src/parsec/disk-image/parsec/parsec-benchmark/pkgs/libs/gsl/src/doc/specfunc-legendre.texi - public/gem5-resources - Git at Google

 @cindex Legendre functions
 @cindex spherical harmonics
 @cindex conical functions
 @cindex hyperbolic space

 The Legendre Functions and Legendre Polynomials are described in
 Abramowitz & Stegun, Chapter 8.  These functions are declared in
 the header file @file{gsl_sf_legendre.h}.

 @menu
 * Legendre Polynomials::
 * Associated Legendre Polynomials and Spherical Harmonics::
 * Conical Functions::
 * Radial Functions for Hyperbolic Space::
 @end menu

 @node Legendre Polynomials
 @subsection Legendre Polynomials

 @deftypefun double gsl_sf_legendre_P1 (double @var{x})
 @deftypefunx double gsl_sf_legendre_P2 (double @var{x})
 @deftypefunx double gsl_sf_legendre_P3 (double @var{x})
 @deftypefunx int gsl_sf_legendre_P1_e (double @var{x}, gsl_sf_result * @var{result})
 @deftypefunx int gsl_sf_legendre_P2_e (double @var{x}, gsl_sf_result * @var{result})
 @deftypefunx int gsl_sf_legendre_P3_e (double @var{x}, gsl_sf_result * @var{result})
 These functions evaluate the Legendre polynomials
 @c{$P_l(x)$}
 @math{P_l(x)} using explicit
 representations for @math{l=1, 2, 3}.
 @comment Exceptional Return Values: none
 @end deftypefun

 @deftypefun double gsl_sf_legendre_Pl (int @var{l}, double @var{x})
 @deftypefunx int gsl_sf_legendre_Pl_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
 These functions evaluate the Legendre polynomial @c{$P_l(x)$}
 @math{P_l(x)} for a specific value of @var{l},
 @var{x} subject to @c{$l \ge 0$}
 @math{l >= 0},
 @c{$|x| \le 1$}
 @math{|x| <= 1}
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun int gsl_sf_legendre_Pl_array (int @var{lmax}, double @var{x}, double @var{result_array}[])
 @deftypefunx int gsl_sf_legendre_Pl_deriv_array (int @var{lmax}, double @var{x}, double @var{result_array}[], double @var{result_deriv_array}[])

 These functions compute an array of Legendre polynomials
 @math{P_l(x)}, and optionally their derivatives @math{dP_l(x)/dx},
 for @math{l = 0, \dots, lmax},
 @c{$|x| \le 1$}
 @math{|x| <= 1}
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun


 @deftypefun double gsl_sf_legendre_Q0 (double @var{x})
 @deftypefunx int gsl_sf_legendre_Q0_e (double @var{x}, gsl_sf_result * @var{result})
 These routines compute the Legendre function @math{Q_0(x)} for @math{x >
 -1}, @c{$x \ne 1$}
 @math{x != 1}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun


 @deftypefun double gsl_sf_legendre_Q1 (double @var{x})
 @deftypefunx int gsl_sf_legendre_Q1_e (double @var{x}, gsl_sf_result * @var{result})
 These routines compute the Legendre function @math{Q_1(x)} for @math{x >
 -1}, @c{$x \ne 1$}
 @math{x != 1}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun double gsl_sf_legendre_Ql (int @var{l}, double @var{x})
 @deftypefunx int gsl_sf_legendre_Ql_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
 These routines compute the Legendre function @math{Q_l(x)} for @math{x >
 -1}, @c{$x \ne 1$}
 @math{x != 1} and @c{$l \ge 0$}
 @math{l >= 0}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun


 @node Associated Legendre Polynomials and Spherical Harmonics
 @subsection Associated Legendre Polynomials and Spherical Harmonics

 The following functions compute the associated Legendre Polynomials
 @math{P_l^m(x)}.  Note that this function grows combinatorially with
 @math{l} and can overflow for @math{l} larger than about 150.  There is
 no trouble for small @math{m}, but overflow occurs when @math{m} and
 @math{l} are both large.  Rather than allow overflows, these functions
 refuse to calculate @math{P_l^m(x)} and return @code{GSL_EOVRFLW} when
 they can sense that @math{l} and @math{m} are too big.

