4.38

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
4.38.E4	${\displaystyle{\displaystyle\operatorname{arccosh}z=(2(z-1))^{1/2}\*{\left(1+\sum_{n=1}^{\infty}(-1)^{n}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}n!(2n+1)}(z-1)^{n}\right)}}}$ \acosh@@{z} = (2(z-1))^{1/2}\*{\left(1+\sum_{n=1}^{\infty}(-1)^{n}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}n!(2n+1)}(z-1)^{n}\right)}	${\displaystyle{\displaystyle\Re z>0,|z-1|\leq 2}}$	arccosh(z) = (2*(z - 1))^(1/2)*(1 + sum((- 1)^(n)*(1 * 3 * 5*(2*n - 1))/((2)^(2*n)* factorial(n)*(2*n + 1))*(z - 1)^(n), n = 1..infinity))	ArcCosh[z] == (2*(z - 1))^(1/2)*(1 + Sum[(- 1)^(n)*Divide[1 * 3 * 5*(2*n - 1),(2)^(2*n)* (n)!*(2*n + 1)]*(z - 1)^(n), {n, 1, Infinity}, GenerateConditions->None])	Failure	Failure	Failed [5 / 5] Result: -.5552108774+.3065228369*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: -1.819822265-.3498215011*I Test Values: {z = 1/2-1/2*I*3^(1/2)} Result: .5204832489 Test Values: {z = 1.5} Result: -.651724541*I Test Values: {z = .5} ... skip entries to safe data	Failed [5 / 5] Result: Complex[-0.5552108781095244, 0.30652283644847583] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.8198222655846492, -0.34982149976378074] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.38.E8	${\displaystyle{\displaystyle x^{2}-y^{2}=\tfrac{1}{2}}}$ 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	-	-
4.38.E9	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsinh}z=(1+z^{2})^{-1/2}}}$ \deriv{}{z}\asinh@@{z} = (1+z^{2})^{-1/2}		diff(arcsinh(z), z) = (1 + (z)^(2))^(- 1/2)	D[ArcSinh[z], z] == (1 + (z)^(2))^(- 1/2)	Successful	Successful	-	Successful [Tested: 7]
4.38.E10	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccosh}z=+(z^{2}-1)^{-1/2}}}$ \deriv{}{z}\acosh@@{z} = +(z^{2}-1)^{-1/2}		diff(arccosh(z), z) = +((z)^(2)- 1)^(- 1/2)	D[ArcCosh[z], z] == +((z)^(2)- 1)^(- 1/2)	Failure	Failure	Failed [2 / 7] Result: -.3933198932-1.467889825*I Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2} Result: -1.000000000+1.732050808*I Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}	Successful [Tested: 1]
4.38.E10	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccosh}z=-(z^{2}-1)^{-1/2}}}$ \deriv{}{z}\acosh@@{z} = -(z^{2}-1)^{-1/2}		diff(arccosh(z), z) = -((z)^(2)- 1)^(- 1/2)	D[ArcCosh[z], z] == -((z)^(2)- 1)^(- 1/2)	Failure	Failure	Failed [5 / 7] Result: 1.000000000-1.732050808*I Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2} Result: .3933198932+1.467889825*I Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2} Result: 1.788854382 Test Values: {z = 1.5, z = 1/2} Result: -2.309401076*I Test Values: {z = .5, z = 1/2} ... skip entries to safe data	Failed [1 / 1] Result: Complex[0.0, -2.3094010767585034] Test Values: {Rule[z, Rational[1, 2]]}
4.38.E11	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arctanh}z=\frac{1}{1-z^{2}}}}$ \deriv{}{z}\atanh@@{z} = \frac{1}{1-z^{2}}		diff(arctanh(z), z) = (1)/(1 - (z)^(2))	D[ArcTanh[z], z] == Divide[1,1 - (z)^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.38.E12	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsch}z=-\frac{1}{z(1+z^{2})^{1/2}}}}$ \deriv{}{z}\acsch@@{z} = -\frac{1}{z(1+z^{2})^{1/2}}		diff(arccsch(z), z) = -(1)/(z*(1 + (z)^(2))^(1/2))	D[ArcCsch[z], z] == -Divide[1,z*(1 + (z)^(2))^(1/2)]	Failure	Failure	Failed [2 / 7] Result: .6696152420e-9-2.000000000*I Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2} Result: -1.074569932+1.074569932*I Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}	Successful [Tested: 1]
