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  27 <a name="math_toolkit.tr1_ref"></a><a class="link" href="tr1_ref.html" title="TR1 C Functions Quick Reference">TR1 C Functions Quick Reference</a>
  28 </h2></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.tr1_ref.h0"></a>
  31       <span class="phrase"><a name="math_toolkit.tr1_ref.supported_tr1_functions"></a></span><a class="link" href="tr1_ref.html#math_toolkit.tr1_ref.supported_tr1_functions">Supported
  32       TR1 Functions</a>
  33     </h5>
  34 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">tr1</span><span class="special">{</span> <span class="keyword">extern</span> <span class="string">"C"</span><span class="special">{</span>
  35
  36 <span class="comment">// [5.2.1.1] associated Laguerre polynomials:</span>
  37 <span class="keyword">double</span> <span class="identifier">assoc_laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
  38 <span class="keyword">float</span> <span class="identifier">assoc_laguerref</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
  39 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">assoc_laguerrel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
  40
  41 <span class="comment">// [5.2.1.2] associated Legendre functions:</span>
  42 <span class="keyword">double</span> <span class="identifier">assoc_legendre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
  43 <span class="keyword">float</span> <span class="identifier">assoc_legendref</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
  44 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">assoc_legendrel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
  45
  46 <span class="comment">// [5.2.1.3] beta function:</span>
  47 <span class="keyword">double</span> <span class="identifier">beta</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">y</span><span class="special">);</span>
  48 <span class="keyword">float</span> <span class="identifier">betaf</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">y</span><span class="special">);</span>
  49 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">betal</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">y</span><span class="special">);</span>
  50
  51 <span class="comment">// [5.2.1.4] (complete) elliptic integral of the first kind:</span>
  52 <span class="keyword">double</span> <span class="identifier">comp_ellint_1</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">k</span><span class="special">);</span>
  53 <span class="keyword">float</span> <span class="identifier">comp_ellint_1f</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">k</span><span class="special">);</span>
  54 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">comp_ellint_1l</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">k</span><span class="special">);</span>
  55
  56 <span class="comment">// [5.2.1.5] (complete) elliptic integral of the second kind:</span>
  57 <span class="keyword">double</span> <span class="identifier">comp_ellint_2</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">k</span><span class="special">);</span>
  58 <span class="keyword">float</span> <span class="identifier">comp_ellint_2f</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">k</span><span class="special">);</span>
  59 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">comp_ellint_2l</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">k</span><span class="special">);</span>
  60
  61 <span class="comment">// [5.2.1.6] (complete) elliptic integral of the third kind:</span>
  62 <span class="keyword">double</span> <span class="identifier">comp_ellint_3</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">);</span>
  63 <span class="keyword">float</span> <span class="identifier">comp_ellint_3f</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">nu</span><span class="special">);</span>
  64 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">comp_ellint_3l</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">);</span>
  65
  66 <span class="comment">// [5.2.1.8] regular modified cylindrical Bessel functions:</span>
  67 <span class="keyword">double</span> <span class="identifier">cyl_bessel_i</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
  68 <span class="keyword">float</span> <span class="identifier">cyl_bessel_if</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
  69 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">cyl_bessel_il</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
  70
  71 <span class="comment">// [5.2.1.9] cylindrical Bessel functions (of the first kind):</span>
  72 <span class="keyword">double</span> <span class="identifier">cyl_bessel_j</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
  73 <span class="keyword">float</span> <span class="identifier">cyl_bessel_jf</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
  74 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">cyl_bessel_jl</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
  75
  76 <span class="comment">// [5.2.1.10] irregular modified cylindrical Bessel functions:</span>
  77 <span class="keyword">double</span> <span class="identifier">cyl_bessel_k</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
  78 <span class="keyword">float</span> <span class="identifier">cyl_bessel_kf</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
  79 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">cyl_bessel_kl</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
  80
  81 <span class="comment">// [5.2.1.11] cylindrical Neumann functions;</span>
  82 <span class="comment">// cylindrical Bessel functions (of the second kind):</span>
  83 <span class="keyword">double</span> <span class="identifier">cyl_neumann</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
  84 <span class="keyword">float</span> <span class="identifier">cyl_neumannf</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
  85 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">cyl_neumannl</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
  86
  87 <span class="comment">// [5.2.1.12] (incomplete) elliptic integral of the first kind:</span>
  88 <span class="keyword">double</span> <span class="identifier">ellint_1</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">phi</span><span class="special">);</span>