 If you want to calculate a spherical harmonic, then @emph{do not} use
 these functions.  Instead use @code{gsl_sf_legendre_sphPlm} below,
 which uses a similar recursion, but with the normalized functions.

 @deftypefun double gsl_sf_legendre_Plm (int @var{l}, int @var{m}, double @var{x})
 @deftypefunx int gsl_sf_legendre_Plm_e (int @var{l}, int @var{m}, double @var{x}, gsl_sf_result * @var{result})
 These routines compute the associated Legendre polynomial
 @math{P_l^m(x)} for @c{$m \ge 0$}
 @math{m >= 0}, @c{$l \ge m$}
 @math{l >= m}, @c{$|x| \le 1$}
 @math{|x| <= 1}.
 @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
 @end deftypefun

 @deftypefun int gsl_sf_legendre_Plm_array (int @var{lmax}, int @var{m}, double @var{x}, double @var{result_array}[])
 @deftypefunx int gsl_sf_legendre_Plm_deriv_array (int @var{lmax}, int @var{m}, double @var{x}, double @var{result_array}[], double @var{result_deriv_array}[])
 These functions compute an array of Legendre polynomials
 @math{P_l^m(x)}, and optionally their derivatives @math{dP_l^m(x)/dx},
 for @c{$m \ge 0$}
 @math{m >= 0}, @c{$l = |m|, \dots, lmax$}
 @math{l = |m|, ..., lmax}, @c{$|x| \le 1$}
 @math{|x| <= 1}.
 @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
 @end deftypefun


 @deftypefun double gsl_sf_legendre_sphPlm (int @var{l}, int @var{m}, double @var{x})
 @deftypefunx int gsl_sf_legendre_sphPlm_e (int @var{l}, int @var{m}, double @var{x}, gsl_sf_result * @var{result})
 These routines compute the normalized associated Legendre polynomial
 @c{$\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$}
 @math{$\sqrt@{(2l+1)/(4\pi)@} \sqrt@{(l-m)!/(l+m)!@} P_l^m(x)$} suitable
 for use in spherical harmonics.  The parameters must satisfy @c{$m \ge 0$}
 @math{m >= 0}, @c{$l \ge m$}
 @math{l >= m}, @c{$|x| \le 1$}
 @math{|x| <= 1}. Theses routines avoid the overflows
 that occur for the standard normalization of @math{P_l^m(x)}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun int gsl_sf_legendre_sphPlm_array (int @var{lmax}, int @var{m}, double @var{x}, double @var{result_array}[])
 @deftypefunx int gsl_sf_legendre_sphPlm_deriv_array (int @var{lmax}, int @var{m}, double @var{x}, double @var{result_array}[], double @var{result_deriv_array}[])
 These functions compute an array of normalized associated Legendre functions
 @c{$\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$}
 @math{$\sqrt@{(2l+1)/(4\pi)@} \sqrt@{(l-m)!/(l+m)!@} P_l^m(x)$},
 and optionally their derivatives,
 for @c{$m \ge 0$}
 @math{m >= 0}, @c{$l = |m|, \dots, lmax$}
 @math{l = |m|, ..., lmax}, @c{$|x| \le 1$}
 @math{|x| <= 1.0}
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun int gsl_sf_legendre_array_size (const int @var{lmax}, const int @var{m})
 This function returns the size of @var{result_array}[] needed for the array
 versions of @math{P_l^m(x)}, @math{@var{lmax} - @var{m} + 1}.
 @comment Exceptional Return Values: none
 @end deftypefun

 @node Conical Functions
 @subsection Conical Functions

 The Conical Functions @c{$P^\mu_{-(1/2)+i\lambda}(x)$}
 @math{P^\mu_@{-(1/2)+i\lambda@}(x)} and @c{$Q^\mu_{-(1/2)+i\lambda}$}
 @math{Q^\mu_@{-(1/2)+i\lambda@}}
 are described in Abramowitz & Stegun, Section 8.12.

