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 $zn(u,k)$

:$\Theta(u)=\Theta_\left(\frac\right)$
:$Z(u)=\frac\ln\Theta(u)$ $=\frac$
:$Z(\phi, m)=E(\phi, m)-\fracF(\phi, m)$
: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.

:$zn(u,k)=Z(u)=\int_^dn^v-\fracdv$
:This relates Jacobi's common notation of, $dn =\sqrt$, $sn u= \sin$, $cn u= \cos$. to Jacobi's Zeta function.
:Some additional relations include ,
:$zn(u,k)=\frac\frac-\frac$
:$zn(u,k)=\frac\frac-\frac$
:$zn(u,k)=\frac\frac-k^2\frac$
:$zn(u,k)=\frac\frac$
:
:

References

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

Special functions

{{mathanalysis-stub