  89 <span class="keyword">float</span> <span class="identifier">ellint_1f</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">phi</span><span class="special">);</span>
  90 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">ellint_1l</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">phi</span><span class="special">);</span>
  91
  92 <span class="comment">// [5.2.1.13] (incomplete) elliptic integral of the second kind:</span>
  93 <span class="keyword">double</span> <span class="identifier">ellint_2</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">phi</span><span class="special">);</span>
  94 <span class="keyword">float</span> <span class="identifier">ellint_2f</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">phi</span><span class="special">);</span>
  95 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">ellint_2l</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">phi</span><span class="special">);</span>
  96
  97 <span class="comment">// [5.2.1.14] (incomplete) elliptic integral of the third kind:</span>
  98 <span class="keyword">double</span> <span class="identifier">ellint_3</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">phi</span><span class="special">);</span>
  99 <span class="keyword">float</span> <span class="identifier">ellint_3f</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">phi</span><span class="special">);</span>
 100 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">ellint_3l</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">phi</span><span class="special">);</span>
 101
 102 <span class="comment">// [5.2.1.15] exponential integral:</span>
 103 <span class="keyword">double</span> <span class="identifier">expint</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 104 <span class="keyword">float</span> <span class="identifier">expintf</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 105 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">expintl</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 106
 107 <span class="comment">// [5.2.1.16] Hermite polynomials:</span>
 108 <span class="keyword">double</span> <span class="identifier">hermite</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 109 <span class="keyword">float</span> <span class="identifier">hermitef</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 110 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">hermitel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 111
 112 <span class="comment">// [5.2.1.18] Laguerre polynomials:</span>
 113 <span class="keyword">double</span> <span class="identifier">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 114 <span class="keyword">float</span> <span class="identifier">laguerref</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 115 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">laguerrel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 116
 117 <span class="comment">// [5.2.1.19] Legendre polynomials:</span>
 118 <span class="keyword">double</span> <span class="identifier">legendre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 119 <span class="keyword">float</span> <span class="identifier">legendref</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 120 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">legendrel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 121
 122 <span class="comment">// [5.2.1.20] Riemann zeta function:</span>
 123 <span class="keyword">double</span> <span class="identifier">riemann_zeta</span><span class="special">(</span><span class="keyword">double</span><span class="special">);</span>
 124 <span class="keyword">float</span> <span class="identifier">riemann_zetaf</span><span class="special">(</span><span class="keyword">float</span><span class="special">);</span>
 125 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">riemann_zetal</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">);</span>
 126
 127 <span class="comment">// [5.2.1.21] spherical Bessel functions (of the first kind):</span>
 128 <span class="keyword">double</span> <span class="identifier">sph_bessel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 129 <span class="keyword">float</span> <span class="identifier">sph_besself</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 130 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">sph_bessell</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 131
 132 <span class="comment">// [5.2.1.22] spherical associated Legendre functions:</span>
 133 <span class="keyword">double</span> <span class="identifier">sph_legendre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">theta</span><span class="special">);</span>
 134 <span class="keyword">float</span> <span class="identifier">sph_legendref</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">theta</span><span class="special">);</span>
 135 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">sph_legendrel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">theta</span><span class="special">);</span>
 136
 137 <span class="comment">// [5.2.1.23] spherical Neumann functions;</span>
 138 <span class="comment">// spherical Bessel functions (of the second kind):</span>
 139 <span class="keyword">double</span> <span class="identifier">sph_neumann</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 140 <span class="keyword">float</span> <span class="identifier">sph_neumannf</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 141 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">sph_neumannl</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 142
 143 <span class="special">}}}}</span> <span class="comment">// namespaces</span>
 144 </pre>
 145 <p>
 146       In addition sufficient additional overloads of the <code class="computeroutput"><span class="keyword">double</span></code>
 147       versions of the above functions are provided, so that calling the function
 148       with any mixture of <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>, <code class="computeroutput"><span class="keyword">long</span>
 149       <span class="keyword">double</span></code>, or <span class="emphasis"><em>integer</em></span>
 150       arguments is supported, with the return type determined by the <a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
 151       type calculation rules</em></span></a>.
