19.5

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
19.5.E1	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}}}$ \compellintKk@{k} = \frac{\pi}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}k^{2m}		EllipticK(k) = (Pi)/(2)*sum((pochhammer((1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity)	EllipticK[(k)^2] == Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Error	Successful [Tested: 3]
19.5.E1	${\displaystyle{\displaystyle\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2}F_{1}}\left(\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)}}$ \frac{\pi}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}k^{2m} = \frac{\pi}{2}\genhyperF{2}{1}@{\tfrac{1}{2},\tfrac{1}{2}}{1}{k^{2}}		(Pi)/(2)*sum((pochhammer((1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity) = (Pi)/(2)*hypergeom([(1)/(2),(1)/(2)], [1], (k)^(2))	Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None] == Divide[Pi,2]*HypergeometricPFQ[{Divide[1,2],Divide[1,2]}, {1}, (k)^(2)]	Failure	Successful	Failed [3 / 3] Result: Float(infinity)+Float(infinity)*I Test Values: {k = 1} Result: Float(infinity)+1.078257824*I Test Values: {k = 2} ... skip entries to safe data	Successful [Tested: 3]
19.5.E2	${\displaystyle{\displaystyle E\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}}}$ \compellintEk@{k} = \frac{\pi}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{-\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}k^{2m}		EllipticE(k) = (Pi)/(2)*sum((pochhammer(-(1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity)	EllipticE[(k)^2] == Divide[Pi,2]*Sum[Divide[Pochhammer[-Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Failed [2 / 3] Result: Float(infinity)+1.343854231*I Test Values: {k = 2} Result: Float(infinity)+2.498348128*I Test Values: {k = 3}	Successful [Tested: 3]
19.5.E2	${\displaystyle{\displaystyle\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2}F_{1}}\left(-\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)}}$ \frac{\pi}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{-\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}k^{2m} = \frac{\pi}{2}\genhyperF{2}{1}@{-\tfrac{1}{2},\tfrac{1}{2}}{1}{k^{2}}		(Pi)/(2)*sum((pochhammer(-(1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity) = (Pi)/(2)*hypergeom([-(1)/(2),(1)/(2)], [1], (k)^(2))	Divide[Pi,2]*Sum[Divide[Pochhammer[-Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None] == Divide[Pi,2]*HypergeometricPFQ[{-Divide[1,2],Divide[1,2]}, {1}, (k)^(2)]	Failure	Successful	Failed [2 / 3] Result: Float(-infinity)-1.343854232*I Test Values: {k = 2} Result: Float(-infinity)-2.498348127*I Test Values: {k = 3}	Successful [Tested: 3]
19.5.E3	${\displaystyle{\displaystyle D\left(k\right)=\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{3}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;m!}k^{2m}}}$ \compellintDk@{k} = \frac{\pi}{4}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{3}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{(m+1)!\;m!}k^{2m}		(EllipticK(k) - EllipticE(k))/(k)^2 = (Pi)/(4)*sum((pochhammer((3)/(2), m)*pochhammer((1)/(2), m))/(factorial(m + 1)*factorial(m))*(k)^(2*m), m = 0..infinity)	Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4] == Divide[Pi,4]*Sum[Divide[Pochhammer[Divide[3,2], m]*Pochhammer[Divide[1,2], m],(m + 1)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-0.08185805455243848, 0.4541460103381727] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.5.E3	${\displaystyle{\displaystyle\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{3}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;m!