Jacobi zeta function explained

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

\operatorname{zn}(u,k) ^[1] \Theta(u) = \Theta _{4} \left( (\pi u)/(2 K) \right) Z(u) = (\partial)/(\partial u) ln \Theta(u) = (\Theta'(u))/(\Theta(u)) ^[2] Z(\phi|m) = E(\phi|m) - (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.

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

This relates Jacobi's common notation of,

\operatorname{dn}{u} = \sqrt{ 1- m \sin{\theta} ^{2}} , \operatorname{sn}u = \sin{\theta} , \operatorname{cn}u = \cos{\theta} . to Jacobi's Zeta function.

Some additional relations include,

\operatorname{zn}(u,k) = (\pi)/(2 K) (\Theta _{1 ' (\pi u)/(2 K)})/(- \Theta _{1 (\pi u)/(2 K)}) (\operatorname{cn)/(u) \operatorname{dn}{u}}{\operatorname{sn}{u}} \operatorname{zn}(u,k) = (\pi)/(2 K) (\Theta _{2 ' (\pi u)/(2 K)})/(- \Theta _{2 (\pi u)/(2 K)}) (\operatorname{sn)/(u) \operatorname{dn}{u}}{\operatorname{cn}{u}} \operatorname{zn}(u,k) = (\pi)/(2 K) (\Theta _{3 ' (\pi u)/(2 K)})/(\Theta _{3 (\pi u)/(2 K)}) - k ^{2 (\operatorname{sn)/(u)} \operatorname{cn}{u}}{\operatorname{dn}{u}} \operatorname{zn}(u,k) = (\pi)/(2 K) (\Theta _{4 ' (\pi u)/(2 K)})/(\Theta _{4 (\pi u)/(2 K)})

References

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

Notes and References

1. Web site: Table of Integrals, Series, and Products. Gradshteyn, Ryzhik. I.S., I.M.. booksite.com.
2. Book: Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Abramowitz. Milton. Stegun. Irene A.. 2012-04-30. Courier Corporation. 978-0-486-15824-2. en.
3. Web site: Jacobi Zeta Function. Weisstein. Eric W.. mathworld.wolfram.com. en. 2019-12-02.