 152     </p>
 153 <p>
 154       For example:
 155     </p>
 156 <pre class="programlisting"><span class="identifier">expintf</span><span class="special">(</span><span class="number">2.0f</span><span class="special">);</span>  <span class="comment">// float version, returns float.</span>
 157 <span class="identifier">expint</span><span class="special">(</span><span class="number">2.0f</span><span class="special">);</span>   <span class="comment">// also calls the float version and returns float.</span>
 158 <span class="identifier">expint</span><span class="special">(</span><span class="number">2.0</span><span class="special">);</span>    <span class="comment">// double version, returns double.</span>
 159 <span class="identifier">expintl</span><span class="special">(</span><span class="number">2.0L</span><span class="special">);</span>  <span class="comment">// long double version, returns a long double.</span>
 160 <span class="identifier">expint</span><span class="special">(</span><span class="number">2.0L</span><span class="special">);</span>   <span class="comment">// also calls the long double version.</span>
 161 <span class="identifier">expint</span><span class="special">(</span><span class="number">2</span><span class="special">);</span>      <span class="comment">// integer argument is treated as a double, returns double.</span>
 162 </pre>
 163 <h5>
 164 <a name="math_toolkit.tr1_ref.h1"></a>
 165       <span class="phrase"><a name="math_toolkit.tr1_ref.quick_reference"></a></span><a class="link" href="tr1_ref.html#math_toolkit.tr1_ref.quick_reference">Quick
 166       Reference</a>
 167     </h5>
 168 <pre class="programlisting"><span class="comment">// [5.2.1.1] associated Laguerre polynomials:</span>
 169 <span class="keyword">double</span> <span class="identifier">assoc_laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 170 <span class="keyword">float</span> <span class="identifier">assoc_laguerref</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 171 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">assoc_laguerrel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 172 </pre>
 173 <p>
 174       The assoc_laguerre functions return:
 175     </p>
 176 <div class="blockquote"><blockquote class="blockquote"><p>
 177         <span class="inlinemediaobject"><img src="../../equations/laguerre_1.svg"></span>
 178
 179       </p></blockquote></div>
 180 <p>
 181       See also <a class="link" href="sf_poly/laguerre.html" title="Laguerre (and Associated) Polynomials">laguerre</a> for
 182       the full template (header only) version of this function.
 183     </p>
 184 <pre class="programlisting"><span class="comment">// [5.2.1.2] associated Legendre functions:</span>
 185 <span class="keyword">double</span> <span class="identifier">assoc_legendre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 186 <span class="keyword">float</span> <span class="identifier">assoc_legendref</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 187 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">assoc_legendrel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 188 </pre>
 189 <p>
 190       The assoc_legendre functions return:
 191     </p>
 192 <div class="blockquote"><blockquote class="blockquote"><p>
 193         <span class="inlinemediaobject"><img src="../../equations/legendre_1b.svg"></span>
 194
 195       </p></blockquote></div>
 196 <p>
 197       See also <a class="link" href="sf_poly/legendre.html" title="Legendre (and Associated) Polynomials">legendre_p</a> for
 198       the full template (header only) version of this function.
