# Elliptic Integrals¶

Information about the elliptic integrals can be found in Abramowitz & Stegun, Chapter 17.

## Definition of Legendre Forms¶

The Legendre forms of elliptic integrals $$F(\phi,k)$$, $$E(\phi,k)$$ and $$\Pi(\phi,k,n)$$ are defined by,

$\begin{split}F(\phi,k) = \int_0^\phi 1/\sqrt{1 - k^2 \sin^2(t)} dt \\ E(\phi,k) = \int_0^\phi \sqrt{1 - k^2 \sin^2(t)} dt \\ \Pi(\phi,k,n) = \int_0^\phi 1/((1 + n \sin^2(t))\sqrt{1 - k^2 \sin^2(t)}) dt\end{split}$

The complete Legendre forms are denoted by $$K(k) = F(\pi/2, k)$$ and $$E(k) = E(\pi/2, k)$$.

The notation used here is based on Carlson, Numerische Mathematik 33 (1979) 1 and differs slightly from that used by Abramowitz & Stegun, where the functions are given in terms of the parameter $$m = k^2$$ and $$n$$ is replaced by $$-n$$.

## Definition of Carlson Forms¶

The Carlson symmetric forms of elliptical integrals $$RC(x,y)$$, $$RD(x,y,z)$$, $$RF(x,y,z)$$ and $$RJ(x,y,z,p)$$ are defined by,

$\begin{split}RC(x,y) = 1/2 \int_0^\infty (t+x)^{-1/2} (t+y)^{-1} dt \\ RD(x,y,z) = 3/2 \int_0^\infty (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-3/2} dt \\ RF(x,y,z) = 1/2 \int_0^\infty (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2} dt \\ RJ(x,y,z,p) = 3/2 \int_0^\infty (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2} (t+p)^{-1} dt\end{split}$

## Legendre Form of Complete Elliptic Integrals¶

gsl_sf_ellint_Kcomp(k)

This routine computes the complete elliptic integral $$K(k)$$. Note that Abramowitz & Stegun define this function in terms of the parameter $$m = k^2$$.

gsl_sf_ellint_Ecomp(k)

This routine computes the complete elliptic integral $$E(k)$$. Note that Abramowitz & Stegun define this function in terms of the parameter $$m = k^2$$.

gsl_sf_ellint_Pcomp(k, n)

This routine computes the complete elliptic integral $$\Pi(k,n)$$. Note that Abramowitz & Stegun define this function in terms of the parameters $$m = k^2$$ and $$\sin^2(\alpha) = k^2$$, with the change of sign $$n \to -n$$.

## Legendre Form of Incomplete Elliptic Integrals¶

gsl_sf_ellint_F(phi, k)

This routine computes the incomplete elliptic integral $$F(\phi,k)$$. Note that Abramowitz & Stegun define this function in terms of the parameter $$m = k^2$$.

gsl_sf_ellint_E(phi, k)

This routine computes the incomplete elliptic integral $$E(\phi,k)$$. Note that Abramowitz & Stegun define this function in terms of the parameter $$m = k^2$$.

gsl_sf_ellint_P(phi, k, n)

This routine computes the incomplete elliptic integral $$\Pi(\phi,k,n)$$. Note that Abramowitz & Stegun define this function in terms of the parameters $$m = k^2$$ and $$\sin^2(\alpha) = k^2$$, with the change of sign $$n \to -n$$.

gsl_sf_ellint_D(phi, k)

This routine computes the incomplete elliptic integral $$D(\phi,k)$$ which is defined through the Carlson form $$RD(x,y,z)$$ by the following relation,

$D(\phi,k) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).$

## Carlson Forms¶

gsl_sf_ellint_RC(x, y)

This routine computes the incomplete elliptic integral $$RC(x,y)$$.

gsl_sf_ellint_RD(x, y, z)

This routine computes the incomplete elliptic integral $$RD(x,y,z)$$.

gsl_sf_ellint_RF(x, y, z)

This routine computes the incomplete elliptic integral $$RF(x,y,z)$$.

gsl_sf_ellint_RJ(x, y, z, p)

This routine computes the incomplete elliptic integral $$RJ(x,y,z,p)$$.