4.38.E12	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsch}z=+\frac{1}{z(1+z^{2})^{1/2}}}}$ \deriv{}{z}\acsch@@{z} = +\frac{1}{z(1+z^{2})^{1/2}}		diff(arccsch(z), z) = +(1)/(z*(1 + (z)^(2))^(1/2))	D[ArcCsch[z], z] == +Divide[1,z*(1 + (z)^(2))^(1/2)]	Failure	Failure	Failed [5 / 7] Result: -1.074569932+1.074569932*I Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2} Result: .6696152420e-9-2.000000000*I Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2} Result: -.7396002616 Test Values: {z = 1.5, z = 1/2} Result: -3.577708764 Test Values: {z = .5, z = 1/2} ... skip entries to safe data	Failed [1 / 1] Result: -3.5777087639996634 Test Values: {Rule[z, Rational[1, 2]]}
4.38.E13	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsech}z=-\frac{1}{z(1-z^{2})^{1/2}}}}$ \deriv{}{z}\asech@@{z} = -\frac{1}{z(1-z^{2})^{1/2}}		diff(arcsech(z), z) = -(1)/(z*(1 - (z)^(2))^(1/2))	D[ArcSech[z], z] == -Divide[1,z*(1 - (z)^(2))^(1/2)]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.38.E14	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccoth}z=\frac{1}{1-z^{2}}}}$ \deriv{}{z}\acoth@@{z} = \frac{1}{1-z^{2}}		diff(arccoth(z), z) = (1)/(1 - (z)^(2))	D[ArcCoth[z], z] == Divide[1,1 - (z)^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.38.E15	${\displaystyle{\displaystyle\operatorname{Arcsinh}u+\operatorname{Arcsinh}v=\operatorname{Arcsinh}\left(u(1+v^{2})^{1/2}+v(1+u^{2})^{1/2}\right)}}$ \Asinh@@{u}+\Asinh@@{v} = \Asinh@{u(1+v^{2})^{1/2}+ v(1+u^{2})^{1/2}}		Error	ArcSinh[u]+ ArcSinh[v] == ArcSinh[u*(1 + (v)^(2))^(1/2)+ v*(1 + (u)^(2))^(1/2)]	Missing Macro Error	Failure	-	Failed [1 / 100] Result: Complex[-2.633915793849633, 4.440892098500626*^-16] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
4.38.E15	${\displaystyle{\displaystyle\operatorname{Arcsinh}u-\operatorname{Arcsinh}v=\operatorname{Arcsinh}\left(u(1+v^{2})^{1/2}-v(1+u^{2})^{1/2}\right)}}$ \Asinh@@{u}-\Asinh@@{v} = \Asinh@{u(1+v^{2})^{1/2}- v(1+u^{2})^{1/2}}		Error	ArcSinh[u]- ArcSinh[v] == ArcSinh[u*(1 + (v)^(2))^(1/2)- v*(1 + (u)^(2))^(1/2)]	Missing Macro Error	Failure	-	Successful [Tested: 100]
4.38.E16	${\displaystyle{\displaystyle\operatorname{Arccosh}u+\operatorname{Arccosh}v=\operatorname{Arccosh}\left(uv+((u^{2}-1)(v^{2}-1))^{1/2}\right)}}$ \Acosh@@{u}+\Acosh@@{v} = \Acosh@{uv+((u^{2}-1)(v^{2}-1))^{1/2}}		Error	ArcCosh[u]+ ArcCosh[v] == ArcCosh[u*v +(((u)^(2)- 1)*((v)^(2)- 1))^(1/2)]	Missing Macro Error	Failure	-	Failed [60 / 100] Result: Complex[1.3169578969248166, 1.5707963267948966] 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[1.316957896924817, 1.5707963267948966] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.38.E16	${\displaystyle{\displaystyle\operatorname{Arccosh}u-\operatorname{Arccosh}v=\operatorname{Arccosh}\left(uv-((u^{2}-1)(v^{2}-1))^{1/2}\right)}}$ \Acosh@@{u}-\Acosh@@{v} = \Acosh@{uv-((u^{2}-1)(v^{2}-1))^{1/2}}		Error	ArcCosh[u]- ArcCosh[v] == ArcCosh[u*v -(((u)^(2)- 1)*((v)^(2)- 1))^(1/2)]	Missing Macro Error	Failure	-	Failed [80 / 100] Result: Complex[-1.3169578969248166, -1.5707963267948966] 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[-1.6628858910586213, -3.8910615190072733] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.38.E17	${\displaystyle{\displaystyle\operatorname{Arctanh}u+\operatorname{Arctanh}v=\operatorname{Arctanh}\left(\frac{u+v}{1+uv}\right)}}$ \Atanh@@{u}+\Atanh@@{v} = \Atanh@{\frac{u+ v}{1+ uv}}		Error	ArcTanh[u]+ ArcTanh[v] == ArcTanh[Divide[u + v,1 + u*v]]	Missing Macro Error	Failure	-	Successful [Tested: 1]
4.38.E17	${\displaystyle{\displaystyle\operatorname{Arctanh}u-\operatorname{Arctanh}v=\operatorname{Arctanh}\left(\frac{u-v}{1-uv}\right)}}$ \Atanh@@{u}-\Atanh@@{v} = \Atanh@{\frac{u- v}{1- uv}}		Error	ArcTanh[u]- ArcTanh[v] == ArcTanh[Divide[u - v,1 - u*v]]	Missing Macro Error	Failure	-	Successful [Tested: 1]
