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diff --git a/gsl-1.9/doc/specfunc-ellint.texi b/gsl-1.9/doc/specfunc-ellint.texi new file mode 100644 index 0000000..601f9de --- /dev/null +++ b/gsl-1.9/doc/specfunc-ellint.texi @@ -0,0 +1,198 @@ +@cindex 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. + +@menu +* Definition of Legendre Forms:: +* Definition of Carlson Forms:: +* Legendre Form of Complete Elliptic Integrals:: +* Legendre Form of Incomplete Elliptic Integrals:: +* Carlson Forms:: +@end menu + +@node Definition of Legendre Forms +@subsection Definition of Legendre Forms +@cindex 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, +@tex +\beforedisplay +$$ +\eqalign{ +F(\phi,k) &= \int_0^\phi dt {1 \over \sqrt{(1 - k^2 \sin^2(t))}}\cr +E(\phi,k) &= \int_0^\phi dt \sqrt{(1 - k^2 \sin^2(t))}\cr +\Pi(\phi,k,n) &= \int_0^\phi dt {1 \over (1 + n \sin^2(t)) \sqrt{1 - k^2 \sin^2(t)}} +} +$$ +\afterdisplay +@end tex +@ifinfo + +@example + 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))) +@end example + +@end ifinfo +@noindent +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, @cite{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}. + +@node Definition of Carlson Forms +@subsection Definition of Carlson Forms +@cindex 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, +@tex +\beforedisplay +$$ +\eqalign{ +RC(x,y) &= 1/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1}\cr +RD(x,y,z) &= 3/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-3/2}\cr +RF(x,y,z) &= 1/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2}\cr +RJ(x,y,z,p) &= 3/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2} (t+p)^{-1} +} +$$ +\afterdisplay +@end tex +@ifinfo + +@example + 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) +@end example +@end ifinfo + +@node Legendre Form of Complete Elliptic Integrals +@subsection Legendre Form of Complete Elliptic Integrals + +@deftypefun double gsl_sf_ellint_Kcomp (double @var{k}, gsl_mode_t @var{mode}) +@deftypefunx int gsl_sf_ellint_Kcomp_e (double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) +These routines compute the complete elliptic integral @math{K(k)} to +the accuracy specified by the mode variable @var{mode}. +Note that Abramowitz & Stegun define this function in terms of the +parameter @math{m = k^2}. +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun + +@deftypefun double gsl_sf_ellint_Ecomp (double @var{k}, gsl_mode_t @var{mode}) +@deftypefunx int gsl_sf_ellint_Ecomp_e (double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) +These routines compute the complete elliptic integral @math{E(k)} to the +accuracy specified by the mode variable @var{mode}. +Note that Abramowitz & Stegun define this function in terms of the +parameter @math{m = k^2}. +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun + +@deftypefun double gsl_sf_ellint_Pcomp (double @var{k}, double @var{n}, gsl_mode_t @var{mode}) +@deftypefunx int gsl_sf_ellint_Pcomp_e (double @var{k}, double @var{n}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) +These routines compute the complete elliptic integral @math{\Pi(k,n)} to the +accuracy specified by the mode variable @var{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}. +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun + +@node Legendre Form of Incomplete Elliptic Integrals +@subsection Legendre Form of Incomplete Elliptic Integrals + +@deftypefun double gsl_sf_ellint_F (double @var{phi}, double @var{k}, gsl_mode_t @var{mode}) +@deftypefunx int gsl_sf_ellint_F_e (double @var{phi}, double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) +These routines compute the incomplete elliptic integral @math{F(\phi,k)} +to the accuracy specified by the mode variable @var{mode}. +Note that Abramowitz & Stegun define this function in terms of the +parameter @math{m = k^2}. +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun + +@deftypefun double gsl_sf_ellint_E (double @var{phi}, double @var{k}, gsl_mode_t @var{mode}) +@deftypefunx int gsl_sf_ellint_E_e (double @var{phi}, double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) +These routines compute the incomplete elliptic integral @math{E(\phi,k)} +to the accuracy specified by the mode variable @var{mode}. +Note that Abramowitz & Stegun define this function in terms of the +parameter @math{m = k^2}. +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun + +@deftypefun double gsl_sf_ellint_P (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode}) +@deftypefunx int gsl_sf_ellint_P_e (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) +These routines compute the incomplete elliptic integral @math{\Pi(\phi,k,n)} +to the accuracy specified by the mode variable @var{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}. +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun + +@deftypefun double gsl_sf_ellint_D (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode}) +@deftypefunx int gsl_sf_ellint_D_e (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode}, gsl_sf_result * @var{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, +@tex +\beforedisplay +$$ +D(\phi,k,n) = {1 \over 3} (\sin \phi)^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1). +$$ +\afterdisplay +@end tex +@ifinfo + +@example +D(\phi,k,n) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1). +@end example +@end ifinfo +The argument @var{n} is not used and will be removed in a future release. + +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun + + +@node Carlson Forms +@subsection Carlson Forms + +@deftypefun double gsl_sf_ellint_RC (double @var{x}, double @var{y}, gsl_mode_t @var{mode}) +@deftypefunx int gsl_sf_ellint_RC_e (double @var{x}, double @var{y}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) +These routines compute the incomplete elliptic integral @math{RC(x,y)} +to the accuracy specified by the mode variable @var{mode}. +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun + +@deftypefun double gsl_sf_ellint_RD (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode}) +@deftypefunx int gsl_sf_ellint_RD_e (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) +These routines compute the incomplete elliptic integral @math{RD(x,y,z)} +to the accuracy specified by the mode variable @var{mode}. +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun + +@deftypefun double gsl_sf_ellint_RF (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode}) +@deftypefunx int gsl_sf_ellint_RF_e (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) +These routines compute the incomplete elliptic integral @math{RF(x,y,z)} +to the accuracy specified by the mode variable @var{mode}. +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun + +@deftypefun double gsl_sf_ellint_RJ (double @var{x}, double @var{y}, double @var{z}, double @var{p}, gsl_mode_t @var{mode}) +@deftypefunx int gsl_sf_ellint_RJ_e (double @var{x}, double @var{y}, double @var{z}, double @var{p}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result}) +These routines compute the incomplete elliptic integral @math{RJ(x,y,z,p)} +to the accuracy specified by the mode variable @var{mode}. +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun |