JacobiZeta - Maple Help

JacobiZeta

Jacobi's Zeta function

 Calling Sequence JacobiZeta(z, k)

Parameters

 z - algebraic expression k - algebraic expression

Description

 • JacobiZeta is defined by:
 $\left[{\mathrm{JacobiZeta}}{}\left({z}{,}{k}\right){=}{\mathrm{Diff}}{}\left({\mathrm{ln}}{}\left({\mathrm{JacobiTheta4}}{}\left(\frac{{1}}{{2}}{}\frac{{\mathrm{π}}{}{z}}{{\mathrm{EllipticK}}{}\left({k}\right)}{,}{\mathrm{EllipticNome}}{}\left({k}\right)\right)\right){,}{z}\right){,}{\mathrm{with no restrictions on}}{}\left({z}{,}{k}\right)\right]$ (1)
 which is essentially the logarithmic derivative of JacobiTheta4.
 • JacobiZeta(z,k) is a periodic function of $z$ with period $2\mathrm{EllipticK}\left(k\right)$

Examples

 > $\mathrm{JacobiZeta}\left(1.0,0.5\right)$
 ${0.06347769531}$ (2)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{JacobiZeta}\right)$
 $\left[{\mathrm{JacobiZeta}}{}\left({-}{z}{,}{k}\right){=}{-}{\mathrm{JacobiZeta}}{}\left({z}{,}{k}\right){,}{\mathrm{JacobiZeta}}{}\left({z}{,}{-}{k}\right){=}{\mathrm{JacobiZeta}}{}\left({z}{,}{k}\right){,}{\mathrm{JacobiZeta}}{}\left({0}{,}{k}\right){=}{0}{,}{\mathrm{JacobiZeta}}{}\left({z}{,}{0}\right){=}{0}{,}{\mathrm{JacobiZeta}}{}\left({z}{,}{1}\right){=}{\mathrm{tanh}}{}\left({z}\right){,}{\mathrm{JacobiZeta}}{}\left({z}{,}{\mathrm{\infty }}\right){=}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}{,}\left[{\mathrm{JacobiZeta}}{}\left({2}{}{\mathrm{_n1}}{}{\mathrm{EllipticK}}{}\left({k}\right){,}{k}\right){=}{0}{,}{\mathrm{_n1}}{::}{'}{\mathrm{integer}}{'}\right]\right]$ (3)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{JacobiZeta}\right)$
 $\left[{\mathrm{JacobiZeta}}{}\left({z}{,}{k}\right){=}{\sum }_{{\mathrm{_k1}}{=}{1}}^{{\mathrm{\infty }}}{}\left({-}\frac{{2}{}{\mathrm{\pi }}{}{{\mathrm{EllipticNome}}{}\left({k}\right)}^{{\mathrm{_k1}}}{}{\mathrm{sin}}{}\left(\frac{{\mathrm{_k1}}{}{\mathrm{\pi }}{}{z}}{{\mathrm{EllipticK}}{}\left({k}\right)}\right)}{{\mathrm{EllipticK}}{}\left({k}\right){}\left({{\mathrm{EllipticNome}}{}\left({k}\right)}^{{2}{}{\mathrm{_k1}}}{-}{1}\right)}\right){,}{\mathrm{with no restrictions on}}{}\left({z}{,}{k}\right)\right]$ (4)
 >