 199     </p>
 200 <pre class="programlisting"><span class="comment">// [5.2.1.3] beta function:</span>
 201 <span class="keyword">double</span> <span class="identifier">beta</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">y</span><span class="special">);</span>
 202 <span class="keyword">float</span> <span class="identifier">betaf</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">y</span><span class="special">);</span>
 203 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">betal</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">y</span><span class="special">);</span>
 204 </pre>
 205 <p>
 206       Returns the beta function of <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>y</em></span>:
 207     </p>
 208 <div class="blockquote"><blockquote class="blockquote"><p>
 209         <span class="inlinemediaobject"><img src="../../equations/beta1.svg"></span>
 210
 211       </p></blockquote></div>
 212 <p>
 213       See also <a class="link" href="sf_beta/beta_function.html" title="Beta">beta</a> for
 214       the full template (header only) version of this function.
 215     </p>
 216 <pre class="programlisting"><span class="comment">// [5.2.1.4] (complete) elliptic integral of the first kind:</span>
 217 <span class="keyword">double</span> <span class="identifier">comp_ellint_1</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">k</span><span class="special">);</span>
 218 <span class="keyword">float</span> <span class="identifier">comp_ellint_1f</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">k</span><span class="special">);</span>
 219 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">comp_ellint_1l</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">k</span><span class="special">);</span>
 220 </pre>
 221 <p>
 222       Returns the complete elliptic integral of the first kind of <span class="emphasis"><em>k</em></span>:
 223     </p>
 224 <div class="blockquote"><blockquote class="blockquote"><p>
 225         <span class="inlinemediaobject"><img src="../../equations/ellint6.svg"></span>
 226
 227       </p></blockquote></div>
 228 <p>
 229       See also <a class="link" href="ellint/ellint_1.html" title="Elliptic Integrals of the First Kind - Legendre Form">ellint_1</a> for the
 230       full template (header only) version of this function.
 231     </p>
 232 <pre class="programlisting"><span class="comment">// [5.2.1.5] (complete) elliptic integral of the second kind:</span>
 233 <span class="keyword">double</span> <span class="identifier">comp_ellint_2</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">k</span><span class="special">);</span>
 234 <span class="keyword">float</span> <span class="identifier">comp_ellint_2f</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">k</span><span class="special">);</span>
 235 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">comp_ellint_2l</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">k</span><span class="special">);</span>
 236 </pre>
 237 <p>
 238       Returns the complete elliptic integral of the second kind of <span class="emphasis"><em>k</em></span>:
 239     </p>
 240 <div class="blockquote"><blockquote class="blockquote"><p>
 241         <span class="inlinemediaobject"><img src="../../equations/ellint7.svg"></span>
 242
 243       </p></blockquote></div>
 244 <p>
 245       See also <a class="link" href="ellint/ellint_2.html" title="Elliptic Integrals of the Second Kind - Legendre Form">ellint_2</a> for the
 246       full template (header only) version of this function.
 247     </p>
 248 <pre class="programlisting"><span class="comment">// [5.2.1.6] (complete) elliptic integral of the third kind:</span>
 249 <span class="keyword">double</span> <span class="identifier">comp_ellint_3</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">);</span>
 250 <span class="keyword">float</span> <span class="identifier">comp_ellint_3f</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">nu</span><span class="special">);</span>
 251 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">comp_ellint_3l</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">);</span>
 252 </pre>
 253 <p>
 254       Returns the complete elliptic integral of the third kind of <span class="emphasis"><em>k</em></span>
 255       and <span class="emphasis"><em>nu</em></span>:
 256     </p>
 257 <div class="blockquote"><blockquote class="blockquote"><p>
 258         <span class="inlinemediaobject"><img src="../../equations/ellint8.svg"></span>
 259
 260       </p></blockquote></div>
 261 <p>
 262       See also <a class="link" href="ellint/ellint_3.html" title="Elliptic Integrals of the Third Kind - Legendre Form">ellint_3</a> for the
 263       full template (header only) version of this function.
