.. index:: elliptic integrals
The functions described in this section are declared in the header
file :file:`gsl_sf_ellint.h`. Further information about the elliptic
integrals can be found in Abramowitz & Stegun, Chapter 17.
Definition of Legendre Forms
----------------------------
.. index:: Legendre forms of elliptic integrals
The Legendre forms of elliptic integrals :math:`F(\phi,k)`,
:math:`E(\phi,k)` and :math:`\Pi(\phi,k,n)` are defined by,
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.. math::
F(\phi,k) &= \int_0^\phi dt {1 \over \sqrt{(1 - k^2 \sin^2(t))}} \\
E(\phi,k) &= \int_0^\phi dt \sqrt{(1 - k^2 \sin^2(t))} \\
\Pi(\phi,k,n) &= \int_0^\phi dt {1 \over (1 + n \sin^2(t)) \sqrt{1 - k^2 \sin^2(t)}}
.. only:: texinfo
::
F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))
E(\phi,k) = \int_0^\phi dt \sqrt((1 - k^2 \sin^2(t)))
Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))
The complete Legendre forms are denoted by :math:`K(k) = F(\pi/2, k)` and
:math:`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 :math:`m = k^2` and :math:`n` is replaced by :math:`-n`.
Definition of Carlson Forms
---------------------------
.. index:: Carlson forms of Elliptic integrals
The Carlson symmetric forms of elliptical integrals :math:`RC(x,y)`,
:math:`RD(x,y,z)`, :math:`RF(x,y,z)` and :math:`RJ(x,y,z,p)` are defined
by,
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.. math::
RC(x,y) &= 1/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1} \\
RD(x,y,z) &= 3/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-3/2} \\
RF(x,y,z) &= 1/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2} \\
RJ(x,y,z,p) &= 3/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2} (t+p)^{-1}
.. only:: texinfo
::
RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)
RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)
RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)
RJ(x,y,z,p) = 3/2 \int_0^\infty dt
(t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)
Legendre Form of Complete Elliptic Integrals
--------------------------------------------
.. function:: double gsl_sf_ellint_Kcomp (double k, gsl_mode_t mode)
int gsl_sf_ellint_Kcomp_e (double k, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the complete elliptic integral :math:`K(k)` to
the accuracy specified by the mode variable :data:`mode`.
Note that Abramowitz & Stegun define this function in terms of the
parameter :math:`m = k^2`.
.. Exceptional Return Values: GSL_EDOM
.. function:: double gsl_sf_ellint_Ecomp (double k, gsl_mode_t mode)
int gsl_sf_ellint_Ecomp_e (double k, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the complete elliptic integral :math:`E(k)` to the
accuracy specified by the mode variable :data:`mode`.
Note that Abramowitz & Stegun define this function in terms of the
parameter :math:`m = k^2`.
.. Exceptional Return Values: GSL_EDOM
.. function:: double gsl_sf_ellint_Pcomp (double k, double n, gsl_mode_t mode)
int gsl_sf_ellint_Pcomp_e (double k, double n, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the complete elliptic integral :math:`\Pi(k,n)` to the
accuracy specified by the mode variable :data:`mode`.
Note that Abramowitz & Stegun define this function in terms of the
parameters :math:`m = k^2` and :math:`\sin^2(\alpha) = k^2`, with the
change of sign :math:`n \to -n`.
.. Exceptional Return Values: GSL_EDOM
Legendre Form of Incomplete Elliptic Integrals
----------------------------------------------
.. function:: double gsl_sf_ellint_F (double phi, double k, gsl_mode_t mode)
int gsl_sf_ellint_F_e (double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral :math:`F(\phi,k)`
to the accuracy specified by the mode variable :data:`mode`.
Note that Abramowitz & Stegun define this function in terms of the
parameter :math:`m = k^2`.
.. Exceptional Return Values: GSL_EDOM
.. function:: double gsl_sf_ellint_E (double phi, double k, gsl_mode_t mode)
int gsl_sf_ellint_E_e (double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral :math:`E(\phi,k)`
to the accuracy specified by the mode variable :data:`mode`.
Note that Abramowitz & Stegun define this function in terms of the
parameter :math:`m = k^2`.
.. Exceptional Return Values: GSL_EDOM
.. function:: double gsl_sf_ellint_P (double phi, double k, double n, gsl_mode_t mode)
int gsl_sf_ellint_P_e (double phi, double k, double n, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral :math:`\Pi(\phi,k,n)`
to the accuracy specified by the mode variable :data:`mode`.
Note that Abramowitz & Stegun define this function in terms of the
parameters :math:`m = k^2` and :math:`\sin^2(\alpha) = k^2`, with the
change of sign :math:`n \to -n`.
.. Exceptional Return Values: GSL_EDOM
.. function:: double gsl_sf_ellint_D (double phi, double k, gsl_mode_t mode)
int gsl_sf_ellint_D_e (double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
These functions compute the incomplete elliptic integral
:math:`D(\phi,k)` which is defined through the Carlson form :math:`RD(x,y,z)`
by the following relation,
.. only:: not texinfo
.. math:: D(\phi,k) = {1 \over 3} (\sin \phi)^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1)
.. only:: texinfo
::
D(\phi,k) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).
.. Exceptional Return Values: GSL_EDOM
Carlson Forms
-------------
.. function:: double gsl_sf_ellint_RC (double x, double y, gsl_mode_t mode)
int gsl_sf_ellint_RC_e (double x, double y, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral :math:`RC(x,y)`
to the accuracy specified by the mode variable :data:`mode`.
.. Exceptional Return Values: GSL_EDOM
.. function:: double gsl_sf_ellint_RD (double x, double y, double z, gsl_mode_t mode)
int gsl_sf_ellint_RD_e (double x, double y, double z, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral :math:`RD(x,y,z)`
to the accuracy specified by the mode variable :data:`mode`.
.. Exceptional Return Values: GSL_EDOM
.. function:: double gsl_sf_ellint_RF (double x, double y, double z, gsl_mode_t mode)
int gsl_sf_ellint_RF_e (double x, double y, double z, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral :math:`RF(x,y,z)`
to the accuracy specified by the mode variable :data:`mode`.
.. Exceptional Return Values: GSL_EDOM
.. function:: double gsl_sf_ellint_RJ (double x, double y, double z, double p, gsl_mode_t mode)
int gsl_sf_ellint_RJ_e (double x, double y, double z, double p, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral :math:`RJ(x,y,z,p)`
to the accuracy specified by the mode variable :data:`mode`.
.. Exceptional Return Values: GSL_EDOM