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@@ -79,8 +79,9 @@
 <p>
         The second argument is the precision required, summation will stop when the
         next term is less than <span class="emphasis"><em>tolerance</em></span> times the result. The
-        deprecated versions of sum_series take an integer number of bits here - internally
-        they just convert this to a tolerance and forward the call.
+        deprecated versions of <code class="computeroutput"><span class="identifier">sum_series</span></code>
+        take an integer number of bits here - internally they just convert this to
+        a tolerance and forward the call.
       </p>
 <p>
         The third argument <span class="emphasis"><em>max_terms</em></span> sets an upper limit on
@@ -122,15 +123,19 @@
       </p>
 <h5>
 <a name="math_toolkit.internals.series_evaluation.h2"></a>
-        <span class="phrase"><a name="math_toolkit.internals.series_evaluation.example"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals.series_evaluation.example">Example</a>
+        <span class="phrase"><a name="math_toolkit.internals.series_evaluation.examples"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals.series_evaluation.examples">Examples</a>
       </h5>
+<p>
+        These examples are all in <a href="../../../../example/series.cpp" target="_top">../../example/series.cpp</a>
+      </p>
 <p>
         Let's suppose we want to implement <span class="emphasis"><em>log(1+x)</em></span> via its
         infinite series,
       </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../equations/log1pseries.svg"></span>
-      </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/log1pseries.svg"></span>
+
+        </p></blockquote></div>
 <p>
         We begin by writing a small function object to return successive terms of
         the series:
@@ -142,13 +147,13 @@
    <span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">result_type</span><span class="special">;</span>
 
    <span class="identifier">log1p_series</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span>
-      <span class="special">:</span> <span class="identifier">k</span><span class="special">(</span><span class="number">0</span><span class="special">),</span> <span class="identifier">m_mult</span><span class="special">(-</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">m_prod</span><span class="special">(-</span><span class="number">1</span><span class="special">){}</span>
+      <span class="special">:</span> <span class="identifier">k</span><span class="special">(</span><span class="number">0</span><span class="special">),</span> <span class="identifier">m_mult</span><span class="special">(-</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">m_prod</span><span class="special">(-</span><span class="number">1</span><span class="special">)</span> <span class="special">{}</span>
 
    <span class="identifier">T</span> <span class="keyword">operator</span><span class="special">()()</span>
    <span class="special">{</span>
       <span class="comment">// This is the function operator invoked by the summation</span>
       <span class="comment">// algorithm, the first call to this operator should return</span>
-      <span class="comment">// the first term of the series, the second call the second </span>
+      <span class="comment">// the first term of the series, the second call the second</span>
       <span class="comment">// term and so on.</span>
       <span class="identifier">m_prod</span> <span class="special">*=</span> <span class="identifier">m_mult</span><span class="special">;</span>
       <span class="keyword">return</span> <span class="identifier">m_prod</span> <span class="special">/</span> <span class="special">++</span><span class="identifier">k</span><span class="special">;</span>
@@ -167,24 +172,75 @@
 <span class="identifier">T</span> <span class="identifier">log1p</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span>
 <span class="special">{</span>
    <span class="comment">// We really should add some error checking on x here!</span>
-   <span class="identifier">assert</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">fabs</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">&lt;</span> <span class="number">1</span><span class="special">);</span>
+   <span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">fabs</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">&lt;</span> <span class="number">1</span><span class="special">);</span>
 
    <span class="comment">// Construct the series functor:</span>
    <span class="identifier">log1p_series</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">s</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
    <span class="comment">// Set a limit on how many iterations we permit:</span>
    <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">max_iter</span> <span class="special">=</span> <span class="number">1000</span><span class="special">;</span>
    <span class="comment">// Add it up, with enough precision for full machine precision:</span>
-   <span class="keyword">return</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">(),</span> <span class="identifier">max_iter</span><span class="special">);</span>
+   <span class="keyword">return</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">(),</span> <span class="identifier">max_iter</span><span class="special">);</span>
 <span class="special">}</span>
 </pre>
+<p>
+        We can almost use the code above for complex numbers as well - unfortunately
+        we need a slightly different definition for epsilon, and within the functor,
+        mixed complex and integer arithmetic is sadly not supported (as of C++17),
+        so we need to cast out integers to floats:
+      </p>
+<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
+<span class="keyword">struct</span> <span class="identifier">log1p_series</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&gt;</span>
+<span class="special">{</span>
+   <span class="comment">// we must define a result_type typedef:</span>
+   <span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">result_type</span><span class="special">;</span>
+
+   <span class="identifier">log1p_series</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">x</span><span class="special">)</span>
+      <span class="special">:</span> <span class="identifier">k</span><span class="special">(</span><span class="number">0</span><span class="special">),</span> <span class="identifier">m_mult</span><span class="special">(-</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">m_prod</span><span class="special">(-</span><span class="number">1</span><span class="special">)</span> <span class="special">{}</span>
+
+   <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span><span class="special">()()</span>
+   <span class="special">{</span>
+      <span class="comment">// This is the function operator invoked by the summation</span>
+      <span class="comment">// algorithm, the first call to this operator should return</span>
+      <span class="comment">// the first term of the series, the second call the second</span>
+      <span class="comment">// term and so on.</span>
+      <span class="identifier">m_prod</span> <span class="special">*=</span> <span class="identifier">m_mult</span><span class="special">;</span>
+      <span class="keyword">return</span> <span class="identifier">m_prod</span> <span class="special">/</span> <span class="identifier">T</span><span class="special">(++</span><span class="identifier">k</span><span class="special">);</span>
+   <span class="special">}</span>
+
+<span class="keyword">private</span><span class="special">:</span>
+   <span class="keyword">int</span> <span class="identifier">k</span><span class="special">;</span>
+   <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">m_mult</span><span class="special">;</span>
+   <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">m_prod</span><span class="special">;</span>
+<span class="special">};</span>
+
+
+<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
+<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">log1p</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">x</span><span class="special">)</span>
+<span class="special">{</span>
+   <span class="comment">// We really should add some error checking on x here!</span>
+   <span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">abs</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">&lt;</span> <span class="number">1</span><span class="special">);</span>
+
+   <span class="comment">// Construct the series functor:</span>
+   <span class="identifier">log1p_series</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="identifier">s</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
+   <span class="comment">// Set a limit on how many iterations we permit:</span>
+   <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">max_iter</span> <span class="special">=</span> <span class="number">1000</span><span class="special">;</span>
+   <span class="comment">// Add it up, with enough precision for full machine precision:</span>
+   <span class="keyword">return</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">()),</span> <span class="identifier">max_iter</span><span class="special">);</span>
+<span class="special">}</span>
+</pre>
+<p>
+        Of course with a few traits classes and a bit of meta-programming we could
+        fold these two implementations into one, but that's beyond the scope of these
+        examples.
+      </p>
 </div>
 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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-<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
-      Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
-      Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
-      Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
-      and Xiaogang Zhang<p>
+<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
+      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
+      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
+      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
+      Walker and Xiaogang Zhang<p>
         Distributed under the Boost Software License, Version 1.0. (See accompanying
         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>)
       </p>
@@ -192,7 +248,7 @@
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