 264     </p>
 265 <pre class="programlisting"><span class="comment">// [5.2.1.8] regular modified cylindrical Bessel functions:</span>
 266 <span class="keyword">double</span> <span class="identifier">cyl_bessel_i</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 267 <span class="keyword">float</span> <span class="identifier">cyl_bessel_if</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 268 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">cyl_bessel_il</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 269 </pre>
 270 <p>
 271       Returns the modified bessel function of the first kind of <span class="emphasis"><em>nu</em></span>
 272       and <span class="emphasis"><em>x</em></span>:
 273     </p>
 274 <div class="blockquote"><blockquote class="blockquote"><p>
 275         <span class="inlinemediaobject"><img src="../../equations/mbessel2.svg"></span>
 276
 277       </p></blockquote></div>
 278 <p>
 279       See also <a class="link" href="bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a> for
 280       the full template (header only) version of this function.
 281     </p>
 282 <pre class="programlisting"><span class="comment">// [5.2.1.9] cylindrical Bessel functions (of the first kind):</span>
 283 <span class="keyword">double</span> <span class="identifier">cyl_bessel_j</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 284 <span class="keyword">float</span> <span class="identifier">cyl_bessel_jf</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 285 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">cyl_bessel_jl</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 286 </pre>
 287 <p>
 288       Returns the bessel function of the first kind of <span class="emphasis"><em>nu</em></span> and
 289       <span class="emphasis"><em>x</em></span>:
 290     </p>
 291 <div class="blockquote"><blockquote class="blockquote"><p>
 292         <span class="inlinemediaobject"><img src="../../equations/bessel2.svg"></span>
 293
 294       </p></blockquote></div>
 295 <p>
 296       See also <a class="link" href="bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
 297       for the full template (header only) version of this function.
 298     </p>
 299 <pre class="programlisting"><span class="comment">// [5.2.1.10] irregular modified cylindrical Bessel functions:</span>
 300 <span class="keyword">double</span> <span class="identifier">cyl_bessel_k</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 301 <span class="keyword">float</span> <span class="identifier">cyl_bessel_kf</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 302 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">cyl_bessel_kl</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 303 </pre>
 304 <p>
 305       Returns the modified bessel function of the second kind of <span class="emphasis"><em>nu</em></span>
 306       and <span class="emphasis"><em>x</em></span>:
 307     </p>
 308 <div class="blockquote"><blockquote class="blockquote"><p>
 309         <span class="inlinemediaobject"><img src="../../equations/mbessel3.svg"></span>
 310
 311       </p></blockquote></div>
 312 <p>
 313       See also <a class="link" href="bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_k</a> for
 314       the full template (header only) version of this function.
 315     </p>
 316 <pre class="programlisting"><span class="comment">// [5.2.1.11] cylindrical Neumann functions;</span>
 317 <span class="comment">// cylindrical Bessel functions (of the second kind):</span>
 318 <span class="keyword">double</span> <span class="identifier">cyl_neumann</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 319 <span class="keyword">float</span> <span class="identifier">cyl_neumannf</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 320 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">cyl_neumannl</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 321 </pre>
 322 <p>
 323       Returns the bessel function of the second kind (Neumann function) of <span class="emphasis"><em>nu</em></span>
 324       and <span class="emphasis"><em>x</em></span>:
 325     </p>
 326 <div class="blockquote"><blockquote class="blockquote"><p>
 327         <span class="inlinemediaobject"><img src="../../equations/bessel3.svg"></span>
 328
 329       </p></blockquote></div>
 330 <p>
 331       See also <a class="link" href="bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>
 332       for the full template (header only) version of this function.
 333     </p>
 334 <pre class="programlisting"><span class="comment">// [5.2.1.12] (incomplete) elliptic integral of the first kind:</span>
 335 <span class="keyword">double</span> <span class="identifier">ellint_1</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">phi</span><span class="special">);</span>
 336 <span class="keyword">float</span> <span class="identifier">ellint_1f</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">phi</span><span class="special">);</span>
 337 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">ellint_1l</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">phi</span><span class="special">);</span>
 338 </pre>
 339 <p>
 340       Returns the incomplete elliptic integral of the first kind of <span class="emphasis"><em>k</em></span>
 341       and <span class="emphasis"><em>phi</em></span>:
 342     </p>
 343 <div class="blockquote"><blockquote class="blockquote"><p>
 344         <span class="inlinemediaobject"><img src="../../equations/ellint2.svg"></span>
 345
 346       </p></blockquote></div>
 347 <p>
 348       See also <a class="link" href="ellint/ellint_1.html" title="Elliptic Integrals of the First Kind - Legendre Form">ellint_1</a> for the
 349       full template (header only) version of this function.