}k^{2m}=\frac{\pi}{4}{{}_{2}F_{1}}\left(\tfrac{3}{2},\tfrac{1}{2};2;k^{2}\right)}}$ \frac{\pi}{4}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{3}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{(m+1)!\;m!}k^{2m} = \frac{\pi}{4}\genhyperF{2}{1}@{\tfrac{3}{2},\tfrac{1}{2}}{2}{k^{2}}		(Pi)/(4)*sum((pochhammer((3)/(2), m)*pochhammer((1)/(2), m))/(factorial(m + 1)*factorial(m))*(k)^(2*m), m = 0..infinity) = (Pi)/(4)*hypergeom([(3)/(2),(1)/(2)], [2], (k)^(2))	Divide[Pi,4]*Sum[Divide[Pochhammer[Divide[3,2], m]*Pochhammer[Divide[1,2], m],(m + 1)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None] == Divide[Pi,4]*HypergeometricPFQ[{Divide[3,2],Divide[1,2]}, {2}, (k)^(2)]	Failure	Successful	Failed [3 / 3] Result: Float(infinity)+Float(infinity)*I Test Values: {k = 1} Result: Float(infinity)+.6055280139*I Test Values: {k = 2} ... skip entries to safe data	Failed [1 / 3] Result: Indeterminate Test Values: {Rule[k, 1]}
19.5.E4	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{n}}}{n!}\sum_{m=0}^{n}\frac{{\left(\tfrac{1}{2}\right)_{m}}}{m!}k^{2m}\alpha^{2n-2m}}}$ \compellintPik@{\alpha^{2}}{k} = \frac{\pi}{2}\sum_{n=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{n}}{n!}\sum_{m=0}^{n}\frac{\Pochhammersym{\tfrac{1}{2}}{m}}{m!}k^{2m}\alpha^{2n-2m}		EllipticPi((alpha)^(2), k) = (Pi)/(2)*sum((pochhammer((1)/(2), n))/(factorial(n))*sum((pochhammer((1)/(2), m))/(factorial(m))*(k)^(2*m)* (alpha)^(2*n - 2*m), m = 0..n), n = 0..infinity)	EllipticPi[\[Alpha]^(2), (k)^2] == Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], n],(n)!]*Sum[Divide[Pochhammer[Divide[1,2], m],(m)!]*(k)^(2*m)* \[Alpha]^(2*n - 2*m), {m, 0, n}, GenerateConditions->None], {n, 0, Infinity}, GenerateConditions->None]	Aborted	Failure	Error	Skipped - Because timed out
19.5.E4	${\displaystyle{\displaystyle\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{n}}}{n!}\sum_{m=0}^{n}\frac{{\left(\tfrac{1}{2}\right)_{m}}}{m!}k^{2m}\alpha^{2n-2m}=\frac{\pi}{2}{F_{1}}\left(\tfrac{1}{2};\tfrac{1}{2},1;1;k^{2},\alpha^{2}\right)}}$ \frac{\pi}{2}\sum_{n=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{n}}{n!}\sum_{m=0}^{n}\frac{\Pochhammersym{\tfrac{1}{2}}{m}}{m!}k^{2m}\alpha^{2n-2m} = \frac{\pi}{2}\AppellF{1}@{\tfrac{1}{2}}{\tfrac{1}{2}}{1}{1}{k^{2}}{\alpha^{2}}		Error	Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], n],(n)!]*Sum[Divide[Pochhammer[Divide[1,2], m],(m)!]*(k)^(2*m)* \[Alpha]^(2*n - 2*m), {m, 0, n}, GenerateConditions->None], {n, 0, Infinity}, GenerateConditions->None] == Divide[Pi,2]*AppellF[1, , Divide[1,2], Divide[1,2], 1, 1]*(k)^(2)*\[Alpha]^(2)	Missing Macro Error	Failure	Skip - symbolical successful subtest	Skipped - Because timed out
19.5.E5	${\displaystyle{\displaystyle q=\exp\left(-\pi{K^{\prime}}\left(k\right)/K\left(k\right)\right)}}$ q = \exp@{-\pi\ccompellintKk@{k}/\compellintKk@{k}}	${\displaystyle{\displaystyle r=\frac{1}{16}k^{2},0\leq k,k\leq 1}}$	(exp(- Pi*EllipticCK(k)/EllipticK(k))) = exp(- Pi*EllipticCK(k)/EllipticK(k))	(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]) == Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]	Successful	Successful	-	Successful [Tested: 1]
