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 ) {\displaystyle \operatorname {zn} (u,k)} [1]

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)}}} [2]
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)} [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.
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} [1]
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 }} .[1] 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}}}} [1]
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}}}} [1]
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}}}} [1]
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}}}}} [1]

References

  1. ^ a b c d e f g Gradshteyn, Ryzhik, I.S., I.M. "Table of Integrals, Series, and Products" (PDF). booksite.com.{{cite web}}: CS1 maint: multiple names: authors list (link)
  2. ^ Abramowitz, Milton; Stegun, Irene A. (2012-04-30). Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Courier Corporation. ISBN 978-0-486-15824-2.
  3. ^ Weisstein, Eric W. "Jacobi Zeta Function". mathworld.wolfram.com. Retrieved 2019-12-02.
  • 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


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