 350     </p>
 351 <pre class="programlisting"><span class="comment">// [5.2.1.13] (incomplete) elliptic integral of the second kind:</span>
 352 <span class="keyword">double</span> <span class="identifier">ellint_2</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">phi</span><span class="special">);</span>
 353 <span class="keyword">float</span> <span class="identifier">ellint_2f</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">phi</span><span class="special">);</span>
 354 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">ellint_2l</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">phi</span><span class="special">);</span>
 355 </pre>
 356 <p>
 357       Returns the incomplete elliptic integral of the second kind of <span class="emphasis"><em>k</em></span>
 358       and <span class="emphasis"><em>phi</em></span>:
 359     </p>
 360 <div class="blockquote"><blockquote class="blockquote"><p>
 361         <span class="inlinemediaobject"><img src="../../equations/ellint3.svg"></span>
 362
 363       </p></blockquote></div>
 364 <p>
 365       See also <a class="link" href="ellint/ellint_2.html" title="Elliptic Integrals of the Second Kind - Legendre Form">ellint_2</a> for the
 366       full template (header only) version of this function.
 367     </p>
 368 <pre class="programlisting"><span class="comment">// [5.2.1.14] (incomplete) elliptic integral of the third kind:</span>
 369 <span class="keyword">double</span> <span class="identifier">ellint_3</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">phi</span><span class="special">);</span>
 370 <span class="keyword">float</span> <span class="identifier">ellint_3f</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">phi</span><span class="special">);</span>
 371 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">ellint_3l</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">nu</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">phi</span><span class="special">);</span>
 372 </pre>
 373 <p>
 374       Returns the incomplete elliptic integral of the third kind of <span class="emphasis"><em>k</em></span>,
 375       <span class="emphasis"><em>nu</em></span> and <span class="emphasis"><em>phi</em></span>:
 376     </p>
 377 <div class="blockquote"><blockquote class="blockquote"><p>
 378         <span class="inlinemediaobject"><img src="../../equations/ellint4.svg"></span>
 379
 380       </p></blockquote></div>
 381 <p>
 382       See also <a class="link" href="ellint/ellint_3.html" title="Elliptic Integrals of the Third Kind - Legendre Form">ellint_3</a> for the
 383       full template (header only) version of this function.
 384     </p>
 385 <pre class="programlisting"><span class="comment">// [5.2.1.15] exponential integral:</span>
 386 <span class="keyword">double</span> <span class="identifier">expint</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 387 <span class="keyword">float</span> <span class="identifier">expintf</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 388 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">expintl</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 389 </pre>
 390 <p>
 391       Returns the exponential integral Ei of <span class="emphasis"><em>x</em></span>:
 392     </p>
 393 <div class="blockquote"><blockquote class="blockquote"><p>
 394         <span class="inlinemediaobject"><img src="../../equations/expint_i_1.svg"></span>
 395
 396       </p></blockquote></div>
 397 <p>
 398       See also <a class="link" href="expint/expint_i.html" title="Exponential Integral Ei">expint</a> for the
 399       full template (header only) version of this function.
 400     </p>
 401 <pre class="programlisting"><span class="comment">// [5.2.1.16] Hermite polynomials:</span>
 402 <span class="keyword">double</span> <span class="identifier">hermite</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 403 <span class="keyword">float</span> <span class="identifier">hermitef</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 404 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">hermitel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 405 </pre>
 406 <p>
 407       Returns the n'th Hermite polynomial of <span class="emphasis"><em>x</em></span>:
 408     </p>
 409 <div class="blockquote"><blockquote class="blockquote"><p>
 410         <span class="inlinemediaobject"><img src="../../equations/hermite_0.svg"></span>
 411
 412       </p></blockquote></div>
 413 <p>
 414       See also <a class="link" href="sf_poly/hermite.html" title="Hermite Polynomials">hermite</a> for the
 415       full template (header only) version of this function.
