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<a name="Hypergeometric-Functions"></a>
<div class="header">
<p>
Next: <a href="Laguerre-Functions.html#Laguerre-Functions" accesskey="n" rel="next">Laguerre Functions</a>, Previous: <a href="Gegenbauer-Functions.html#Gegenbauer-Functions" accesskey="p" rel="previous">Gegenbauer Functions</a>, Up: <a href="Special-Functions.html#Special-Functions" accesskey="u" rel="up">Special Functions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Hypergeometric-Functions-1"></a>
<h3 class="section">7.21 Hypergeometric Functions</h3>
<a name="index-hypergeometric-functions"></a>
<a name="index-confluent-hypergeometric-functions"></a>

<p>Hypergeometric functions are described in Abramowitz &amp; Stegun, Chapters
13 and 15.  These functions are declared in the header file
<samp>gsl_sf_hyperg.h</samp>.
</p>
<dl>
<dt><a name="index-gsl_005fsf_005fhyperg_005f0F1"></a>Function: <em>double</em> <strong>gsl_sf_hyperg_0F1</strong> <em>(double <var>c</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fhyperg_005f0F1_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hyperg_0F1_e</strong> <em>(double <var>c</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the hypergeometric function <em>0F1(c,x)</em>.  
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fhyperg_005f1F1_005fint"></a>Function: <em>double</em> <strong>gsl_sf_hyperg_1F1_int</strong> <em>(int <var>m</var>, int <var>n</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fhyperg_005f1F1_005fint_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hyperg_1F1_int_e</strong> <em>(int <var>m</var>, int <var>n</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the confluent hypergeometric function
<em>1F1(m,n,x) = M(m,n,x)</em> for integer parameters <var>m</var>, <var>n</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fhyperg_005f1F1"></a>Function: <em>double</em> <strong>gsl_sf_hyperg_1F1</strong> <em>(double <var>a</var>, double <var>b</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fhyperg_005f1F1_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hyperg_1F1_e</strong> <em>(double <var>a</var>, double <var>b</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the confluent hypergeometric function
<em>1F1(a,b,x) = M(a,b,x)</em> for general parameters <var>a</var>, <var>b</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fhyperg_005fU_005fint"></a>Function: <em>double</em> <strong>gsl_sf_hyperg_U_int</strong> <em>(int <var>m</var>, int <var>n</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fhyperg_005fU_005fint_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hyperg_U_int_e</strong> <em>(int <var>m</var>, int <var>n</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the confluent hypergeometric function
<em>U(m,n,x)</em> for integer parameters <var>m</var>, <var>n</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fhyperg_005fU_005fint_005fe10_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hyperg_U_int_e10_e</strong> <em>(int <var>m</var>, int <var>n</var>, double <var>x</var>, gsl_sf_result_e10 * <var>result</var>)</em></dt>
<dd><p>This routine computes the confluent hypergeometric function
<em>U(m,n,x)</em> 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></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fhyperg_005fU"></a>Function: <em>double</em> <strong>gsl_sf_hyperg_U</strong> <em>(double <var>a</var>, double <var>b</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fhyperg_005fU_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hyperg_U_e</strong> <em>(double <var>a</var>, double <var>b</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the confluent hypergeometric function <em>U(a,b,x)</em>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fhyperg_005fU_005fe10_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hyperg_U_e10_e</strong> <em>(double <var>a</var>, double <var>b</var>, double <var>x</var>, gsl_sf_result_e10 * <var>result</var>)</em></dt>
<dd><p>This routine computes the confluent hypergeometric function
<em>U(a,b,x)</em> using the <code>gsl_sf_result_e10</code> type to return a
result with extended range. 
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fhyperg_005f2F1"></a>Function: <em>double</em> <strong>gsl_sf_hyperg_2F1</strong> <em>(double <var>a</var>, double <var>b</var>, double <var>c</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fhyperg_005f2F1_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hyperg_2F1_e</strong> <em>(double <var>a</var>, double <var>b</var>, double <var>c</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the Gauss hypergeometric function 
<em>2F1(a,b,c,x) = F(a,b,c,x)</em> for <em>|x| &lt; 1</em>.  
</p>
<p>If the arguments <em>(a,b,c,x)</em> 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
<em>x=1</em>, <em>c - a - b = m</em> for integer m.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fhyperg_005f2F1_005fconj"></a>Function: <em>double</em> <strong>gsl_sf_hyperg_2F1_conj</strong> <em>(double <var>aR</var>, double <var>aI</var>, double <var>c</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fhyperg_005f2F1_005fconj_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hyperg_2F1_conj_e</strong> <em>(double <var>aR</var>, double <var>aI</var>, double <var>c</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the Gauss hypergeometric function
<em>2F1(a_R + i a_I, a_R - i a_I, c, x)</em> with complex parameters 
for <em>|x| &lt; 1</em>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fhyperg_005f2F1_005frenorm"></a>Function: <em>double</em> <strong>gsl_sf_hyperg_2F1_renorm</strong> <em>(double <var>a</var>, double <var>b</var>, double <var>c</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fhyperg_005f2F1_005frenorm_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hyperg_2F1_renorm_e</strong> <em>(double <var>a</var>, double <var>b</var>, double <var>c</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the renormalized Gauss hypergeometric function
<em>2F1(a,b,c,x) / \Gamma(c)</em> for <em>|x| &lt; 1</em>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fhyperg_005f2F1_005fconj_005frenorm"></a>Function: <em>double</em> <strong>gsl_sf_hyperg_2F1_conj_renorm</strong> <em>(double <var>aR</var>, double <var>aI</var>, double <var>c</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fhyperg_005f2F1_005fconj_005frenorm_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hyperg_2F1_conj_renorm_e</strong> <em>(double <var>aR</var>, double <var>aI</var>, double <var>c</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the renormalized Gauss hypergeometric function
<em>2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c)</em> for <em>|x| &lt; 1</em>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fhyperg_005f2F0"></a>Function: <em>double</em> <strong>gsl_sf_hyperg_2F0</strong> <em>(double <var>a</var>, double <var>b</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fhyperg_005f2F0_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hyperg_2F0_e</strong> <em>(double <var>a</var>, double <var>b</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the hypergeometric function <em>2F0(a,b,x)</em>.  The series representation
is a divergent hypergeometric series.  However, for <em>x &lt; 0</em> we
have 
<em>2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)</em>
</p></dd></dl>

<hr>
<div class="header">
<p>
Next: <a href="Laguerre-Functions.html#Laguerre-Functions" accesskey="n" rel="next">Laguerre Functions</a>, Previous: <a href="Gegenbauer-Functions.html#Gegenbauer-Functions" accesskey="p" rel="previous">Gegenbauer Functions</a>, Up: <a href="Special-Functions.html#Special-Functions" accesskey="u" rel="up">Special Functions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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