Elliptic Integrals - 19.36 Methods of Computation

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
19.36.E3	${\displaystyle{\displaystyle R_{F}\left(1,2,4\right)=R_{F}\left(z_{1},z_{2},z_{3}\right)}}$ \CarlsonsymellintRF@{1}{2}{4} = \CarlsonsymellintRF@{z_{1}}{z_{2}}{z_{3}}		0.5*int(1/(sqrt(t+1)*sqrt(t+2)*sqrt(t+4)), t = 0..infinity) = 0.5*int(1/(sqrt(t+z[1])*sqrt(t+z[2])*sqrt(t+z[3])), t = 0..infinity)	EllipticF[ArcCos[Sqrt[1/4]],(4-2)/(4-1)]/Sqrt[4-1] == EllipticF[ArcCos[Sqrt[Subscript[z, 1]/Subscript[z, 3]]],(Subscript[z, 3]-Subscript[z, 2])/(Subscript[z, 3]-Subscript[z, 1])]/Sqrt[Subscript[z, 3]-Subscript[z, 1]]	Aborted	Failure	Skipped - Because timed out	Failed [300 / 300] Result: Indeterminate Test Values: {Rule[Subscript[z, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[z, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[z, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.6113291272616378, 0.7460602493090597] Test Values: {Rule[Subscript[z, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[z, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[z, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.36.E4	\begin{aligned} \displaystyle z_{1}&\displaystyle = 2.10985\;99098\;8,\\ \displaystyle z_{3}&\displaystyle				Skipped - no semantic math	Skipped - no semantic math	-	-
19.36.E5	${\displaystyle{\displaystyle R_{F}\left(1,2,4\right)=0.68508\;58166\dots}}$ \CarlsonsymellintRF@{1}{2}{4} = 0.68508\;58166\dots		0.5*int(1/(sqrt(t+1)*sqrt(t+2)*sqrt(t+4)), t = 0..infinity) = 0.6850858166	EllipticF[ArcCos[Sqrt[1/4]],(4-2)/(4-1)]/Sqrt[4-1] == 0.6850858166	Failure	Failure	Successful [Tested: 0]	Successful [Tested: 1]
19.36#Ex1	${\displaystyle{\displaystyle 2a_{n+1}=a_{n}+\sqrt{a_{n}^{2}-c_{n}^{2}}}}$ 2a_{n+1} = a_{n}+\sqrt{a_{n}^{2}-c_{n}^{2}}		2*a[n + 1] = a[n]+sqrt((a[n])^(2)- (c[n])^(2))	2*Subscript[a, n + 1] == Subscript[a, n]+Sqrt[(Subscript[a, n])^(2)- (Subscript[c, n])^(2)]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.36#Ex2	${\displaystyle{\displaystyle 2c_{n+1}=a_{n}-\sqrt{a_{n}^{2}-c_{n}^{2}}}}$ 2c_{n+1} = a_{n}-\sqrt{a_{n}^{2}-c_{n}^{2}}		2*c[n + 1] = a[n]-sqrt((a[n])^(2)- (c[n])^(2))	2*Subscript[c, n + 1] == Subscript[a, n]-Sqrt[(Subscript[a, n])^(2)- (Subscript[c, n])^(2)]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.36#Ex3	${\displaystyle{\displaystyle 2t_{n+1}=t_{n}+\sqrt{t_{n}^{2}+\theta c_{n}^{2}}}}$ 2t_{n+1} = t_{n}+\sqrt{t_{n}^{2}+\theta c_{n}^{2}}		2*t[n + 1] = t[n]+sqrt((t[n])^(2)+ theta*(c[n])^(2))	2*Subscript[t, n + 1] == Subscript[t, n]+Sqrt[(Subscript[t, n])^(2)+ \[Theta]*(Subscript[c, n])^(2)]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.36#Ex4	${\displaystyle{\displaystyle 0<c_{0}}}$ 0 < c_{0}		0 < c[0]	0 < Subscript[c, 0]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.36#Ex5	${\displaystyle{\displaystyle t_{0}\geq 0}}$ t_{0} \geq 0		t[0] >= 0	Subscript[t, 0] >= 0	Skipped - no semantic math	Skipped - no semantic math	-	-
19.36#Ex6	${\displaystyle{\displaystyle t_{0}^{2}+\theta a_{0}^{2}\geq 0}}$ t_{0}^{2}+\theta a_{0}^{2} \geq 0		(t[0])^(2)+ theta*(a[0])^(2) >= 0	(Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2) >= 0	Skipped - no semantic math	Skipped - no semantic math	-	-
19.36#Ex7	${\displaystyle{\displaystyle\theta=+1}}$ \theta = + 1		theta = + 1	\[Theta] == + 1	Skipped - no semantic math	Skipped - no semantic math	-	-
