Theta Functions - 20.9 Relations to Other Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
20.9#Ex1 K ⁡ ( k ) = 1 2 ⁢ π ⁢ θ 3 2 ⁡ ( 0 | τ ) complete-elliptic-integral-first-kind-K 𝑘 1 2 𝜋 Jacobi-theta-tau 3 2 0 𝜏 {\displaystyle{\displaystyle K\left(k\right)=\tfrac{1}{2}\pi{\theta_{3}^{2}}% \left(0\middle|\tau\right)}}
\compellintKk@{k} = \tfrac{1}{2}\pi\Jacobithetatau{3}^{2}@{0}{\tau}		 

EllipticK((JacobiTheta2(0,exp(I*Pi*tau)))^(2)/(JacobiTheta3(0,exp(I*Pi*tau)))^(2)) = (1)/(2)*Pi*(JacobiTheta3(0,exp(I*Pi*tau)))^(2)

EllipticK[((EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]])^(2)/(EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2))^2] == Divide[1,2]*Pi*(EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2)
Failure Failure Error Successful [Tested: 10]
20.9#Ex2 K ′ ⁡ ( k ) = - i ⁢ τ ⁢ K ⁡ ( k ) diffop complete-elliptic-integral-first-kind-K 1 𝑘 𝑖 𝜏 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K'\left(k\right)=-i\tau K\left(k\right)}}
\compellintKk'@{k} = -i\tau\compellintKk@{k}		 

subs( temp=((JacobiTheta2(0,exp(I*Pi*tau)))^(2)/(JacobiTheta3(0,exp(I*Pi*tau)))^(2)), diff( EllipticK(temp), temp$(1) ) ) = - I*tau*EllipticK((JacobiTheta2(0,exp(I*Pi*tau)))^(2)/(JacobiTheta3(0,exp(I*Pi*tau)))^(2))

(D[EllipticK[(temp)^2], {temp, 1}]/.temp-> ((EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]])^(2)/(EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2))) == - I*\[Tau]*EllipticK[((EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]])^(2)/(EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2))^2]
Failure Failure
Failed [10 / 10]
Result: -.6481210221+.3604335389*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I}


Result: -1.647990213-1.212940701*I
Test Values: {tau = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [2 / 10]
Result: Complex[2.220446049250313*^-16, 0.6473902356608235]
Test Values: {Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.220446049250313*^-16, -0.8272591738499964]
Test Values: {Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

20.9.E3 R F ⁡ ( θ 2 2 ⁡ ( z , q ) θ 2 2 ⁡ ( 0 , q ) , θ 3 2 ⁡ ( z , q ) θ 3 2 ⁡ ( 0 , q ) , θ 4 2 ⁡ ( z , q ) θ 4 2 ⁡ ( 0 , q ) ) = θ 1 ′ ⁡ ( 0 , q ) θ 1 ⁡ ( z , q ) ⁢ z Carlson-integral-RF Jacobi-theta 2 2 𝑧 𝑞 Jacobi-theta 2 2 0 𝑞 Jacobi-theta 3 2 𝑧 𝑞 Jacobi-theta 3 2 0 𝑞 Jacobi-theta 4 2 𝑧 𝑞 Jacobi-theta 4 2 0 𝑞 diffop Jacobi-theta 1 1 0 𝑞 Jacobi-theta 1 𝑧 𝑞 𝑧 {\displaystyle{\displaystyle R_{F}\left(\frac{{\theta_{2}^{2}}\left(z,q\right)% }{{\theta_{2}^{2}}\left(0,q\right)},\frac{{\theta_{3}^{2}}\left(z,q\right)}{{% \theta_{3}^{2}}\left(0,q\right)},\frac{{\theta_{4}^{2}}\left(z,q\right)}{{% \theta_{4}^{2}}\left(0,q\right)}\right)=\frac{\theta_{1}'\left(0,q\right)}{% \theta_{1}\left(z,q\right)}z}}
\CarlsonsymellintRF@{\frac{\Jacobithetaq{2}^{2}@{z}{q}}{\Jacobithetaq{2}^{2}@{0}{q}}}{\frac{\Jacobithetaq{3}^{2}@{z}{q}}{\Jacobithetaq{3}^{2}@{0}{q}}}{\frac{\Jacobithetaq{4}^{2}@{z}{q}}{\Jacobithetaq{4}^{2}@{0}{q}}} = \frac{\Jacobithetaq{1}'@{0}{q}}{\Jacobithetaq{1}@{z}{q}}z		 

