Theta Functions - 20.6 Power Series

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
20.6.E3	${\displaystyle{\displaystyle\theta_{2}\left(\pi z\middle|\tau\right)=\theta_{2}\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j}\alpha_{2j}(\tau)z^{2j}\right)}}$ \Jacobithetatau{2}@{\pi z}{\tau} = \Jacobithetatau{2}@{0}{\tau}\exp@{-\sum_{j=1}^{\infty}\frac{1}{2j}\alpha_{2j}(\tau)z^{2j}}		JacobiTheta2(Pi*z,exp(I*Pi*tau)) = JacobiTheta2(0,exp(I*Pi*tau))*exp(- sum((1)/(2*j)*(sum(sum((m -(1)/(2)+ n*tau)^(- 2*j), m = - infinity..infinity), n = - infinity..infinity))*(z)^(2*j), j = 1..infinity))	EllipticTheta[2, Pi*z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]]*Exp[- Sum[Divide[1,2*j]*(Sum[Sum[(m -Divide[1,2]+ n*\[Tau])^(- 2*j), {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None])*(z)^(2*j), {j, 1, Infinity}, GenerateConditions->None]]	Aborted	Aborted	Skipped - Because timed out	Skipped - Because timed out
20.6.E4	${\displaystyle{\displaystyle\theta_{3}\left(\pi z\middle|\tau\right)=\theta_{3}\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j}\beta_{2j}(\tau)z^{2j}\right)}}$ \Jacobithetatau{3}@{\pi z}{\tau} = \Jacobithetatau{3}@{0}{\tau}\exp@{-\sum_{j=1}^{\infty}\frac{1}{2j}\beta_{2j}(\tau)z^{2j}}		JacobiTheta3(Pi*z,exp(I*Pi*tau)) = JacobiTheta3(0,exp(I*Pi*tau))*exp(- sum((1)/(2*j)*(sum(sum((m -(1)/(2)+(n -(1)/(2))*tau)^(- 2*j), m = - infinity..infinity), n = - infinity..infinity))*(z)^(2*j), j = 1..infinity))	EllipticTheta[3, Pi*z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]]*Exp[- Sum[Divide[1,2*j]*(Sum[Sum[(m -Divide[1,2]+(n -Divide[1,2])*\[Tau])^(- 2*j), {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None])*(z)^(2*j), {j, 1, Infinity}, GenerateConditions->None]]	Aborted	Aborted	Skipped - Because timed out	Skipped - Because timed out
20.6.E5	${\displaystyle{\displaystyle\theta_{4}\left(\pi z\middle|\tau\right)=\theta_{4}\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j}\gamma_{2j}(\tau)z^{2j}\right)}}$ \Jacobithetatau{4}@{\pi z}{\tau} = \Jacobithetatau{4}@{0}{\tau}\exp@{-\sum_{j=1}^{\infty}\frac{1}{2j}\gamma_{2j}(\tau)z^{2j}}		JacobiTheta4(Pi*z,exp(I*Pi*tau)) = JacobiTheta4(0,exp(I*Pi*tau))*exp(- sum((1)/(2*j)*(sum(sum((m +(n -(1)/(2))*tau)^(- 2*j), m = - infinity..infinity), n = - infinity..infinity))*(z)^(2*j), j = 1..infinity))	EllipticTheta[4, Pi*z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[4, 0, Exp[I*Pi*(\[Tau])]]*Exp[- Sum[Divide[1,2*j]*(Sum[Sum[(m +(n -Divide[1,2])*\[Tau])^(- 2*j), {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None])*(z)^(2*j), {j, 1, Infinity}, GenerateConditions->None]]	Aborted	Aborted	Skipped - Because timed out	Skipped - Because timed out
20.6#Ex2	${\displaystyle{\displaystyle\beta_{2j}(\tau)=2^{2j}\gamma_{2j}(2\tau)-\gamma_{2j}(\tau)}}$ \beta_{2j}(\tau) = 2^{2j}\gamma_{2j}(2\tau)-\gamma_{2j}(\tau)		(sum(sum((m -(1)/(2)+(n -(1)/(2))*tau)^(- 2*j), m = - infinity..infinity), n = - infinity..infinity)) = (2)^(2*j)* gamma[2*j](2*tau)-(sum(sum((m +(n -(1)/(2))*tau)^(- 2*j), m = - infinity..infinity), n = - infinity..infinity))	(Sum[Sum[(m -Divide[1,2]+(n -Divide[1,2])*\[Tau])^(- 2*j), {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None]) == (2)^(2*j)* Subscript[\[Gamma], 2*j][2*\[Tau]]-(Sum[Sum[(m +(n -Divide[1,2])*\[Tau])^(- 2*j), {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None])	Skipped - no semantic math	Skipped - no semantic math	-	-