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 \(RJ(x,y,z,p)\).

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