Jacobian zeta function

In mathematics, the Jacobian zeta function is the logarithmic derivative of the Jacobian theta function . It is named after the German mathematician Carl Gustav Jacob Jacobi .

definition

The Jacobian zeta function is defined as

${\displaystyle zn(u,k)=\mathrm {Z} (u)={\frac {\partial }{\partial u}}\ln \Theta (u)={\frac {\pi }{2K}}{\frac {\vartheta '_{4}\left({\frac {\pi u}{2K}}\right)}{\vartheta _{4}\left({\frac {\pi u}{2K}}\right)}}}$,

applies , with the theta series through ${\displaystyle \Theta (u)=\vartheta ({\frac {\pi u}{2K}})=\vartheta _{4}({\frac {\pi u}{2K}})}$

${\displaystyle \vartheta _{4}(z):=\sum _{n=-\infty }^{\infty }(-1)^{n}e^{i\pi n^{2}\tau }e^{2niz}}$

is defined.

Representation using elliptic integrals

The Jacobian zeta function can also be equivalent to elliptic integrals by

${\displaystyle \mathrm {Z} (m,u)=E(m,u)-{\frac {F(m,u)E(u)}{K(u)}}}$

being an incomplete first-order elliptic integral, an incomplete second-order elliptic integral, a complete first-order elliptic integral, and a complete second-order elliptic integral. ${\displaystyle F(m,u)}$ ${\displaystyle E(m,u)}$ ${\displaystyle K(u)=F({\frac {\pi }{2}},u)}$ ${\displaystyle E(u)=E({\frac {\pi }{2}},u)}$

literature

• Christian Houzel Elliptic Functions and Abelian Integrals , in Jean Dieudonné (Ed.) Geschichte der Mathematik 1700-1900 , Vieweg 1985, p. 462 (Chapter 7.1.10)
• Leo Koenigsberger On the history of the elliptical transcendent in the years 1826 to 1829 , Teubner 1879, p. 78, gutenberg

See also

• Zeta function

Web links

• Eric W. Weisstein : Jacobi Zeta Function . In: MathWorld (English).

Individual evidence

1. ↑ ^{a b} Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 16", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , New York: Dover, p. 578, ISBN 978-0486612720 , MR 0167642.
2. ↑ Gradshteyn, IS and Ryzhik, IM Tables of Integrals, Series, and Products , 6th ed. San Diego, CA: Academic Press, 2000, p. xxxiv.