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<h3 class="section">7.21 Hypergeometric Functions</h3>
<p><a name="index-hypergeometric-functions-658"></a><a name="index-confluent-hypergeometric-functions-659"></a>
Hypergeometric functions are described in Abramowitz & Stegun, Chapters
13 and 15. These functions are declared in the header file
<samp><span class="file">gsl_sf_hyperg.h</span></samp>.
<div class="defun">
— Function: double <b>gsl_sf_hyperg_0F1</b> (<var>double c, double x</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f0F1-660"></a></var><br>
— Function: int <b>gsl_sf_hyperg_0F1_e</b> (<var>double c, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f0F1_005fe-661"></a></var><br>
<blockquote><p>These routines compute the hypergeometric function <!-- {${}_0F_1(c,x)$} -->
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 -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_hyperg_1F1_int</b> (<var>int m, int n, double x</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f1F1_005fint-662"></a></var><br>
— Function: int <b>gsl_sf_hyperg_1F1_int_e</b> (<var>int m, int n, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f1F1_005fint_005fe-663"></a></var><br>
<blockquote><p>These routines compute the confluent hypergeometric function
<!-- {${}_1F_1(m,n,x) = M(m,n,x)$} -->
1F1(m,n,x) = M(m,n,x) for integer parameters <var>m</var>, <var>n</var>.
<!-- exceptions: -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_hyperg_1F1</b> (<var>double a, double b, double x</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f1F1-664"></a></var><br>
— Function: int <b>gsl_sf_hyperg_1F1_e</b> (<var>double a, double b, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f1F1_005fe-665"></a></var><br>
<blockquote><p>These routines compute the confluent hypergeometric function
<!-- {${}_1F_1(a,b,x) = M(a,b,x)$} -->
1F1(a,b,x) = M(a,b,x) for general parameters <var>a</var>, <var>b</var>.
<!-- exceptions: -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_hyperg_U_int</b> (<var>int m, int n, double x</var>)<var><a name="index-gsl_005fsf_005fhyperg_005fU_005fint-666"></a></var><br>
— Function: int <b>gsl_sf_hyperg_U_int_e</b> (<var>int m, int n, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fhyperg_005fU_005fint_005fe-667"></a></var><br>
<blockquote><p>These routines compute the confluent hypergeometric function
U(m,n,x) for integer parameters <var>m</var>, <var>n</var>.
<!-- exceptions: -->
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_sf_hyperg_U_int_e10_e</b> (<var>int m, int n, double x, gsl_sf_result_e10 * result</var>)<var><a name="index-gsl_005fsf_005fhyperg_005fU_005fint_005fe10_005fe-668"></a></var><br>
<blockquote><p>This routine computes the confluent hypergeometric function
U(m,n,x) for integer parameters <var>m</var>, <var>n</var> using the
<code>gsl_sf_result_e10</code> type to return a result with extended range.
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_hyperg_U</b> (<var>double a, double b, double x</var>)<var><a name="index-gsl_005fsf_005fhyperg_005fU-669"></a></var><br>
— Function: int <b>gsl_sf_hyperg_U_e</b> (<var>double a, double b, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fhyperg_005fU_005fe-670"></a></var><br>
<blockquote><p>These routines compute the confluent hypergeometric function U(a,b,x).
<!-- exceptions: -->
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_sf_hyperg_U_e10_e</b> (<var>double a, double b, double x, gsl_sf_result_e10 * result</var>)<var><a name="index-gsl_005fsf_005fhyperg_005fU_005fe10_005fe-671"></a></var><br>
<blockquote><p>This routine computes the confluent hypergeometric function
U(a,b,x) using the <code>gsl_sf_result_e10</code> type to return a
result with extended range.
<!-- exceptions: -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_hyperg_2F1</b> (<var>double a, double b, double c, double x</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f2F1-672"></a></var><br>
— Function: int <b>gsl_sf_hyperg_2F1_e</b> (<var>double a, double b, double c, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f2F1_005fe-673"></a></var><br>
<blockquote><p>These routines compute the Gauss hypergeometric function
<!-- {${}_2F_1(a,b,c,x) = F(a,b,c,x)$} -->
2F1(a,b,c,x) = F(a,b,c,x) for |x| < 1.
<p>If the arguments (a,b,c,x) are too close to a singularity then
the function can return the error code <code>GSL_EMAXITER</code> when the
series approximation converges too slowly. This occurs in the region of
x=1, c - a - b = m for integer m.
<!-- exceptions: -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_hyperg_2F1_conj</b> (<var>double aR, double aI, double c, double x</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f2F1_005fconj-674"></a></var><br>
— Function: int <b>gsl_sf_hyperg_2F1_conj_e</b> (<var>double aR, double aI, double c, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f2F1_005fconj_005fe-675"></a></var><br>
<blockquote><p>These routines compute the Gauss hypergeometric function
<!-- {${}_2F_1(a_R + i a_I, aR - i aI, c, x)$} -->
2F1(a_R + i a_I, a_R - i a_I, c, x) with complex parameters
for |x| < 1.
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_hyperg_2F1_renorm</b> (<var>double a, double b, double c, double x</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f2F1_005frenorm-676"></a></var><br>
— Function: int <b>gsl_sf_hyperg_2F1_renorm_e</b> (<var>double a, double b, double c, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f2F1_005frenorm_005fe-677"></a></var><br>
<blockquote><p>These routines compute the renormalized Gauss hypergeometric function
<!-- {${}_2F_1(a,b,c,x) / \Gamma(c)$} -->
2F1(a,b,c,x) / \Gamma(c) for |x| < 1.
<!-- exceptions: -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_hyperg_2F1_conj_renorm</b> (<var>double aR, double aI, double c, double x</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f2F1_005fconj_005frenorm-678"></a></var><br>
— Function: int <b>gsl_sf_hyperg_2F1_conj_renorm_e</b> (<var>double aR, double aI, double c, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f2F1_005fconj_005frenorm_005fe-679"></a></var><br>
<blockquote><p>These routines compute the renormalized Gauss hypergeometric function
<!-- {${}_2F_1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c)$} -->
2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c) for |x| < 1.
<!-- exceptions: -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_hyperg_2F0</b> (<var>double a, double b, double x</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f2F0-680"></a></var><br>
— Function: int <b>gsl_sf_hyperg_2F0_e</b> (<var>double a, double b, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fhyperg_005f2F0_005fe-681"></a></var><br>
<blockquote><p>These routines compute the hypergeometric function <!-- {${}_2F_0(a,b,x)$} -->
2F0(a,b,x). The series representation
is a divergent hypergeometric series. However, for x < 0 we
have
<!-- {${}_2F_0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)$} -->
2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)
<!-- exceptions: GSL_EDOM -->
</p></blockquote></div>
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