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<title>Associated Legendre Polynomials and Spherical Harmonics - GNU Scientific Library -- Reference Manual</title>
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Next: <a rel="next" accesskey="n" href="Conical-Functions.html">Conical Functions</a>,
Previous: <a rel="previous" accesskey="p" href="Legendre-Polynomials.html">Legendre Polynomials</a>,
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<h4 class="subsection">7.24.2 Associated Legendre Polynomials and Spherical Harmonics</h4>
<p>The following functions compute the associated Legendre Polynomials
P_l^m(x). Note that this function grows combinatorially with
l and can overflow for l larger than about 150. There is
no trouble for small m, but overflow occurs when m and
l are both large. Rather than allow overflows, these functions
refuse to calculate P_l^m(x) and return <code>GSL_EOVRFLW</code> when
they can sense that l and m are too big.
<p>If you want to calculate a spherical harmonic, then <em>do not</em> use
these functions. Instead use <code>gsl_sf_legendre_sphPlm</code> below,
which uses a similar recursion, but with the normalized functions.
<div class="defun">
— Function: double <b>gsl_sf_legendre_Plm</b> (<var>int l, int m, double x</var>)<var><a name="index-gsl_005fsf_005flegendre_005fPlm-719"></a></var><br>
— Function: int <b>gsl_sf_legendre_Plm_e</b> (<var>int l, int m, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005flegendre_005fPlm_005fe-720"></a></var><br>
<blockquote><p>These routines compute the associated Legendre polynomial
P_l^m(x) for <!-- {$m \ge 0$} -->
m >= 0, <!-- {$l \ge m$} -->
l >= m, <!-- {$|x| \le 1$} -->
|x| <= 1.
<!-- Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW -->
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_sf_legendre_Plm_array</b> (<var>int lmax, int m, double x, double result_array</var>[])<var><a name="index-gsl_005fsf_005flegendre_005fPlm_005farray-721"></a></var><br>
— Function: int <b>gsl_sf_legendre_Plm_deriv_array</b> (<var>int lmax, int m, double x, double result_array</var>[]<var>, double result_deriv_array</var>[])<var><a name="index-gsl_005fsf_005flegendre_005fPlm_005fderiv_005farray-722"></a></var><br>
<blockquote><p>These functions compute arrays of Legendre polynomials
P_l^m(x) and derivatives dP_l^m(x)/dx,
for <!-- {$m \ge 0$} -->
m >= 0, <!-- {$l = |m|, \dots, lmax$} -->
l = |m|, ..., lmax, <!-- {$|x| \le 1$} -->
|x| <= 1.
<!-- Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_legendre_sphPlm</b> (<var>int l, int m, double x</var>)<var><a name="index-gsl_005fsf_005flegendre_005fsphPlm-723"></a></var><br>
— Function: int <b>gsl_sf_legendre_sphPlm_e</b> (<var>int l, int m, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005flegendre_005fsphPlm_005fe-724"></a></var><br>
<blockquote><p>These routines compute the normalized associated Legendre polynomial
<!-- {$\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$} -->
\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x) suitable
for use in spherical harmonics. The parameters must satisfy <!-- {$m \ge 0$} -->
m >= 0, <!-- {$l \ge m$} -->
l >= m, <!-- {$|x| \le 1$} -->
|x| <= 1. Theses routines avoid the overflows
that occur for the standard normalization of P_l^m(x).
<!-- Exceptional Return Values: GSL_EDOM -->
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_sf_legendre_sphPlm_array</b> (<var>int lmax, int m, double x, double result_array</var>[])<var><a name="index-gsl_005fsf_005flegendre_005fsphPlm_005farray-725"></a></var><br>
— Function: int <b>gsl_sf_legendre_sphPlm_deriv_array</b> (<var>int lmax, int m, double x, double result_array</var>[]<var>, double result_deriv_array</var>[])<var><a name="index-gsl_005fsf_005flegendre_005fsphPlm_005fderiv_005farray-726"></a></var><br>
<blockquote><p>These functions compute arrays of normalized associated Legendre functions
<!-- {$\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$} -->
\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x),
and derivatives,
for <!-- {$m \ge 0$} -->
m >= 0, <!-- {$l = |m|, \dots, lmax$} -->
l = |m|, ..., lmax, <!-- {$|x| \le 1$} -->
|x| <= 1.0
<!-- Exceptional Return Values: GSL_EDOM -->
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_sf_legendre_array_size</b> (<var>const int lmax, const int m</var>)<var><a name="index-gsl_005fsf_005flegendre_005farray_005fsize-727"></a></var><br>
<blockquote><p>This function returns the size of <var>result_array</var>[] needed for the array
versions of P_l^m(x), <var>lmax</var> - <var>m</var> + 1. An inline version of this function is used when <code>HAVE_INLINE</code> is defined.
<!-- Exceptional Return Values: none -->
</p></blockquote></div>
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