 @deftypefun double gsl_sf_conicalP_half (double @var{lambda}, double @var{x})
 @deftypefunx int gsl_sf_conicalP_half_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
 These routines compute the irregular Spherical Conical Function
 @c{$P^{1/2}_{-1/2 + i \lambda}(x)$}
 @math{P^@{1/2@}_@{-1/2 + i \lambda@}(x)} for @math{x > -1}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun double gsl_sf_conicalP_mhalf (double @var{lambda}, double @var{x})
 @deftypefunx int gsl_sf_conicalP_mhalf_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
 These routines compute the regular Spherical Conical Function
 @c{$P^{-1/2}_{-1/2 + i \lambda}(x)$}
 @math{P^@{-1/2@}_@{-1/2 + i \lambda@}(x)} for @math{x > -1}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun double gsl_sf_conicalP_0 (double @var{lambda}, double @var{x})
 @deftypefunx int gsl_sf_conicalP_0_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
 These routines compute the conical function
 @c{$P^0_{-1/2 + i \lambda}(x)$}
 @math{P^0_@{-1/2 + i \lambda@}(x)}
 for @math{x > -1}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun


 @deftypefun double gsl_sf_conicalP_1 (double @var{lambda}, double @var{x})
 @deftypefunx int gsl_sf_conicalP_1_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
 These routines compute the conical function
 @c{$P^1_{-1/2 + i \lambda}(x)$}
 @math{P^1_@{-1/2 + i \lambda@}(x)} for @math{x > -1}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun


 @deftypefun double gsl_sf_conicalP_sph_reg (int @var{l}, double @var{lambda}, double @var{x})
 @deftypefunx int gsl_sf_conicalP_sph_reg_e (int @var{l}, double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
 These routines compute the Regular Spherical Conical Function
 @c{$P^{-1/2-l}_{-1/2 + i \lambda}(x)$}
 @math{P^@{-1/2-l@}_@{-1/2 + i \lambda@}(x)} for @math{x > -1}, @c{$l \ge -1$}
 @math{l >= -1}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun


 @deftypefun double gsl_sf_conicalP_cyl_reg (int @var{m}, double @var{lambda}, double @var{x})
 @deftypefunx int gsl_sf_conicalP_cyl_reg_e (int @var{m}, double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
 These routines compute the Regular Cylindrical Conical Function
 @c{$P^{-m}_{-1/2 + i \lambda}(x)$}
 @math{P^@{-m@}_@{-1/2 + i \lambda@}(x)} for @math{x > -1}, @c{$m \ge -1$}
 @math{m >= -1}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun


 @node Radial Functions for Hyperbolic Space
 @subsection Radial Functions for Hyperbolic Space

 The following spherical functions are specializations of Legendre
 functions which give the regular eigenfunctions of the Laplacian on a
 3-dimensional hyperbolic space @math{H3d}.  Of particular interest is
 the flat limit, @math{\lambda \to \infty}, @math{\eta \to 0},
 @math{\lambda\eta} fixed.