19.36.E9	${\displaystyle{\displaystyle R_{F}\left(t_{0}^{2},t_{0}^{2}+\theta c_{0}^{2},t_{0}^{2}+\theta a_{0}^{2}\right)=R_{F}\left(T^{2},T^{2},T^{2}+\theta M^{2}\right)}}$ \CarlsonsymellintRF@{t_{0}^{2}}{t_{0}^{2}+\theta c_{0}^{2}}{t_{0}^{2}+\theta a_{0}^{2}} = \CarlsonsymellintRF@{T^{2}}{T^{2}}{T^{2}+\theta M^{2}}		0.5*int(1/(sqrt(t+(t[0])^(2))*sqrt(t+(t[0])^(2)+ theta*(c[0])^(2))*sqrt(t+(t[0])^(2)+ theta*(a[0])^(2))), t = 0..infinity) = 0.5*int(1/(sqrt(t+(T)^(2))*sqrt(t+(T)^(2))*sqrt(t+(T)^(2)+ theta*(M)^(2))), t = 0..infinity)	EllipticF[ArcCos[Sqrt[(Subscript[t, 0])^(2)/(Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)]],((Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)-(Subscript[t, 0])^(2)+ \[Theta]*(Subscript[c, 0])^(2))/((Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)-(Subscript[t, 0])^(2))]/Sqrt[(Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)-(Subscript[t, 0])^(2)] == EllipticF[ArcCos[Sqrt[(T)^(2)/(T)^(2)+ \[Theta]*(M)^(2)]],((T)^(2)+ \[Theta]*(M)^(2)-(T)^(2))/((T)^(2)+ \[Theta]*(M)^(2)-(T)^(2))]/Sqrt[(T)^(2)+ \[Theta]*(M)^(2)-(T)^(2)]	Error	Failure	-	Failed [300 / 300] Result: Plus[Complex[0.041390391732804066, 0.9969018367602411], Times[2.8284271247461903, Power[Times[Complex[0.0, 1.0], a], Rational[-1, 2]], EllipticF[ArcCos[Power[Plus[Complex[-0.031249999999999986, 0.05412658773652742], Times[Complex[0.0, 0.125], a]], Rational[1, 2]]], Times[Complex[0.0, -8.0], Power[a, -1], Plus[Times[Complex[0.0, 0.125], a], Times[Complex[0.0, 0.125], c]]]]]] Test Values: {Rule[M, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[T, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[c, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[t, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Plus[Complex[0.041390391732804066, 0.9969018367602411], Times[2.8284271247461903, Power[Times[Complex[0.0, 1.0], a], Rational[-1, 2]], EllipticF[ArcCos[Power[Plus[Complex[-0.031249999999999986, 0.05412658773652742], Times[Complex[0.0, 0.125], a]], Rational[1, 2]]], Times[Complex[0.0, -8.0], Power[a, -1], Plus[Times[Complex[0.0, 0.125], a], Times[Complex[0.0, 0.125], c]]]]]] Test Values: {Rule[M, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[T, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[c, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[t, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.36.E9	${\displaystyle{\displaystyle R_{F}\left(T^{2},T^{2},T^{2}+\theta M^{2}\right)=R_{C}\left(T^{2}+\theta M^{2},T^{2}\right)}}$ \CarlsonsymellintRF@{T^{2}}{T^{2}}{T^{2}+\theta M^{2}} = \CarlsonellintRC@{T^{2}+\theta M^{2}}{T^{2}}		Error	EllipticF[ArcCos[Sqrt[(T)^(2)/(T)^(2)+ \[Theta]*(M)^(2)]],((T)^(2)+ \[Theta]*(M)^(2)-(T)^(2))/((T)^(2)+ \[Theta]*(M)^(2)-(T)^(2))]/Sqrt[(T)^(2)+ \[Theta]*(M)^(2)-(T)^(2)] == 1/Sqrt[(T)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((T)^(2)+ \[Theta]*(M)^(2))/((T)^(2))]	Missing Macro Error	Failure	-	Failed [300 / 300] Result: Complex[-1.634056915706757, -0.008820605997006181] Test Values: {Rule[M, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[T, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-1.6914869520542948, 0.13073697514602478] Test Values: {Rule[M, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[T, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.36#Ex9	${\displaystyle{\displaystyle a_{3}^{2}=2.46209\;30206\;0}}$ a_{3}^{2} = 2.46209\;30206\;0		(a[3])^(2) = 2.46209302060	(Subscript[a, 3])^(2) == 2.46209302060	Skipped - no semantic math	Skipped - no semantic math	-	-
19.36#Ex10	${\displaystyle{\displaystyle t_{3}^{2}=1.46971\;53173\;1}}$ t_{3}^{2} = 1.46971\;53173\;1		(t[3])^(2) = 1.46971531731	(Subscript[t, 3])^(2) == 1.46971531731	Skipped - no semantic math	Skipped - no semantic math	-	-
