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 ) {\displaystyle \operatorname {zn} (u,k)}
Θ ( u ) = Θ 4 ( π u 2 K ) {\displaystyle \Theta (u)=\Theta _{4}\left({\frac {\pi u}{2K}}\right)}
Z ( u ) = ∂ ∂ u ln Θ ( u ) {\displaystyle Z(u)={\frac {\partial }{\partial u}}\ln \Theta (u)} = Θ ′ ( u ) Θ ( u ) {\displaystyle ={\frac {\Theta '(u)}{\Theta (u)}}}
Z ( ϕ | m ) = E ( ϕ | m ) − E ( m ) K ( m ) F ( ϕ | m ) {\displaystyle Z(\phi |m)=E(\phi |m)-{\frac {E(m)}{K(m)}}F(\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 ) = ∫ 0 u dn 2 v − E K d v {\displaystyle \operatorname {zn} (u,k)=Z(u)=\int _{0}^{u}\operatorname {dn} ^{2}v-{\frac {E}{K}}dv}
This relates Jacobi's common notation of, dn u = 1 − m sin θ 2 {\displaystyle \operatorname {dn} {u}={\sqrt {1-m\sin {\theta }^{2}}}} , sn u = sin θ {\displaystyle \operatorname {sn} u=\sin {\theta }} , cn u = cos θ {\displaystyle \operatorname {cn} u=\cos {\theta }} . to Jacobi's Zeta function. Some additional relations include ,
zn ( u , k ) = π 2 K Θ 1 ′ π u 2 K Θ 1 π u 2 K − cn u dn u sn u {\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}}}}
zn ( u , k ) = π 2 K Θ 2 ′ π u 2 K Θ 2 π u 2 K − sn u dn u cn u {\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}}}}
zn ( u , k ) = π 2 K Θ 3 ′ π u 2 K Θ 3 π u 2 K − k 2 sn u cn u dn u {\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}}}}
zn ( u , k ) = π 2 K Θ 4 ′ π u 2 K Θ 4 π u 2 K {\displaystyle \operatorname {zn} (u,k)={\frac {\pi }{2K}}{\frac {\Theta _{4}'{\frac {\pi u}{2K}}}{\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) [June 1964]. "Chapter 16". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 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. p. 578. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. http://mathworld.wolfram.com/JacobiZetaFunction.html