Jacobi zeta function

In mathematics, the Jacobi zeta function Z(u) is the logarithmic derivative of the Jacobi theta function Θ(u). It is also commonly denoted as $\mathrm{zn}(u,k)$

$\mathrm{\Theta }(u)={\mathrm{\Theta }}_4\left(\frac{\pi u}{2K}\right)$

$Z(u)=\frac{\partial }{\partial u}\ln \mathrm{\Theta }(u)$ $=\frac{{\mathrm{\Theta }}^{\prime }(u)}{\mathrm{\Theta }(u)}$

$Z(\varphi |m)=E(\varphi |m)-\frac{E(m)}{K(m)}F(\varphi |m)$

Where E, K, and F are generic Incomplete Elliptical Integrals of the first and second kind. Jacobi Zeta Functions being kinds of Jacobi theta functions have applications to all their relevant fields and application.

$\mathrm{zn}(u,k)=Z(u)={\int }_0^u{\mathrm{dn}}^{2}v-\frac{E}{K}dv$

This relates Jacobi's common notation of, $\mathrm{dn}u=\sqrt{1-m\sin {\theta }^2}$, $\mathrm{sn}u=\sin \theta$, $\mathrm{cn}u=\cos \theta$. to Jacobi's Zeta function.

Some additional relations include ,

$\mathrm{zn}(u,k)=\frac{\pi }{2K}\frac{{{\mathrm{\Theta }}_1}^{\prime }\frac{\pi u}{2K}}{{\mathrm{\Theta }}_1\frac{\pi u}{2K}}-\frac{\mathrm{cn}u\mathrm{dn}u}{\mathrm{sn}u}$

$\mathrm{zn}(u,k)=\frac{\pi }{2K}\frac{{{\mathrm{\Theta }}_2}^{\prime }\frac{\pi u}{2K}}{{\mathrm{\Theta }}_2\frac{\pi u}{2K}}-\frac{\mathrm{sn}u\mathrm{dn}u}{\mathrm{cn}u}$

$\mathrm{zn}(u,k)=\frac{\pi }{2K}\frac{{{\mathrm{\Theta }}_3}^{\prime }\frac{\pi u}{2K}}{{\mathrm{\Theta }}_3\frac{\pi u}{2K}}-k^2\frac{\mathrm{sn}u\mathrm{cn}u}{\mathrm{dn}u}$

$\mathrm{zn}(u,k)=\frac{\pi }{2K}\frac{{{\mathrm{\Theta }}_4}^{\prime }\frac{\pi u}{2K}}{{\mathrm{\Theta }}_4\frac{\pi u}{2K}}$

References

• https://booksite.elsevier.com/samplechapters/9780123736376/Sample_Chapters/01~Front_Matter.pdf Pg.xxxiv
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• http://mathworld.wolfram.com/JacobiZetaFunction.html

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