19.36.E11	${\displaystyle{\displaystyle R_{F}\left(1,2,4\right)=R_{C}\left(T^{2}+M^{2},T^{2}\right)}}$ \CarlsonsymellintRF@{1}{2}{4} = \CarlsonellintRC@{T^{2}+M^{2}}{T^{2}}		Error	EllipticF[ArcCos[Sqrt[1/4]],(4-2)/(4-1)]/Sqrt[4-1] == 1/Sqrt[(T)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((T)^(2)+ (M)^(2))/((T)^(2))]	Missing Macro Error	Failure	-	Failed [100 / 100] Result: Complex[-0.841498016533642, 0.8813735870195429] Test Values: {Rule[M, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[T, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.8857105197615976, -2.720699010523131] Test Values: {Rule[M, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[T, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.36.E11	${\displaystyle{\displaystyle R_{C}\left(T^{2}+M^{2},T^{2}\right)=0.68508\;58166}}$ \CarlsonellintRC@{T^{2}+M^{2}}{T^{2}} = 0.68508\;58166		Error	1/Sqrt[(T)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((T)^(2)+ (M)^(2))/((T)^(2))] == 0.6850858166	Missing Macro Error	Failure	-	Failed [100 / 100] Result: Complex[0.8414980165670778, -0.8813735870195429] Test Values: {Rule[M, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[T, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.8857105197950335, 2.720699010523131] Test Values: {Rule[M, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[T, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.36#Ex11	${\displaystyle{\displaystyle h_{n}=\sqrt{t_{n}^{2}+\theta a_{n}^{2}}}}$ h_{n} = \sqrt{t_{n}^{2}+\theta a_{n}^{2}}		h[n] = sqrt((t[n])^(2)+ theta*(a[n])^(2))	Subscript[h, n] == Sqrt[(Subscript[t, n])^(2)+ \[Theta]*(Subscript[a, n])^(2)]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.36#Ex12	${\displaystyle{\displaystyle h_{n}=h_{n-1}\frac{t_{n}}{\sqrt{t_{n}^{2}+\theta c_{n}^{2}}}}}$ h_{n} = h_{n-1}\frac{t_{n}}{\sqrt{t_{n}^{2}+\theta c_{n}^{2}}}		h[n] = h[n - 1]*(t[n])/(sqrt((t[n])^(2)+ theta*(c[n])^(2)))	Subscript[h, n] == Subscript[h, n - 1]*Divide[Subscript[t, n],Sqrt[(Subscript[t, n])^(2)+ \[Theta]*(Subscript[c, n])^(2)]]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.36.E13	${\displaystyle{\displaystyle 2R_{G}\left(t_{0}^{2},t_{0}^{2}+\theta c_{0}^{2},t_{0}^{2}+\theta a_{0}^{2}\right)=\left(t_{0}^{2}+\theta\sum_{m=0}^{\infty}2^{m-1}c_{m}^{2}\right)R_{C}\left(T^{2}+\theta M^{2},T^{2}\right)+h_{0}+\sum_{m=1}^{\infty}2^{m}(h_{m}-h_{m-1})}}$ 2\CarlsonsymellintRG@{t_{0}^{2}}{t_{0}^{2}+\theta c_{0}^{2}}{t_{0}^{2}+\theta a_{0}^{2}} = \left(t_{0}^{2}+\theta\sum_{m=0}^{\infty}2^{m-1}c_{m}^{2}\right)\CarlsonellintRC@{T^{2}+\theta M^{2}}{T^{2}}+h_{0}+\sum_{m=1}^{\infty}2^{m}(h_{m}-h_{m-1})		Error	2*Sqrt[(Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)-(Subscript[t, 0])^(2)]*(EllipticE[ArcCos[Sqrt[(Subscript[t, 0])^(2)/(Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)]],((Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)-(Subscript[t, 0])^(2)+ \[Theta]*(Subscript[c, 0])^(2))/((Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)-(Subscript[t, 0])^(2))]+(Cot[ArcCos[Sqrt[(Subscript[t, 0])^(2)/(Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)]]])^2*EllipticF[ArcCos[Sqrt[(Subscript[t, 0])^(2)/(Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)]],((Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)-(Subscript[t, 0])^(2)+ \[Theta]*(Subscript[c, 0])^(2))/((Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)-(Subscript[t, 0])^(2))]+Cot[ArcCos[Sqrt[(Subscript[t, 0])^(2)/(Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[(Subscript[t, 0])^(2)/(Subscript[t, 0])^(2)+ \[Theta]*(Subscript[a, 0])^(2)]]]^2]) == ((Subscript[t, 0])^(2)+ \[Theta]*Sum[(2)^(m - 1)* (Subscript[c, m])^(2), {m, 0, Infinity}, GenerateConditions->None])*1/Sqrt[(T)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((T)^(2)+ \[Theta]*(M)^(2))/((T)^(2))]+ Subscript[h, 0]+ Sum[(2)^(m)*(Subscript[h, m]- Subscript[h, m - 1]), {m, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Failed [1 / 1]