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<title>Gamma Functions - GNU Scientific Library -- Reference Manual</title>
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<a name="Gamma-Functions"></a>
Next: <a rel="next" accesskey="n" href="Factorials.html">Factorials</a>,
Up: <a rel="up" accesskey="u" href="Gamma-and-Beta-Functions.html">Gamma and Beta Functions</a>
<hr>
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<h4 class="subsection">7.19.1 Gamma Functions</h4>
<p><a name="index-gamma-functions-559"></a>
The Gamma function is defined by the following integral,
<pre class="example"> \Gamma(x) = \int_0^\infty dt t^{x-1} \exp(-t)
</pre>
<p class="noindent">It is related to the factorial function by \Gamma(n)=(n-1)!
for positive integer n. Further information on the Gamma function
can be found in Abramowitz & Stegun, Chapter 6. The functions
described in this section are declared in the header file
<samp><span class="file">gsl_sf_gamma.h</span></samp>.
<div class="defun">
— Function: double <b>gsl_sf_gamma</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005fgamma-560"></a></var><br>
— Function: int <b>gsl_sf_gamma_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fgamma_005fe-561"></a></var><br>
<blockquote><p>These routines compute the Gamma function \Gamma(x), subject to x
not being a negative integer or zero. The function is computed using the real
Lanczos method. The maximum value of x such that \Gamma(x) is not
considered an overflow is given by the macro <code>GSL_SF_GAMMA_XMAX</code>
and is 171.0.
<!-- exceptions: GSL_EDOM, GSL_EOVRFLW, GSL_EROUND -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_lngamma</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005flngamma-562"></a></var><br>
— Function: int <b>gsl_sf_lngamma_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005flngamma_005fe-563"></a></var><br>
<blockquote><p><a name="index-logarithm-of-Gamma-function-564"></a>These routines compute the logarithm of the Gamma function,
\log(\Gamma(x)), subject to x not being a negative
integer or zero. For x<0 the real part of \log(\Gamma(x)) is
returned, which is equivalent to \log(|\Gamma(x)|). The function
is computed using the real Lanczos method.
<!-- exceptions: GSL_EDOM, GSL_EROUND -->
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_sf_lngamma_sgn_e</b> (<var>double x, gsl_sf_result * result_lg, double * sgn</var>)<var><a name="index-gsl_005fsf_005flngamma_005fsgn_005fe-565"></a></var><br>
<blockquote><p>This routine computes the sign of the gamma function and the logarithm of
its magnitude, subject to x not being a negative integer or zero. The
function is computed using the real Lanczos method. The value of the
gamma function can be reconstructed using the relation \Gamma(x) =
sgn * \exp(resultlg).
<!-- exceptions: GSL_EDOM, GSL_EROUND -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_gammastar</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005fgammastar-566"></a></var><br>
— Function: int <b>gsl_sf_gammastar_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fgammastar_005fe-567"></a></var><br>
<blockquote><p><a name="index-Regulated-Gamma-function-568"></a>These routines compute the regulated Gamma Function \Gamma^*(x)
for x > 0. The regulated gamma function is given by,
<pre class="example"> \Gamma^*(x) = \Gamma(x)/(\sqrt{2\pi} x^{(x-1/2)} \exp(-x))
= (1 + (1/12x) + ...) for x \to \infty
</pre>
<p>and is a useful suggestion of Temme.
<!-- exceptions: GSL_EDOM -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_gammainv</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005fgammainv-569"></a></var><br>
— Function: int <b>gsl_sf_gammainv_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fgammainv_005fe-570"></a></var><br>
<blockquote><p><a name="index-Reciprocal-Gamma-function-571"></a>These routines compute the reciprocal of the gamma function,
1/\Gamma(x) using the real Lanczos method.
<!-- exceptions: GSL_EUNDRFLW, GSL_EROUND -->
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_sf_lngamma_complex_e</b> (<var>double zr, double zi, gsl_sf_result * lnr, gsl_sf_result * arg</var>)<var><a name="index-gsl_005fsf_005flngamma_005fcomplex_005fe-572"></a></var><br>
<blockquote><p><a name="index-Complex-Gamma-function-573"></a>This routine computes \log(\Gamma(z)) for complex z=z_r+i
z_i and z not a negative integer or zero, using the complex Lanczos
method. The returned parameters are lnr = \log|\Gamma(z)| and
arg = \arg(\Gamma(z)) in (-\pi,\pi]. Note that the phase
part (<var>arg</var>) is not well-determined when |z| is very large,
due to inevitable roundoff in restricting to (-\pi,\pi]. This
will result in a <code>GSL_ELOSS</code> error when it occurs. The absolute
value part (<var>lnr</var>), however, never suffers from loss of precision.
<!-- exceptions: GSL_EDOM, GSL_ELOSS -->
</p></blockquote></div>
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