19.5.E7	${\displaystyle{\displaystyle\lambda=(1-\sqrt{k^{\prime}})/(2(1+\sqrt{k^{\prime}}))}}$ \lambda = (1-\sqrt{k^{\prime}})/(2(1+\sqrt{k^{\prime}}))		lambda = (1 -sqrt(sqrt(1 - (k)^(2))))/(2*(1 +sqrt(sqrt(1 - (k)^(2)))))	\[Lambda] == (1 -Sqrt[Sqrt[1 - (k)^(2)]])/(2*(1 +Sqrt[Sqrt[1 - (k)^(2)]]))	Skipped - no semantic math	Skipped - no semantic math	-	-
19.5.E8	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\left(1+2\sum_{n=1}^{\infty}q^{n^{2}}\right)^{2}}}$ \compellintKk@{k} = \frac{\pi}{2}\left(1+2\sum_{n=1}^{\infty}q^{n^{2}}\right)^{2}	${\displaystyle{\displaystyle|q|<1}}$	EllipticK(k) = (Pi)/(2)*(1 + 2*sum((exp(- Pi*EllipticCK(k)/EllipticK(k)))^((n)^(2)), n = 1..infinity))^(2)	EllipticK[(k)^2] == Divide[Pi,2]*((1 + 2*Sum[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^((n)^(2)), {n, 1, Infinity}, GenerateConditions->None]))^(2)	Failure	Failure	Error	Failed [1 / 3] Result: DirectedInfinity[] Test Values: {Rule[k, 1]}
19.5.E9	${\displaystyle{\displaystyle E\left(k\right)=K\left(k\right)+\frac{2\pi^{2}}{K\left(k\right)}\,\frac{\sum_{n=1}^{\infty}(-1)^{n}n^{2}q^{n^{2}}}{1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2}}}}}$ \compellintEk@{k} = \compellintKk@{k}+\frac{2\pi^{2}}{\compellintKk@{k}}\,\frac{\sum_{n=1}^{\infty}(-1)^{n}n^{2}q^{n^{2}}}{1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2}}}	${\displaystyle{\displaystyle|q|<1}}$	EllipticE(k) = EllipticK(k)+(2*(Pi)^(2))/(EllipticK(k))*(sum((- 1)^(n)* (n)^(2)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^((n)^(2)), n = 1..infinity))/(1 + 2*sum((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^((n)^(2)), n = 1..infinity))	EllipticE[(k)^2] == EllipticK[(k)^2]+Divide[2*(Pi)^(2),EllipticK[(k)^2]]*Divide[Sum[(- 1)^(n)* (n)^(2)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^((n)^(2)), {n, 1, Infinity}, GenerateConditions->None],1 + 2*Sum[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^((n)^(2)), {n, 1, Infinity}, GenerateConditions->None]]	Failure	Failure	Error	Skipped - Because timed out
19.5.E10	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\prod_{m=1}^{\infty}(1+k_{m})}}$ \compellintKk@{k} = \frac{\pi}{2}\prod_{m=1}^{\infty}(1+k_{m})		EllipticK(k) = (Pi)/(2)*product(1 + k[m], m = 1..infinity)	EllipticK[(k)^2] == Divide[Pi,2]*Product[1 + Subscript[k, m], {m, 1, Infinity}, GenerateConditions->None]	Failure	Failure	Error	Failed [30 / 30] Result: Plus[DirectedInfinity[], Times[-1.5707963267948966, NProduct[Plus[1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {m, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[k, 1], Rule[Subscript[k, m], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Plus[Complex[0.8428751774062981, -1.0782578237498217], Times[-1.5707963267948966, NProduct[Plus[1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {m, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[k, 2], Rule[Subscript[k, m], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.5.E11	${\displaystyle{\displaystyle k_{m+1}=\frac{1-\sqrt{1-k_{m}^{2}}}{1+\sqrt{1-k_{m}^{2}}}}}$ k_{m+1} = \frac{1-\sqrt{1-k_{m}^{2}}}{1+\sqrt{1-k_{m}^{2}}}		k[m + 1] = (1 -sqrt(1 - (k[m])^(2)))/(1 +sqrt(1 - (k[m])^(2)))	Subscript[k, m + 1] == Divide[1 -Sqrt[1 - (Subscript[k, m])^(2)],1 +Sqrt[1 - (Subscript[k, m])^(2)]]	Skipped - no semantic math	Skipped - no semantic math	-	-