# Elliptic integrals

1. Feb 20, 2008

### rsr_life

Hello,

I have the following elliptic integral:

$$\phi_{1}$$
$$\int \frac{d\theta}{\sqrt{1-\frac{sin^{2}\theta}{cos^{2}\phi_{1}}}}$$ .... (1)
$$\phi$$

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

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

(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 $$\phi$$?

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

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

Is there another method that I could use?

Thanks

Last edited: Feb 20, 2008