4.38.E18	${\displaystyle{\displaystyle\operatorname{Arcsinh}u+\operatorname{Arccosh}v=\operatorname{Arcsinh}\left(uv+((1+u^{2})(v^{2}-1))^{1/2}\right)}}$ \Asinh@@{u}+\Acosh@@{v} = \Asinh@{uv+((1+u^{2})(v^{2}-1))^{1/2}}		Error	ArcSinh[u]+ ArcCosh[v] == ArcSinh[u*v +((1 + (u)^(2))*((v)^(2)- 1))^(1/2)]	Missing Macro Error	Failure	-	Failed [53 / 100] Result: Complex[1.66288587615746, 3.891061504106112] 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[1.6628858910586204, -2.3921237881723125] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
4.38.E18	${\displaystyle{\displaystyle\operatorname{Arcsinh}u-\operatorname{Arccosh}v=\operatorname{Arcsinh}\left(uv-((1+u^{2})(v^{2}-1))^{1/2}\right)}}$ \Asinh@@{u}-\Acosh@@{v} = \Asinh@{uv-((1+u^{2})(v^{2}-1))^{1/2}}		Error	ArcSinh[u]- ArcCosh[v] == ArcSinh[u*v -((1 + (u)^(2))*((v)^(2)- 1))^(1/2)]	Missing Macro Error	Failure	-	Failed [53 / 100] Result: Complex[1.6628858910586208, -2.392123788172313] 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[1.6628858910586208, 3.8910615190072733] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
4.38.E18	${\displaystyle{\displaystyle\operatorname{Arcsinh}\left(uv+((1+u^{2})(v^{2}-1))^{1/2}\right)=\operatorname{Arccosh}\left(v(1+u^{2})^{1/2}+u(v^{2}-1)^{1/2}\right)}}$ \Asinh@{uv+((1+u^{2})(v^{2}-1))^{1/2}} = \Acosh@{v(1+u^{2})^{1/2}+ u(v^{2}-1)^{1/2}}		Error	ArcSinh[u*v +((1 + (u)^(2))*((v)^(2)- 1))^(1/2)] == ArcCosh[v*(1 + (u)^(2))^(1/2)+ u*((v)^(2)- 1)^(1/2)]	Missing Macro Error	Failure	-	Failed [65 / 100] Result: Complex[1.4901161193847656*^-8, -3.141592638688632] 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[-0.34592799413380415, -2.320265192212377] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
4.38.E18	${\displaystyle{\displaystyle\operatorname{Arcsinh}\left(uv-((1+u^{2})(v^{2}-1))^{1/2}\right)=\operatorname{Arccosh}\left(v(1+u^{2})^{1/2}-u(v^{2}-1)^{1/2}\right)}}$ \Asinh@{uv-((1+u^{2})(v^{2}-1))^{1/2}} = \Acosh@{v(1+u^{2})^{1/2}- u(v^{2}-1)^{1/2}}		Error	ArcSinh[u*v -((1 + (u)^(2))*((v)^(2)- 1))^(1/2)] == ArcCosh[v*(1 + (u)^(2))^(1/2)- u*((v)^(2)- 1)^(1/2)]	Missing Macro Error	Failure	-	Failed [86 / 100] Result: Complex[-3.325771782117242, -1.4989377308349603] 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[-2.9798437879834374, 0.8213274613774169] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
4.38.E19	${\displaystyle{\displaystyle\operatorname{Arctanh}u+\operatorname{Arccoth}v=\operatorname{Arctanh}\left(\frac{uv+1}{v+u}\right)}}$ \Atanh@@{u}+\Acoth@@{v} = \Atanh@{\frac{uv+ 1}{v+ u}}		Error	ArcTanh[u]+ ArcCoth[v] == ArcTanh[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]}
4.38.E19	${\displaystyle{\displaystyle\operatorname{Arctanh}u-\operatorname{Arccoth}v=\operatorname{Arctanh}\left(\frac{uv-1}{v-u}\right)}}$ \Atanh@@{u}-\Acoth@@{v} = \Atanh@{\frac{uv- 1}{v- u}}		Error	ArcTanh[u]- ArcCoth[v] == ArcTanh[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]}
4.38.E19	${\displaystyle{\displaystyle\operatorname{Arctanh}\left(\frac{uv+1}{v+u}\right)=\operatorname{Arccoth}\left(\frac{v+u}{uv+1}\right)}}$ \Atanh@{\frac{uv+ 1}{v+ u}} = \Acoth@{\frac{v+ u}{uv+ 1}}		Error	ArcTanh[Divide[u*v + 1,v + u]] == ArcCoth[Divide[v + u,u*v + 1]]	Missing Macro Error	Successful	-	Failed [8 / 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: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.38.E19	${\displaystyle{\displaystyle\operatorname{Arctanh}\left(\frac{uv-1}{v-u}\right)=\operatorname{Arccoth}\left(\frac{v-u}{uv-1}\right)}}$ \Atanh@{\frac{uv- 1}{v- u}} = \Acoth@{\frac{v- u}{uv- 1}}		Error	ArcTanh[Divide[u*v - 1,v - u]] == ArcCoth[Divide[v - u,u*v - 1]]	Missing Macro Error	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