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 $\backslash operatorname\{zn\}(u,k)${{Cite web|url=https://booksite.elsevier.com/samplechapters/9780123736376/Sample_Chapters/01~Front_Matter.pdf|title=Table of Integrals, Series, and Products|last=Gradshteyn, Ryzhik|first=I.S., I.M.|date=|website=booksite.com|archive-url=}}

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

:$Z(u)=\backslash frac\{\backslash partial\}\{\backslash partial\; u\}\backslash ln\backslash Theta(u)$ $=\backslash frac\{\backslash Theta\text{'}(u)\}\{\backslash Theta(u)\}${{Cite book|url=https://books.google.com/books?id=KiPCAgAAQBAJ&q=importance+Jacobi+zeta+function&pg=PA576|title=Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables|last1=Abramowitz|first1=Milton|last2=Stegun|first2=Irene A.|date=2012-04-30|publisher=Courier Corporation|isbn=978-0-486-15824-2|language=en}}

:$Z(\backslash phi|m)=E(\backslash phi|m)-\backslash frac\{E(m)\}\{K(m)\}F(\backslash phi|m)${{Cite web|url=http://mathworld.wolfram.com/JacobiZetaFunction.html|title=Jacobi Zeta Function|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-02}}

: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.

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

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

:Some additional relations include ,

:$\backslash operatorname\{zn\}(u,k)=\backslash frac\{\backslash pi\}\{2K\}\backslash frac\{\backslash Theta\_1\text{'}\backslash frac\{\backslash pi\; u\}\{2K\}\}\{\backslash Theta\_1\backslash frac\{\backslash pi\; u\}\{2K\}\}-\backslash frac\{\backslash operatorname\{cn\}\{u\}\backslash ,\backslash operatorname\{dn\}\{u\}\}\{\backslash operatorname\{sn\}\{u\}\}$

:$\backslash operatorname\{zn\}(u,k)=\backslash frac\{\backslash pi\}\{2K\}\backslash frac\{\backslash Theta\_2\text{'}\backslash frac\{\backslash pi\; u\}\{2K\}\}\{\backslash Theta\_2\backslash frac\{\backslash pi\; u\}\{2K\}\}-\backslash frac\{\backslash operatorname\{sn\}\{u\}\backslash ,\backslash operatorname\{dn\}\{u\}\}\{\backslash operatorname\{cn\}\{u\}\}$

:$\backslash operatorname\{zn\}(u,k)=\backslash frac\{\backslash pi\}\{2K\}\backslash frac\{\backslash Theta\_3\text{'}\backslash frac\{\backslash pi\; u\}\{2K\}\}\{\backslash Theta\_3\backslash frac\{\backslash pi\; u\}\{2K\}\}-k^2\backslash frac\{\backslash operatorname\{sn\}\{u\}\backslash ,\backslash operatorname\{cn\}\{u\}\}\{\backslash operatorname\{dn\}\{u\}\}$

:$\backslash operatorname\{zn\}(u,k)=\backslash frac\{\backslash pi\}\{2K\}\backslash frac\{\backslash Theta\_4\text{'}\backslash frac\{\backslash pi\; u\}\{2K\}\}\{\backslash Theta\_4\backslash frac\{\backslash pi\; u\}\{2K\}\}$

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