0.5*int(1/(sqrt(t+((JacobiTheta2(z, q))^(2))/((JacobiTheta2(0, q))^(2)))*sqrt(t+((JacobiTheta3(z, q))^(2))/((JacobiTheta3(0, q))^(2)))*sqrt(t+((JacobiTheta4(z, q))^(2))/((JacobiTheta4(0, q))^(2)))), t = 0..infinity) = (diff( JacobiTheta1(0, q), 0$(1) ))/(JacobiTheta1(z, q))*z

EllipticF[ArcCos[Sqrt[Divide[(EllipticTheta[2, z, q])^(2),(EllipticTheta[2, 0, q])^(2)]/Divide[(EllipticTheta[4, z, q])^(2),(EllipticTheta[4, 0, q])^(2)]]],(Divide[(EllipticTheta[4, z, q])^(2),(EllipticTheta[4, 0, q])^(2)]-Divide[(EllipticTheta[3, z, q])^(2),(EllipticTheta[3, 0, q])^(2)])/(Divide[(EllipticTheta[4, z, q])^(2),(EllipticTheta[4, 0, q])^(2)]-Divide[(EllipticTheta[2, z, q])^(2),(EllipticTheta[2, 0, q])^(2)])]/Sqrt[Divide[(EllipticTheta[4, z, q])^(2),(EllipticTheta[4, 0, q])^(2)]-Divide[(EllipticTheta[2, z, q])^(2),(EllipticTheta[2, 0, q])^(2)]] == Divide[D[EllipticTheta[1, 0, q], {0, 1}],EllipticTheta[1, z, q]]*z
Aborted Failure Skipped - Because timed out
Failed [70 / 70]
Result: Plus[Times[Complex[-0.8660254037844387, -0.49999999999999994], D[0.0
Test Values: {0.0, 1.0}], Power[EllipticTheta[1, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], -1]], Times[EllipticF[ArcCos[Power[Times[Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -2], Power[EllipticTheta[2, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], 2], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 2], Power[EllipticTheta[4, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], -2]], Rational[1, 2]]], Times[Power[Plus[Times[-1.0, Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -2], Power[EllipticTheta[2, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], 2]], Times[Power[E<syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[0.4999999999999998, -0.8660254037844387], D[0.0
Test Values: {0.0, 1.0}], Power[EllipticTheta[1, Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]], -1]], Times[EllipticF[ArcCos[Power[Times[Power[EllipticTheta[2, Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]], 2], Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -2], Power[EllipticTheta[4, Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]], -2], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 2]], Rational[1, 2]]], Times[Power[Plus[Times[-1.0, Power[EllipticTheta[2, Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]], 2], Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -2]], Times[Power[EllipticTheta[4, Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]], 2], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -2]]], -1], Plus[Times[-1.0, Power[EllipticTheta[3, Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]], 2], Power[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -2]], Times[Power[EllipticTheta[4, Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]], 2], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -2]]]]], Power[Plus[Times[-1.0, Power[EllipticTheta[2, Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]], 2], Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -2]], Times[Power[EllipticTheta[4, Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]], 2], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -2]]], Rational[-1, 2]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
20.9.E4 R F ⁡ ( 0 , θ 3 4 ⁡ ( 0 , q ) , θ 4 4 ⁡ ( 0 , q ) ) = 1 2 ⁢ π Carlson-integral-RF 0 Jacobi-theta 3 4 0 𝑞 Jacobi-theta 4 4 0 𝑞 1 2 𝜋 {\displaystyle{\displaystyle R_{F}\left(0,{\theta_{3}^{4}}\left(0,q\right),{% \theta_{4}^{4}}\left(0,q\right)\right)=\tfrac{1}{2}\pi}}
\CarlsonsymellintRF@{0}{\Jacobithetaq{3}^{4}@{0}{q}}{\Jacobithetaq{4}^{4}@{0}{q}} = \tfrac{1}{2}\pi		 

0.5*int(1/(sqrt(t+0)*sqrt(t+(JacobiTheta3(0, q))^(4))*sqrt(t+(JacobiTheta4(0, q))^(4))), t = 0..infinity) = (1)/(2)*Pi

EllipticF[ArcCos[Sqrt[0/(EllipticTheta[4, 0, q])^(4)]],((EllipticTheta[4, 0, q])^(4)-(EllipticTheta[3, 0, q])^(4))/((EllipticTheta[4, 0, q])^(4)-0)]/Sqrt[(EllipticTheta[4, 0, q])^(4)-0] == Divide[1,2]*Pi
Aborted Failure Skipped - Because timed out Successful [Tested: 10]
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