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In mathematics, the Jacobi zeta function Z(u) is the logarithmic derivative of the Jacobi theta function

$$Z(\phi|m)=E(\phi|m)-(E(m)F(\phi|m))/(K(m)),$$

where $$\phi$$ is the Jacobi amplitude, m is the parameter, and $$F(\phi|m)$$and K(m) are elliptic integrals of the first kind, and E(m) is an elliptic integral of the second kind.

See Gradshteyn and Ryzhik (2000 ) for expressions in terms of theta functions.

References

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.