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<a name="stirling_002dpkg"></a>
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<h1 class="chapter"> 83. stirling </h1>

<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC386">83.1 Functions and Variables for stirling</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
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<h2 class="section"> 83.1 Functions and Variables for stirling </h2>

<p><a name="Item_003a-stirling"></a>
</p><dl>
<dt><u>Function:</u> <b>stirling</b><i> &nbsp;<br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>stirling</tt> (<var>z</var>,<var>n</var>) &nbsp;<br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>stirling</tt> (<var>z</var>,<var>n</var>,<var>pred</var>)</i>
<a name="IDX2433"></a>
</dt>
<dd><p>Replace <code>gamma(x)</code> with the <em>O(1/x^(2n-1))</em> Stirling formula. when <var>n</var> isn't
a nonnegative integer, signal an error. With the optional third argument <code>pred</code>,
the Stirling formula is applied only when <code>pred</code> is true.
</p>
<p>Reference: Abramowitz &amp; Stegun, &quot; Handbook of mathematical functions&quot;, 6.1.40.
</p>
<p>Examples:
</p><pre class="example">(%i1) load (stirling)$

(%i2) stirling(gamma(%alpha+x)/gamma(x),1);
       1/2 - x             x + %alpha - 1/2
(%o2) x        (x + %alpha)
                                   1           1
                            --------------- - ---- - %alpha
                            12 (x + %alpha)   12 x
                          %e
(%i3) taylor(%,x,inf,1);
                    %alpha       2    %alpha
          %alpha   x       %alpha  - x       %alpha
(%o3)/T/ x       + -------------------------------- + . . .
                                 2 x
(%i4) map('factor,%);
                                       %alpha - 1
         %alpha   (%alpha - 1) %alpha x
(%o4)   x       + -------------------------------
                                  2
</pre>
<p>The function <code>stirling</code> knows the difference between the variable 'gamma' and
the function gamma:
</p>
<pre class="example">(%i5) stirling(gamma + gamma(x),0);
                                    x - 1/2   - x
(%o5)    gamma + sqrt(2) sqrt(%pi) x        %e
(%i6) stirling(gamma(y) + gamma(x),0);
                         y - 1/2   - y
(%o6) sqrt(2) sqrt(%pi) y        %e
                                              x - 1/2   - x
                         + sqrt(2) sqrt(%pi) x        %e
</pre>
<p>To apply the Stirling formula only to terms that involve the variable <code>k</code>,
use an optional third argument; for example
</p><pre class="example">(%i7) makegamma(pochhammer(a,k)/pochhammer(b,k));
(%o7) (gamma(b)*gamma(k+a))/(gamma(a)*gamma(k+b))
(%i8) stirling(%,1, lambda([s], not(freeof(k,s))));
(%o8) (%e^(b-a)*gamma(b)*(k+a)^(k+a-1/2)*(k+b)^(-k-b+1/2))/gamma(a)
</pre><p>The terms <code>gamma(a)</code> and <code>gamma(b)</code> are free of <code>k</code>, so the Stirling formula
was not applied to these two terms.
</p>
<p>To use this function write first <code>load(&quot;stirling&quot;)</code>.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a> &middot;
<a href="maxima_105.html#Category_003a-Share-packages">Share packages</a> &middot;
<a href="maxima_105.html#Category_003a-Package-stirling">Package stirling</a>
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