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.. index::
single: hypergeometric functions
single: confluent hypergeometric functions
Hypergeometric functions are described in Abramowitz & Stegun, Chapters
13 and 15. These functions are declared in the header file
:file:`gsl_sf_hyperg.h`.
.. function:: double gsl_sf_hyperg_0F1 (double c, double x)
int gsl_sf_hyperg_0F1_e (double c, double x, gsl_sf_result * result)
These routines compute the hypergeometric function
.. only:: not texinfo
.. math:: {}_0F_1(c,x)
.. only:: texinfo
.. math:: 0F1(c,x)
.. It is related to Bessel functions
.. 0F1[c,x] =
.. Gamma[c] x^(1/2(1-c)) I_(c-1)(2 Sqrt[x])
.. Gamma[c] (-x)^(1/2(1-c)) J_(c-1)(2 Sqrt[-x])
.. exceptions: GSL_EOVRFLW, GSL_EUNDRFLW
.. function:: double gsl_sf_hyperg_1F1_int (int m, int n, double x)
int gsl_sf_hyperg_1F1_int_e (int m, int n, double x, gsl_sf_result * result)
These routines compute the confluent hypergeometric function
.. only:: not texinfo
.. math:: {}_1F_1(m,n,x) = M(m,n,x)
.. only:: texinfo
.. math:: 1F1(m,n,x) = M(m,n,x)
for integer parameters :data:`m`, :data:`n`.
.. exceptions:
.. function:: double gsl_sf_hyperg_1F1 (double a, double b, double x)
int gsl_sf_hyperg_1F1_e (double a, double b, double x, gsl_sf_result * result)
These routines compute the confluent hypergeometric function
.. only:: not texinfo
.. math:: {}_1F_1(a,b,x) = M(a,b,x)
.. only:: texinfo
.. math:: 1F1(a,b,x) = M(a,b,x)
for general parameters :data:`a`, :data:`b`.
.. exceptions:
.. function:: double gsl_sf_hyperg_U_int (int m, int n, double x)
int gsl_sf_hyperg_U_int_e (int m, int n, double x, gsl_sf_result * result)
These routines compute the confluent hypergeometric function
:math:`U(m,n,x)` for integer parameters :data:`m`, :data:`n`.
.. exceptions:
.. function:: int gsl_sf_hyperg_U_int_e10_e (int m, int n, double x, gsl_sf_result_e10 * result)
This routine computes the confluent hypergeometric function
:math:`U(m,n,x)` for integer parameters :data:`m`, :data:`n` using the
:type:`gsl_sf_result_e10` type to return a result with extended range.
.. function:: double gsl_sf_hyperg_U (double a, double b, double x)
int gsl_sf_hyperg_U_e (double a, double b, double x, gsl_sf_result * result)
These routines compute the confluent hypergeometric function :math:`U(a,b,x)`.
.. exceptions:
.. function:: int gsl_sf_hyperg_U_e10_e (double a, double b, double x, gsl_sf_result_e10 * result)
This routine computes the confluent hypergeometric function
:math:`U(a,b,x)` using the :type:`gsl_sf_result_e10` type to return a
result with extended range.
.. exceptions:
.. function:: double gsl_sf_hyperg_2F1 (double a, double b, double c, double x)
int gsl_sf_hyperg_2F1_e (double a, double b, double c, double x, gsl_sf_result * result)
These routines compute the Gauss hypergeometric function
.. only:: not texinfo
.. math:: {}_2F_1(a,b,c,x) = F(a,b,c,x)
.. only:: texinfo
.. math:: 2F1(a,b,c,x) = F(a,b,c,x)
for :math:`|x| < 1`. If the arguments :math:`(a,b,c,x)` are too close to a singularity then
the function can return the error code :macro:`GSL_EMAXITER` when the
series approximation converges too slowly. This occurs in the region of
:math:`x = 1`, :math:`c - a - b = m` for integer m.
.. exceptions:
.. function:: double gsl_sf_hyperg_2F1_conj (double aR, double aI, double c, double x)
int gsl_sf_hyperg_2F1_conj_e (double aR, double aI, double c, double x, gsl_sf_result * result)
These routines compute the Gauss hypergeometric function
.. only:: not texinfo
.. math:: {}_2F_1(a_R + i a_I, aR - i aI, c, x)
.. only:: texinfo
.. math:: 2F1(a_R + i a_I, aR - i aI, c, x)
with complex parameters for :math:`|x| < 1`.
.. function:: double gsl_sf_hyperg_2F1_renorm (double a, double b, double c, double x)
int gsl_sf_hyperg_2F1_renorm_e (double a, double b, double c, double x, gsl_sf_result * result)
These routines compute the renormalized Gauss hypergeometric function
.. only:: not texinfo
.. math:: {}_2F_1(a,b,c,x) / \Gamma(c)
.. only:: texinfo
.. math:: 2F1(a,b,c,x) / \Gamma(c)
for :math:`|x| < 1`.
.. exceptions:
.. function:: double gsl_sf_hyperg_2F1_conj_renorm (double aR, double aI, double c, double x)
int gsl_sf_hyperg_2F1_conj_renorm_e (double aR, double aI, double c, double x, gsl_sf_result * result)
These routines compute the renormalized Gauss hypergeometric function
.. only:: not texinfo
.. math:: {}_2F_1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c)
.. only:: texinfo
.. math:: 2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c)
for :math:`|x| < 1`.
.. exceptions:
.. function:: double gsl_sf_hyperg_2F0 (double a, double b, double x)
int gsl_sf_hyperg_2F0_e (double a, double b, double x, gsl_sf_result * result)
These routines compute the hypergeometric function
.. only:: not texinfo
.. math:: {}_2F_0(a,b,x)
.. only:: texinfo
.. math:: 2F0(a,b,x)
The series representation is a divergent hypergeometric series.
However, for :math:`x < 0` we have
.. only:: not texinfo
.. math:: {}_2F_0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)
.. only:: texinfo
.. math:: 2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)
.. exceptions: GSL_EDOM
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