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I have the following elliptic integral:

[tex]\phi_{1}[/tex]

[tex]\int \frac{d\theta}{\sqrt{1-\frac{sin^{2}\theta}{cos^{2}\phi_{1}}}}[/tex] .... (1)

[tex]\phi[/tex]

The parameter m = ([tex]\frac{1}{cos^{2}\phi}[/tex]) is greater than one.

So, i know that the first incomplete elliptic integral F([tex]\phi[/tex],m>1) = m[tex]^{-2}[/tex] F([tex]\beta[/tex],m[tex]^{-1}[/tex]) with sin[tex]\beta[/tex] = m[tex]^{1/2}[/tex]sin([tex]\phi[/tex])

(Abramowitz, M. and Stegun, I. A. (Eds.). "Elliptic Integrals." Ch. 17 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587-607, 1972.)

How can I solve the above equation (1) for [tex]\phi[/tex]?

One method is to apply the Jacobian elliptic function sn[F([tex]\phi[/tex],m)|m] = sin([tex]\phi[/tex]).

This though, applies to regular functions where m<1. Since here m>1, how do I solve for [tex]\phi[/tex]?

Is there another method that I could use?

Thanks

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# Elliptic integrals

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