 416     </p>
 417 <pre class="programlisting"><span class="comment">// [5.2.1.18] Laguerre polynomials:</span>
 418 <span class="keyword">double</span> <span class="identifier">laguerre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 419 <span class="keyword">float</span> <span class="identifier">laguerref</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 420 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">laguerrel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 421 </pre>
 422 <p>
 423       Returns the n'th Laguerre polynomial of <span class="emphasis"><em>x</em></span>:
 424     </p>
 425 <div class="blockquote"><blockquote class="blockquote"><p>
 426         <span class="inlinemediaobject"><img src="../../equations/laguerre_0.svg"></span>
 427
 428       </p></blockquote></div>
 429 <p>
 430       See also <a class="link" href="sf_poly/laguerre.html" title="Laguerre (and Associated) Polynomials">laguerre</a> for
 431       the full template (header only) version of this function.
 432     </p>
 433 <pre class="programlisting"><span class="comment">// [5.2.1.19] Legendre polynomials:</span>
 434 <span class="keyword">double</span> <span class="identifier">legendre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 435 <span class="keyword">float</span> <span class="identifier">legendref</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 436 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">legendrel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 437 </pre>
 438 <p>
 439       Returns the l'th Legendre polynomial of <span class="emphasis"><em>x</em></span>:
 440     </p>
 441 <div class="blockquote"><blockquote class="blockquote"><p>
 442         <span class="inlinemediaobject"><img src="../../equations/legendre_0.svg"></span>
 443
 444       </p></blockquote></div>
 445 <p>
 446       See also <a class="link" href="sf_poly/legendre.html" title="Legendre (and Associated) Polynomials">legendre_p</a> for
 447       the full template (header only) version of this function.
 448     </p>
 449 <pre class="programlisting"><span class="comment">// [5.2.1.20] Riemann zeta function:</span>
 450 <span class="keyword">double</span> <span class="identifier">riemann_zeta</span><span class="special">(</span><span class="keyword">double</span><span class="special">);</span>
 451 <span class="keyword">float</span> <span class="identifier">riemann_zetaf</span><span class="special">(</span><span class="keyword">float</span><span class="special">);</span>
 452 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">riemann_zetal</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">);</span>
 453 </pre>
 454 <p>
 455       Returns the Riemann Zeta function of <span class="emphasis"><em>x</em></span>:
 456     </p>
 457 <div class="blockquote"><blockquote class="blockquote"><p>
 458         <span class="inlinemediaobject"><img src="../../equations/zeta1.svg"></span>
 459
 460       </p></blockquote></div>
 461 <p>
 462       See also <a class="link" href="zetas/zeta.html" title="Riemann Zeta Function">zeta</a> for the full template
 463       (header only) version of this function.
 464     </p>
 465 <pre class="programlisting"><span class="comment">// [5.2.1.21] spherical Bessel functions (of the first kind):</span>
 466 <span class="keyword">double</span> <span class="identifier">sph_bessel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 467 <span class="keyword">float</span> <span class="identifier">sph_besself</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 468 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">sph_bessell</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 469 </pre>
 470 <p>
 471       Returns the spherical Bessel function of the first kind of <span class="emphasis"><em>x</em></span>
 472       j<sub>n</sub>(x):
 473     </p>
 474 <div class="blockquote"><blockquote class="blockquote"><p>
 475         <span class="inlinemediaobject"><img src="../../equations/sbessel2.svg"></span>
 476
 477       </p></blockquote></div>
 478 <p>
 479       See also <a class="link" href="bessel/sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_bessel</a> for
 480       the full template (header only) version of this function.