 @deftypefun double gsl_sf_legendre_H3d_0 (double @var{lambda}, double @var{eta})
 @deftypefunx int gsl_sf_legendre_H3d_0_e (double @var{lambda}, double @var{eta}, gsl_sf_result * @var{result})
 These routines compute the zeroth radial eigenfunction of the Laplacian on the
 3-dimensional hyperbolic space,
 @c{$$L^{H3d}_0(\lambda,\eta) := {\sin(\lambda\eta) \over \lambda\sinh(\eta)}$$}
 @math{L^@{H3d@}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta))}
 for @c{$\eta \ge 0$}
 @math{\eta >= 0}.
 In the flat limit this takes the form
 @c{$L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta)$}
 @math{L^@{H3d@}_0(\lambda,\eta) = j_0(\lambda\eta)}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun double gsl_sf_legendre_H3d_1 (double @var{lambda}, double @var{eta})
 @deftypefunx int gsl_sf_legendre_H3d_1_e (double @var{lambda}, double @var{eta}, gsl_sf_result * @var{result})
 These routines compute the first radial eigenfunction of the Laplacian on
 the 3-dimensional hyperbolic space,
 @c{$$L^{H3d}_1(\lambda,\eta) := {1\over\sqrt{\lambda^2 + 1}} {\left(\sin(\lambda \eta)\over \lambda \sinh(\eta)\right)} \left(\coth(\eta) - \lambda \cot(\lambda\eta)\right)$$}
 @math{L^@{H3d@}_1(\lambda,\eta) := 1/\sqrt@{\lambda^2 + 1@} \sin(\lambda \eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda \cot(\lambda\eta))}
 for @c{$\eta \ge 0$}
 @math{\eta >= 0}.
 In the flat limit this takes the form
 @c{$L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta)$}
 @math{L^@{H3d@}_1(\lambda,\eta) = j_1(\lambda\eta)}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun double gsl_sf_legendre_H3d (int @var{l}, double @var{lambda}, double @var{eta})
 @deftypefunx int gsl_sf_legendre_H3d_e (int @var{l}, double @var{lambda}, double @var{eta}, gsl_sf_result * @var{result})
 These routines compute the @var{l}-th radial eigenfunction of the
 Laplacian on the 3-dimensional hyperbolic space @c{$\eta \ge 0$}
 @math{\eta >= 0}, @c{$l \ge 0$}
 @math{l >= 0}. In the flat limit this takes the form
 @c{$L^{H3d}_l(\lambda,\eta) = j_l(\lambda\eta)$}
 @math{L^@{H3d@}_l(\lambda,\eta) = j_l(\lambda\eta)}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun int gsl_sf_legendre_H3d_array (int @var{lmax}, double @var{lambda}, double @var{eta}, double @var{result_array}[])
 This function computes an array of radial eigenfunctions
 @c{$L^{H3d}_l( \lambda, \eta)$}
 @math{L^@{H3d@}_l(\lambda, \eta)}
 for @c{$0 \le l \le lmax$}
 @math{0 <= l <= lmax}.
 @comment Exceptional Return Values:
 @end deftypefun
	@cindex Legendre functions
	@cindex spherical harmonics
	@cindex conical functions
	@cindex hyperbolic space

	The Legendre Functions and Legendre Polynomials are described in
	Abramowitz & Stegun, Chapter 8. These functions are declared in
	the header file @file{gsl_sf_legendre.h}.

	@menu
	* Legendre Polynomials::
	* Associated Legendre Polynomials and Spherical Harmonics::
	* Conical Functions::
	* Radial Functions for Hyperbolic Space::
	@end menu

	@node Legendre Polynomials
	@subsection Legendre Polynomials

	@deftypefun double gsl_sf_legendre_P1 (double @var{x})
	@deftypefunx double gsl_sf_legendre_P2 (double @var{x})
	@deftypefunx double gsl_sf_legendre_P3 (double @var{x})
	@deftypefunx int gsl_sf_legendre_P1_e (double @var{x}, gsl_sf_result * @var{result})
	@deftypefunx int gsl_sf_legendre_P2_e (double @var{x}, gsl_sf_result * @var{result})
	@deftypefunx int gsl_sf_legendre_P3_e (double @var{x}, gsl_sf_result * @var{result})
	These functions evaluate the Legendre polynomials
	@c{$P_l(x)$}
	@math{P_l(x)} using explicit
	representations for @math{l=1, 2, 3}.
	@comment Exceptional Return Values: none
	@end deftypefun

	@deftypefun double gsl_sf_legendre_Pl (int @var{l}, double @var{x})
	@deftypefunx int gsl_sf_legendre_Pl_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
	These functions evaluate the Legendre polynomial @c{$P_l(x)$}
	@math{P_l(x)} for a specific value of @var{l},
	@var{x} subject to @c{$l \ge 0$}
	@math{l >= 0},
	@c{$\|x\| \le 1$}
	@math{\|x\| <= 1}
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun int gsl_sf_legendre_Pl_array (int @var{lmax}, double @var{x}, double @var{result_array}[])
	@deftypefunx int gsl_sf_legendre_Pl_deriv_array (int @var{lmax}, double @var{x}, double @var{result_array}[], double @var{result_deriv_array}[])

