Misplaced Pages

Jacobi zeta function

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

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

  1. ^ 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.


Stub icon

This mathematical analysis–related article is a stub. You can help Misplaced Pages by expanding it.

Categories: