Elliptic Integrals - 19.10 Relations to Other Functions

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
19.10#Ex1	${\displaystyle{\displaystyle\ln\left(x/y\right)=(x-y)R_{C}\left(\tfrac{1}{4}(x+y)^{2},xy\right)}}$ \ln@{x/y} = (x-y)\CarlsonellintRC@{\tfrac{1}{4}(x+y)^{2}}{xy}		Error	Log[x/y] == (x - y)*1/Sqrt[x*y]*Hypergeometric2F1[1/2,1/2,3/2,1-(Divide[1,4]*(x + y)^(2))/(x*y)]	Missing Macro Error	Failure	-	Failed [12 / 18] Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.10#Ex2	${\displaystyle{\displaystyle\operatorname{arctan}\left(x/y\right)=xR_{C}\left(y^{2},y^{2}+x^{2}\right)}}$ \atan@{x/y} = x\CarlsonellintRC@{y^{2}}{y^{2}+x^{2}}		Error	ArcTan[x/y] == x*1/Sqrt[(y)^(2)+ (x)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((y)^(2))/((y)^(2)+ (x)^(2))]	Missing Macro Error	Failure	-	Failed [9 / 18] Result: -1.5707963267948966 Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: -2.498091544796509 Test Values: {Rule[x, 1.5], Rule[y, -0.5]} ... skip entries to safe data
19.10#Ex3	${\displaystyle{\displaystyle\operatorname{arctanh}\left(x/y\right)=xR_{C}\left(y^{2},y^{2}-x^{2}\right)}}$ \atanh@{x/y} = x\CarlsonellintRC@{y^{2}}{y^{2}-x^{2}}		Error	ArcTanh[x/y] == x*1/Sqrt[(y)^(2)- (x)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((y)^(2))/((y)^(2)- (x)^(2))]	Missing Macro Error	Failure	-	Failed [15 / 18] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.10#Ex4	${\displaystyle{\displaystyle\operatorname{arcsin}\left(x/y\right)=xR_{C}\left(y^{2}-x^{2},y^{2}\right)}}$ \asin@{x/y} = x\CarlsonellintRC@{y^{2}-x^{2}}{y^{2}}		Error	ArcSin[x/y] == x*1/Sqrt[(y)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((y)^(2)- (x)^(2))/((y)^(2))]	Missing Macro Error	Failure	-	Failed [9 / 18] Result: -3.141592653589793 Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[-3.141592653589793, 3.525494348078172] Test Values: {Rule[x, 1.5], Rule[y, -0.5]} ... skip entries to safe data
19.10#Ex5	${\displaystyle{\displaystyle\operatorname{arcsinh}\left(x/y\right)=xR_{C}\left(y^{2}+x^{2},y^{2}\right)}}$ \asinh@{x/y} = x\CarlsonellintRC@{y^{2}+x^{2}}{y^{2}}		Error	ArcSinh[x/y] == x*1/Sqrt[(y)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((y)^(2)+ (x)^(2))/((y)^(2))]	Missing Macro Error	Failure	-	Failed [9 / 18] Result: Complex[-1.7627471740390859, 0.0] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[-3.6368929184641337, 0.0] Test Values: {Rule[x, 1.5], Rule[y, -0.5]} ... skip entries to safe data
19.10#Ex6	${\displaystyle{\displaystyle\operatorname{arccos}\left(x/y\right)=(y^{2}-x^{2})^{1/2}R_{C}\left(x^{2},y^{2}\right)}}$ \acos@{x/y} = (y^{2}-x^{2})^{1/2}\CarlsonellintRC@{x^{2}}{y^{2}}		Error	ArcCos[x/y] == ((y)^(2)- (x)^(2))^(1/2)* 1/Sqrt[(y)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((x)^(2))/((y)^(2))]	Missing Macro Error	Failure	-	Failed [12 / 18] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.10#Ex7	${\displaystyle{\displaystyle\operatorname{arccosh}\left(x/y\right)=(x^{2}-y^{2})^{1/2}R_{C}\left(x^{2},y^{2}\right)}}$ \acosh@{x/y} = (x^{2}-y^{2})^{1/2}\CarlsonellintRC@{x^{2}}{y^{2}}		Error	ArcCosh[x/y] == ((x)^(2)- (y)^(2))^(1/2)* 1/Sqrt[(y)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((x)^(2))/((y)^(2))]	Missing Macro Error	Failure	-	Failed [12 / 18] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.10.E2	${\displaystyle{\displaystyle(\sinh\phi)R_{C}\left(1,{\cosh^{2}}\phi\right)=\operatorname{gd}\left(\phi\right)}}$ (\sinh@@{\phi})\CarlsonellintRC@{1}{\cosh^{2}@@{\phi}} = \Gudermannian@{\phi}	${\displaystyle{\displaystyle-\infty<(\phi),(\phi)<\infty}}$	Error	(Sinh[\[Phi]])*1/Sqrt[(Cosh[\[Phi]])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(1)/((Cosh[\[Phi]])^(2))] == Gudermannian[\[Phi]]	Missing Macro Error	Failure	-	Successful [Tested: 6]