	These functions compute an array of Legendre polynomials
	@math{P_l(x)}, and optionally their derivatives @math{dP_l(x)/dx},
	for @math{l = 0, \dots, lmax},
	@c{$\|x\| \le 1$}
	@math{\|x\| <= 1}
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun


	@deftypefun double gsl_sf_legendre_Q0 (double @var{x})
	@deftypefunx int gsl_sf_legendre_Q0_e (double @var{x}, gsl_sf_result * @var{result})
	These routines compute the Legendre function @math{Q_0(x)} for @math{x >
	-1}, @c{$x \ne 1$}
	@math{x != 1}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun


	@deftypefun double gsl_sf_legendre_Q1 (double @var{x})
	@deftypefunx int gsl_sf_legendre_Q1_e (double @var{x}, gsl_sf_result * @var{result})
	These routines compute the Legendre function @math{Q_1(x)} for @math{x >
	-1}, @c{$x \ne 1$}
	@math{x != 1}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun double gsl_sf_legendre_Ql (int @var{l}, double @var{x})
	@deftypefunx int gsl_sf_legendre_Ql_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
	These routines compute the Legendre function @math{Q_l(x)} for @math{x >
	-1}, @c{$x \ne 1$}
	@math{x != 1} and @c{$l \ge 0$}
	@math{l >= 0}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun


	@node Associated Legendre Polynomials and Spherical Harmonics
	@subsection Associated Legendre Polynomials and Spherical Harmonics

	The following functions compute the associated Legendre Polynomials
	@math{P_l^m(x)}. Note that this function grows combinatorially with
	@math{l} and can overflow for @math{l} larger than about 150. There is
	no trouble for small @math{m}, but overflow occurs when @math{m} and
	@math{l} are both large. Rather than allow overflows, these functions
	refuse to calculate @math{P_l^m(x)} and return @code{GSL_EOVRFLW} when
	they can sense that @math{l} and @math{m} are too big.

	If you want to calculate a spherical harmonic, then @emph{do not} use
	these functions. Instead use @code{gsl_sf_legendre_sphPlm} below,
	which uses a similar recursion, but with the normalized functions.

	@deftypefun double gsl_sf_legendre_Plm (int @var{l}, int @var{m}, double @var{x})
	@deftypefunx int gsl_sf_legendre_Plm_e (int @var{l}, int @var{m}, double @var{x}, gsl_sf_result * @var{result})
	These routines compute the associated Legendre polynomial
	@math{P_l^m(x)} for @c{$m \ge 0$}
	@math{m >= 0}, @c{$l \ge m$}
	@math{l >= m}, @c{$\|x\| \le 1$}
	@math{\|x\| <= 1}.
	@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
	@end deftypefun

	@deftypefun int gsl_sf_legendre_Plm_array (int @var{lmax}, int @var{m}, double @var{x}, double @var{result_array}[])
	@deftypefunx int gsl_sf_legendre_Plm_deriv_array (int @var{lmax}, int @var{m}, double @var{x}, double @var{result_array}[], double @var{result_deriv_array}[])
	These functions compute an array of Legendre polynomials
	@math{P_l^m(x)}, and optionally their derivatives @math{dP_l^m(x)/dx},
	for @c{$m \ge 0$}
	@math{m >= 0}, @c{$l = \|m\|, \dots, lmax$}
	@math{l = \|m\|, ..., lmax}, @c{$\|x\| \le 1$}
	@math{\|x\| <= 1}.
	@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
	@end deftypefun


