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<a name="Special-Functions"></a>
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<p>
Next: <a href="Vectors-and-Matrices.html#Vectors-and-Matrices" accesskey="n" rel="next">Vectors and Matrices</a>, Previous: <a href="Polynomials.html#Polynomials" accesskey="p" rel="previous">Polynomials</a>, Up: <a href="index.html#Top" accesskey="u" rel="up">Top</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Special-Functions-1"></a>
<h2 class="chapter">7 Special Functions</h2>
<a name="index-special-functions"></a>

<p>This chapter describes the GSL special function library.  The library
includes routines for calculating the values of Airy functions, Bessel
functions, Clausen functions, Coulomb wave functions, Coupling
coefficients, the Dawson function, Debye functions, Dilogarithms,
Elliptic integrals, Jacobi elliptic functions, Error functions,
Exponential integrals, Fermi-Dirac functions, Gamma functions,
Gegenbauer functions, Hypergeometric functions, Laguerre functions,
Legendre functions and Spherical Harmonics, the Psi (Digamma) Function,
Synchrotron functions, Transport functions, Trigonometric functions and
Zeta functions.  Each routine also computes an estimate of the numerical
error in the calculated value of the function.
</p>
<p>The functions in this chapter are declared in individual header files,
such as <samp>gsl_sf_airy.h</samp>, <samp>gsl_sf_bessel.h</samp>, etc.  The complete
set of header files can be included using the file <samp>gsl_sf.h</samp>.
</p>
<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top">&bull; <a href="Special-Function-Usage.html#Special-Function-Usage" accesskey="1">Special Function Usage</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="The-gsl_005fsf_005fresult-struct.html#The-gsl_005fsf_005fresult-struct" accesskey="2">The gsl_sf_result struct</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Special-Function-Modes.html#Special-Function-Modes" accesskey="3">Special Function Modes</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Airy-Functions-and-Derivatives.html#Airy-Functions-and-Derivatives" accesskey="4">Airy Functions and Derivatives</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Bessel-Functions.html#Bessel-Functions" accesskey="5">Bessel Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Clausen-Functions.html#Clausen-Functions" accesskey="6">Clausen Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Coulomb-Functions.html#Coulomb-Functions" accesskey="7">Coulomb Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Coupling-Coefficients.html#Coupling-Coefficients" accesskey="8">Coupling Coefficients</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Dawson-Function.html#Dawson-Function" accesskey="9">Dawson Function</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Debye-Functions.html#Debye-Functions">Debye Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Dilogarithm.html#Dilogarithm">Dilogarithm</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
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<tr><td align="left" valign="top">&bull; <a href="Elementary-Operations.html#Elementary-Operations">Elementary Operations</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
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<tr><td align="left" valign="top">&bull; <a href="Elliptic-Functions-_0028Jacobi_0029.html#Elliptic-Functions-_0028Jacobi_0029">Elliptic Functions (Jacobi)</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
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<tr><td align="left" valign="top">&bull; <a href="Error-Functions.html#Error-Functions">Error Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
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<tr><td align="left" valign="top">&bull; <a href="Exponential-Functions.html#Exponential-Functions">Exponential Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
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<tr><td align="left" valign="top">&bull; <a href="Exponential-Integrals.html#Exponential-Integrals">Exponential Integrals</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
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<tr><td align="left" valign="top">&bull; <a href="Fermi_002dDirac-Function.html#Fermi_002dDirac-Function">Fermi-Dirac Function</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Gamma-and-Beta-Functions.html#Gamma-and-Beta-Functions">Gamma and Beta Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Gegenbauer-Functions.html#Gegenbauer-Functions">Gegenbauer Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Hypergeometric-Functions.html#Hypergeometric-Functions">Hypergeometric Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Laguerre-Functions.html#Laguerre-Functions">Laguerre Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Lambert-W-Functions.html#Lambert-W-Functions">Lambert W Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Legendre-Functions-and-Spherical-Harmonics.html#Legendre-Functions-and-Spherical-Harmonics">Legendre Functions and Spherical Harmonics</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Logarithm-and-Related-Functions.html#Logarithm-and-Related-Functions">Logarithm and Related Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Mathieu-Functions.html#Mathieu-Functions">Mathieu Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Power-Function.html#Power-Function">Power Function</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Psi-_0028Digamma_0029-Function.html#Psi-_0028Digamma_0029-Function">Psi (Digamma) Function</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Synchrotron-Functions.html#Synchrotron-Functions">Synchrotron Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Transport-Functions.html#Transport-Functions">Transport Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Trigonometric-Functions.html#Trigonometric-Functions">Trigonometric Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Zeta-Functions.html#Zeta-Functions">Zeta Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Special-Functions-Examples.html#Special-Functions-Examples">Special Functions Examples</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Special-Functions-References-and-Further-Reading.html#Special-Functions-References-and-Further-Reading">Special Functions References and Further Reading</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
</table>

<hr>
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<p>
Next: <a href="Vectors-and-Matrices.html#Vectors-and-Matrices" accesskey="n" rel="next">Vectors and Matrices</a>, Previous: <a href="Polynomials.html#Polynomials" accesskey="p" rel="previous">Polynomials</a>, Up: <a href="index.html#Top" accesskey="u" rel="up">Top</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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