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 ${\displaystyle \operatorname {zn} (u,k)}$^[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 their relevant fields and application.

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

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

Some additional relations include ,

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

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

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

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