]]	Missing Macro Error	Failure	-	Successful [Tested: 1]
4.24.E15	${\displaystyle{\displaystyle\operatorname{Arctan}u-\operatorname{Arctan}v=\operatorname{Arctan}\left(\frac{u-v}{1+uv}\right)}}$ \Atan@@{u}-\Atan@@{v} = \Atan@{\frac{u- v}{1+ uv}}		Error	ArcTan[u]- ArcTan[v] == ArcTan[Divide[u - v,1 + u*v]]	Missing Macro Error	Failure	-	Successful [Tested: 1]
4.24.E16	${\displaystyle{\displaystyle\operatorname{Arcsin}u+\operatorname{Arccos}v=\operatorname{Arcsin}\left(uv+((1-u^{2})(1-v^{2}))^{1/2}\right)}}$ \Asin@@{u}+\Acos@@{v} = \Asin@{uv+((1-u^{2})(1-v^{2}))^{1/2}}		Error	ArcSin[u]+ ArcCos[v] == ArcSin[u*v +((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)]	Missing Macro Error	Failure	-	Failed [63 / 100] Result: Complex[2.3202651922123767, -0.3459279941338042] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.8213274613774166, 2.979843787983438] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.24.E16	${\displaystyle{\displaystyle\operatorname{Arcsin}u-\operatorname{Arccos}v=\operatorname{Arcsin}\left(uv-((1-u^{2})(1-v^{2}))^{1/2}\right)}}$ \Asin@@{u}-\Acos@@{v} = \Asin@{uv-((1-u^{2})(1-v^{2}))^{1/2}}		Error	ArcSin[u]- ArcCos[v] == ArcSin[u*v -((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)]	Missing Macro Error	Failure	-	Failed [63 / 100] Result: Complex[-1.5707963267948963, 3.2418051971632305] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -1.5]} Result: Complex[-1.5707963267948966, 3.9508736907744497] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -2]} ... skip entries to safe data
4.24.E16	${\displaystyle{\displaystyle\operatorname{Arcsin}\left(uv+((1-u^{2})(1-v^{2}))^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/2}-u(1-v^{2})^{1/2}\right)}}$ \Asin@{uv+((1-u^{2})(1-v^{2}))^{1/2}} = \Acos@{v(1-u^{2})^{1/2}- u(1-v^{2})^{1/2}}		Error	ArcSin[u*v +((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)] == ArcCos[v*(1 - (u)^(2))^(1/2)- u*(1 - (v)^(2))^(1/2)]	Missing Macro Error	Failure	-	Failed [72 / 100] Result: Complex[-2.3202651922123763, 0.3459279941338042] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.8213274613774164, -2.979843787983438] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.24.E16	${\displaystyle{\displaystyle\operatorname{Arcsin}\left(uv-((1-u^{2})(1-v^{2}))^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/2}+u(1-v^{2})^{1/2}\right)}}$ \Asin@{uv-((1-u^{2})(1-v^{2}))^{1/2}} = \Acos@{v(1-u^{2})^{1/2}+ u(1-v^{2})^{1/2}}		Error	ArcSin[u*v -((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)] == ArcCos[v*(1 - (u)^(2))^(1/2)+ u*(1 - (v)^(2))^(1/2)]	Missing Macro Error	Failure	-	Failed [91 / 100] Result: Complex[-2.3202651922123767, 2.979843787983438] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.8213274613774161, -0.3459279941338048] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.24.E17	${\displaystyle{\displaystyle\operatorname{Arctan}u+\operatorname{Arccot}v=\operatorname{Arctan}\left(\frac{uv+1}{v-u}\right)}}$ \Atan@@{u}+\Acot@@{v} = \Atan@{\frac{uv+ 1}{v- u}}		Error	ArcTan[u]+ ArcCot[v] == ArcTan[Divide[u*v + 1,v - u]]	Missing Macro Error	Failure	-	Failed [1 / 10] Result: Indeterminate Test Values: {Rule[u, Rational[1, 2]], Rule[v, 0.5]}
4.24.E17	${\displaystyle{\displaystyle\operatorname{Arctan}u-\operatorname{Arccot}v=\operatorname{Arctan}\left(\frac{uv-1}{v+u}\right)}}$ \Atan@@{u}-\Acot@@{v} = \Atan@{\frac{uv- 1}{v+ u}}		Error	ArcTan[u]- ArcCot[v] == ArcTan[Divide[u*v - 1,v + u]]	Missing Macro Error	Failure	-	Failed [1 / 10] Result: Indeterminate Test Values: {Rule[u, Rational[1, 2]], Rule[v, -0.5]}
4.24.E17	${\displaystyle{\displaystyle\operatorname{Arctan}\left(\frac{uv+1}{v-u}\right)=\operatorname{Arccot}\left(\frac{v-u}{uv+1}\right)}}$ \Atan@{\frac{uv+ 1}{v- u}} = \Acot@{\frac{v- u}{uv+ 1}}		Error	ArcTan[Divide[u*v + 1,v - u]] == ArcCot[Divide[v - u,u*v + 1]]	Missing Macro Error	Failure	-	Failed [1 / 10] Result: Indeterminate Test Values: {Rule[u, Rational[1, 2]], Rule[v, 0.5]}
4.24.E17	${\displaystyle{\displaystyle\operatorname{Arctan}\left(\frac{uv-1}{v+u}\right)=\operatorname{Arccot}\left(\frac{v+u}{uv-1}\right)}}$ \Atan@{\frac{uv- 1}{v+ u}} = \Acot@{\frac{v+ u}{uv- 1}}		Error	ArcTan[Divide[u*v - 1,v + u]] == ArcCot[Divide[v + u,u*v - 1]]	Missing Macro Error	Failure	-	Failed [1 / 10] Result: Indeterminate Test Values: {Rule[u, Rational[1, 2]], Rule[v, -0.5]}