<title>Polygamma</title>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">polygamma</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
-<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
-<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">polygamma</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
+<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
+<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">polygamma</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
Returns the polygamma function of <span class="emphasis"><em>x</em></span>. Polygamma is defined
as the n'th derivative of the digamma function:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/polygamma1.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/polygamma1.svg"></span>
+
+ </p></blockquote></div>
<p>
The following graphs illustrate the behaviour of the function for odd and
even order:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/polygamma2.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/polygamma3.svg" align="middle"></span>
+
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../graphs/polygamma2.svg" align="middle"></span>
- <span class="inlinemediaobject"><img src="../../../graphs/polygamma3.svg" align="middle"></span>
- </p>
-<p>
- The final <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+ The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
- what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">policy
+ what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
any floating point type that is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero error</a>.
</p>
<div class="table">
-<a name="math_toolkit.sf_gamma.polygamma.table_polygamma"></a><p class="title"><b>Table 7.6. Error rates for polygamma</b></p>
+<a name="math_toolkit.sf_gamma.polygamma.table_polygamma"></a><p class="title"><b>Table 8.6. Error rates for polygamma</b></p>
<div class="table-contents"><table class="table" summary="Error rates for polygamma">
<colgroup>
<col>
<p>
For x < 0 the following reflection formula is used:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/polygamma2.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/polygamma2.svg"></span>
+
+ </p></blockquote></div>
<p>
The n'th derivative of <span class="emphasis"><em>cot(x)</em></span> is tabulated for small
<span class="emphasis"><em>n</em></span>, and for larger n has the general form:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/polygamma3.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/polygamma3.svg"></span>
+
+ </p></blockquote></div>
<p>
The coefficients of the cosine terms can be calculated iteratively starting
from <span class="emphasis"><em>C<sub>1,0</sub> = -1</em></span> and then using
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/polygamma7.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/polygamma7.svg"></span>
+
+ </p></blockquote></div>
<p>
to generate coefficients for n+1.
</p>
Once x is positive then we have two methods available to us, for small x
we use the series expansion:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/polygamma4.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/polygamma4.svg"></span>
+
+ </p></blockquote></div>
<p>
Note that the evaluation of zeta functions at integer values is essentially
a table lookup as <a class="link" href="../zetas/zeta.html" title="Riemann Zeta Function">zeta</a> is
<p>
For large x we use the asymptotic expansion:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/polygamma5.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/polygamma5.svg"></span>
+
+ </p></blockquote></div>
<p>
For x in-between the two extremes we use the relation:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/polygamma6.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/polygamma6.svg"></span>
+
+ </p></blockquote></div>
<p>
to make x large enough for the asymptotic expansion to be used.
</p>
<p>
There are also two special cases:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/polygamma8.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/polygamma9.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/polygamma8.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/polygamma9.svg"></span>
+
+ </p></blockquote></div>
</div>
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<td align="left"></td>
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