 481     </p>
 482 <pre class="programlisting"><span class="comment">// [5.2.1.22] spherical associated Legendre functions:</span>
 483 <span class="keyword">double</span> <span class="identifier">sph_legendre</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">theta</span><span class="special">);</span>
 484 <span class="keyword">float</span> <span class="identifier">sph_legendref</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">theta</span><span class="special">);</span>
 485 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">sph_legendrel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">theta</span><span class="special">);</span>
 486 </pre>
 487 <p>
 488       Returns the spherical associated Legendre function of <span class="emphasis"><em>l</em></span>,
 489       <span class="emphasis"><em>m</em></span> and <span class="emphasis"><em>theta</em></span>:
 490     </p>
 491 <div class="blockquote"><blockquote class="blockquote"><p>
 492         <span class="inlinemediaobject"><img src="../../equations/spherical_3.svg"></span>
 493
 494       </p></blockquote></div>
 495 <p>
 496       See also <a class="link" href="sf_poly/sph_harm.html" title="Spherical Harmonics">spherical_harmonic</a>
 497       for the full template (header only) version of this function.
 498     </p>
 499 <pre class="programlisting"><span class="comment">// [5.2.1.23] spherical Neumann functions;</span>
 500 <span class="comment">// spherical Bessel functions (of the second kind):</span>
 501 <span class="keyword">double</span> <span class="identifier">sph_neumann</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 502 <span class="keyword">float</span> <span class="identifier">sph_neumannf</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 503 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">sph_neumannl</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 504 </pre>
 505 <p>
 506       Returns the spherical Neumann function of <span class="emphasis"><em>x</em></span> y<sub>n</sub>(x):
 507     </p>
 508 <div class="blockquote"><blockquote class="blockquote"><p>
 509         <span class="inlinemediaobject"><img src="../../equations/sbessel2.svg"></span>
 510
 511       </p></blockquote></div>
 512 <p>
 513       See also <a class="link" href="bessel/sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_bessel</a> for
 514       the full template (header only) version of this function.
 515     </p>
 516 <h5>
 517 <a name="math_toolkit.tr1_ref.h2"></a>
 518       <span class="phrase"><a name="math_toolkit.tr1_ref.currently_unsupported_tr1_functi"></a></span><a class="link" href="tr1_ref.html#math_toolkit.tr1_ref.currently_unsupported_tr1_functi">Currently
 519       Unsupported TR1 Functions</a>
 520     </h5>
 521 <pre class="programlisting"><span class="comment">// [5.2.1.7] confluent hypergeometric functions:</span>
 522 <span class="keyword">double</span> <span class="identifier">conf_hyperg</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">c</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 523 <span class="keyword">float</span> <span class="identifier">conf_hypergf</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">c</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 524 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">conf_hypergl</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">c</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 525
 526 <span class="comment">// [5.2.1.17] hypergeometric functions:</span>
 527 <span class="keyword">double</span> <span class="identifier">hyperg</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">c</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 528 <span class="keyword">float</span> <span class="identifier">hypergf</span><span class="special">(</span><span class="keyword">float</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">c</span><span class="special">,</span> <span class="keyword">float</span> <span class="identifier">x</span><span class="special">);</span>
 529 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">hypergl</span><span class="special">(</span><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">c</span><span class="special">,</span>
 530 <span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">x</span><span class="special">);</span>
 531 </pre>
 532 <div class="note"><table border="0" summary="Note">
 533 <tr>
 534 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td>
 535 <th align="left">Note</th>
 536 </tr>
 537 <tr><td align="left" valign="top"><p>
 538         These two functions are not implemented as they are not believed to be numerically
 539         stable.
 540       </p></td></tr>
 541 </table></div>
 542 </div>
 543 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 544 <td align="left"></td>
 545 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 546       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 547       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 548       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 549       Daryle Walker and Xiaogang Zhang<p>
 550         Distributed under the Boost Software License, Version 1.0. (See accompanying
 551         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
 552       </p>
 553 </div></td>
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