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 ${\displaystyle zn(u,k)}$[1]

${\displaystyle \Theta (u)=\Theta _{4}\left({\frac {\pi u}{2K}}\right)}$

${\displaystyle Z(u)={\frac {\partial }{\partial u}}\ln \Theta (u)}$ ${\displaystyle ={\frac {\Theta ^{'}(u)}{\Theta (u)}}}$[2]

${\displaystyle Z(\phi |m)=E(\phi |m)-{\frac {E(m)}{K(m)}}F(\phi |m)}$[3]

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 there relevant fields and application.

${\displaystyle zn(u,k)=Z(u)=\int _{0}^{u}dn^{2}v-{\frac {E}{K}}dv}$[1]

This relates Jacobi's common notation of, ${\displaystyle dn{u}={\sqrt {1-m\sin {\theta }^{2}}}}$, ${\displaystyle snu=\sin {\theta }}$, ${\displaystyle cnu=\cos {\theta }}$.[1] to Jacobi's Zeta function.

Some additional relations include ,

${\displaystyle zn(u,k)={\frac {\pi }{2K}}{\frac {\Theta _{1}^{'}{\frac {\pi u}{2K}}}{\Theta _{1}{\frac {\pi u}{2K}}}}-{\frac {cn{u}*dn{u}}{sn{u}}}}$[1]

${\displaystyle zn(u,k)={\frac {\pi }{2K}}{\frac {\Theta _{2}^{'}{\frac {\pi u}{2K}}}{\Theta _{2}{\frac {\pi u}{2K}}}}-{\frac {sn{u}*dn{u}}{cn{u}}}}$[1]

${\displaystyle zn(u,k)={\frac {\pi }{2K}}{\frac {\Theta _{3}^{'}{\frac {\pi u}{2K}}}{\Theta _{3}{\frac {\pi u}{2K}}}}-k^{2}{\frac {sn{u}*cn{u}}{dn{u}}}}$[1]

${\displaystyle zn(u,k)={\frac {\pi }{2K}}{\frac {\Theta _{4}^{'}{\frac {\pi u}{2K}}}{\Theta _{4}{\frac {\pi u}{2K}}}}}$[1]