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We present a numerical method to invert a general incomplete elliptic integral with respect to its argument and/or amplitude. The method obtains a solution by bisection accelerated by the half argument formulas and the addition theorems to evaluate the incomplete elliptic integrals and Jacobian elliptic functions required in the course. If a faster… (More)

We developed a novel method to calculate an associate complete elliptic integral of the third kind, J(n|m) ≡ [Π (n|m) − K(m)]/n. The key idea is the double argument formula of J(n|m) with respect to n. We derived it from the not-so-popular addition theorem of Jacobi's complete elliptic integral of the third kind, Π 1 (a|m), with respect to a, which is a… (More)

We developed an efficient procedure to evaluate two auxiliary complete elliptic integrals of the second kind B(m) and D(m) by using their Taylor series expansions, the definition of Jacobi's nome, and Legendre's relation. The developed procedure is more precise than the existing ones in the sense that the maximum relative errors are 1-3 machine epsilons,… (More)

We developed the numerical procedures to evaluate the inverse functions of the complete elliptic integrals of the first and second kind, K(m) and E(m), with respect to the parameter m. The evaluation is executed by inverting eight sets of the truncated Taylor series expansions of the integrals in terms of m or of − log(1 − m). The developed procedures are… (More)