Jacobi Zeta Function

Denoted zn$(u,k)$ or $Z(u)$.

$$Z(\phi \vert m)\equiv E(\phi \vert m)-{E(m)F(\phi \vert m)\over K(m)},$$

where $\phi$ is the Amplitude, $m$ is the Parameter, and $F$ and $K$ are Elliptic Integrals of the First Kind, and $E$ is an Elliptic Integral of the Second Kind. See Gradshteyn and Ryzhik (1980, p. xxxi) for expressions in terms of Theta Functions.

See also Zeta Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 595, 1972.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1979.