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.

$\mathrm{zn}(u,k)=Z(u)=\underset{0}{\overset{u}{\int }}{\mathrm{dn}}^{2}v-\frac{E}{K}dv$

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
• Abramowitz, Milton; Stegun, Irene Ann, eds (1983). "Chapter 16". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 578. LCCN 65-12253. ISBN 978-0-486-61272-0. http://www.math.sfu.ca/~cbm/aands/page_578.htm. 
• http://mathworld.wolfram.com/JacobiZetaFunction.html