	@deftypefun double gsl_sf_legendre_sphPlm (int @var{l}, int @var{m}, double @var{x})
	@deftypefunx int gsl_sf_legendre_sphPlm_e (int @var{l}, int @var{m}, double @var{x}, gsl_sf_result * @var{result})
	These routines compute the normalized associated Legendre polynomial
	@c{$\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$}
	@math{$\sqrt@{(2l+1)/(4\pi)@} \sqrt@{(l-m)!/(l+m)!@} P_l^m(x)$} suitable
	for use in spherical harmonics. The parameters must satisfy @c{$m \ge 0$}
	@math{m >= 0}, @c{$l \ge m$}
	@math{l >= m}, @c{$\|x\| \le 1$}
	@math{\|x\| <= 1}. Theses routines avoid the overflows
	that occur for the standard normalization of @math{P_l^m(x)}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun int gsl_sf_legendre_sphPlm_array (int @var{lmax}, int @var{m}, double @var{x}, double @var{result_array}[])
	@deftypefunx int gsl_sf_legendre_sphPlm_deriv_array (int @var{lmax}, int @var{m}, double @var{x}, double @var{result_array}[], double @var{result_deriv_array}[])
	These functions compute an array of normalized associated Legendre functions
	@c{$\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$}
	@math{$\sqrt@{(2l+1)/(4\pi)@} \sqrt@{(l-m)!/(l+m)!@} P_l^m(x)$},
	and optionally their derivatives,
	for @c{$m \ge 0$}
	@math{m >= 0}, @c{$l = \|m\|, \dots, lmax$}
	@math{l = \|m\|, ..., lmax}, @c{$\|x\| \le 1$}
	@math{\|x\| <= 1.0}
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun int gsl_sf_legendre_array_size (const int @var{lmax}, const int @var{m})
	This function returns the size of @var{result_array}[] needed for the array
	versions of @math{P_l^m(x)}, @math{@var{lmax} - @var{m} + 1}.
	@comment Exceptional Return Values: none
	@end deftypefun

	@node Conical Functions
	@subsection Conical Functions

	The Conical Functions @c{$P^\mu_{-(1/2)+i\lambda}(x)$}
	@math{P^\mu_@{-(1/2)+i\lambda@}(x)} and @c{$Q^\mu_{-(1/2)+i\lambda}$}
	@math{Q^\mu_@{-(1/2)+i\lambda@}}
	are described in Abramowitz & Stegun, Section 8.12.

	@deftypefun double gsl_sf_conicalP_half (double @var{lambda}, double @var{x})
	@deftypefunx int gsl_sf_conicalP_half_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
	These routines compute the irregular Spherical Conical Function
	@c{$P^{1/2}_{-1/2 + i \lambda}(x)$}
	@math{P^@{1/2@}_@{-1/2 + i \lambda@}(x)} for @math{x > -1}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun double gsl_sf_conicalP_mhalf (double @var{lambda}, double @var{x})
	@deftypefunx int gsl_sf_conicalP_mhalf_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
	These routines compute the regular Spherical Conical Function
	@c{$P^{-1/2}_{-1/2 + i \lambda}(x)$}
	@math{P^@{-1/2@}_@{-1/2 + i \lambda@}(x)} for @math{x > -1}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun double gsl_sf_conicalP_0 (double @var{lambda}, double @var{x})
	@deftypefunx int gsl_sf_conicalP_0_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
	These routines compute the conical function
	@c{$P^0_{-1/2 + i \lambda}(x)$}
	@math{P^0_@{-1/2 + i \lambda@}(x)}
	for @math{x > -1}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun


	@deftypefun double gsl_sf_conicalP_1 (double @var{lambda}, double @var{x})
	@deftypefunx int gsl_sf_conicalP_1_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
	These routines compute the conical function
	@c{$P^1_{-1/2 + i \lambda}(x)$}
	@math{P^1_@{-1/2 + i \lambda@}(x)} for @math{x > -1}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun


	@deftypefun double gsl_sf_conicalP_sph_reg (int @var{l}, double @var{lambda}, double @var{x})
	@deftypefunx int gsl_sf_conicalP_sph_reg_e (int @var{l}, double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
	These routines compute the Regular Spherical Conical Function
	@c{$P^{-1/2-l}_{-1/2 + i \lambda}(x)$}
	@math{P^@{-1/2-l@}_@{-1/2 + i \lambda@}(x)} for @math{x > -1}, @c{$l \ge -1$}
	@math{l >= -1}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun


