Jacobi's epsilon and zeta function for moduli outside the interval [0, 1]

TL;DR

This paper derives formulas connecting Jacobi's Epsilon and Zeta functions for moduli outside the standard interval to elliptic functions within the interval, expanding their applicability.

Contribution

It introduces new formulas relating Jacobi's Epsilon and Zeta functions for moduli outside [0,1] to elliptic functions with standard moduli.

Findings

Formulas for moduli in (1, ∞) derived

Formulas for pure imaginary moduli derived

Extended the applicability of Jacobi functions

Abstract

The formulas that relate Jacobi's Epsilon and Zeta function with real moduli in the interval (1,inf) or with pure imaginary moduli to elliptic functions with moduli in the interval [0,1] are derived.