	@deftypefun double gsl_sf_conicalP_cyl_reg (int @var{m}, double @var{lambda}, double @var{x})
	@deftypefunx int gsl_sf_conicalP_cyl_reg_e (int @var{m}, double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
	These routines compute the Regular Cylindrical Conical Function
	@c{$P^{-m}_{-1/2 + i \lambda}(x)$}
	@math{P^@{-m@}_@{-1/2 + i \lambda@}(x)} for @math{x > -1}, @c{$m \ge -1$}
	@math{m >= -1}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun



	@node Radial Functions for Hyperbolic Space
	@subsection Radial Functions for Hyperbolic Space

	The following spherical functions are specializations of Legendre
	functions which give the regular eigenfunctions of the Laplacian on a
	3-dimensional hyperbolic space @math{H3d}. Of particular interest is
	the flat limit, @math{\lambda \to \infty}, @math{\eta \to 0},
	@math{\lambda\eta} fixed.

	@deftypefun double gsl_sf_legendre_H3d_0 (double @var{lambda}, double @var{eta})
	@deftypefunx int gsl_sf_legendre_H3d_0_e (double @var{lambda}, double @var{eta}, gsl_sf_result * @var{result})
	These routines compute the zeroth radial eigenfunction of the Laplacian on the
	3-dimensional hyperbolic space,
	@c{$$L^{H3d}_0(\lambda,\eta) := {\sin(\lambda\eta) \over \lambda\sinh(\eta)}$$}
	@math{L^@{H3d@}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta))}
	for @c{$\eta \ge 0$}
	@math{\eta >= 0}.
	In the flat limit this takes the form
	@c{$L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta)$}
	@math{L^@{H3d@}_0(\lambda,\eta) = j_0(\lambda\eta)}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun double gsl_sf_legendre_H3d_1 (double @var{lambda}, double @var{eta})
	@deftypefunx int gsl_sf_legendre_H3d_1_e (double @var{lambda}, double @var{eta}, gsl_sf_result * @var{result})
	These routines compute the first radial eigenfunction of the Laplacian on
	the 3-dimensional hyperbolic space,
	@c{$$L^{H3d}_1(\lambda,\eta) := {1\over\sqrt{\lambda^2 + 1}} {\left(\sin(\lambda \eta)\over \lambda \sinh(\eta)\right)} \left(\coth(\eta) - \lambda \cot(\lambda\eta)\right)$$}
	@math{L^@{H3d@}_1(\lambda,\eta) := 1/\sqrt@{\lambda^2 + 1@} \sin(\lambda \eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda \cot(\lambda\eta))}
	for @c{$\eta \ge 0$}
	@math{\eta >= 0}.
	In the flat limit this takes the form
	@c{$L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta)$}
	@math{L^@{H3d@}_1(\lambda,\eta) = j_1(\lambda\eta)}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun double gsl_sf_legendre_H3d (int @var{l}, double @var{lambda}, double @var{eta})
	@deftypefunx int gsl_sf_legendre_H3d_e (int @var{l}, double @var{lambda}, double @var{eta}, gsl_sf_result * @var{result})
	These routines compute the @var{l}-th radial eigenfunction of the
	Laplacian on the 3-dimensional hyperbolic space @c{$\eta \ge 0$}
	@math{\eta >= 0}, @c{$l \ge 0$}
	@math{l >= 0}. In the flat limit this takes the form
	@c{$L^{H3d}_l(\lambda,\eta) = j_l(\lambda\eta)$}
	@math{L^@{H3d@}_l(\lambda,\eta) = j_l(\lambda\eta)}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun int gsl_sf_legendre_H3d_array (int @var{lmax}, double @var{lambda}, double @var{eta}, double @var{result_array}[])
	This function computes an array of radial eigenfunctions
	@c{$L^{H3d}_l( \lambda, \eta)$}
	@math{L^@{H3d@}_l(\lambda, \eta)}
	for @c{$0 \le l \le lmax$}
	@math{0 <= l <= lmax}.
	@comment Exceptional Return Values:
	@end deftypefun