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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.jacobi_theta.jacobi_theta2"></a><a class="link" href="jacobi_theta2.html" title="Jacobi Theta Function θ2">Jacobi Theta
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Function θ<sub>2</sub></a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.jacobi_theta.jacobi_theta2.h0"></a>
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<span class="phrase"><a name="math_toolkit.jacobi_theta.jacobi_theta2.synopsis"></a></span><a class="link" href="jacobi_theta2.html#math_toolkit.jacobi_theta.jacobi_theta2.synopsis">Synopsis</a>
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</h5>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">jacobi_theta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<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>
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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> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
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<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">jacobi_theta2</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">q</span><span class="special">);</span>
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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> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span>
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<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">jacobi_theta2</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&);</span>
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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> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
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<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">jacobi_theta2tau</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">tau</span><span class="special">);</span>
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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> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span>
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<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">jacobi_theta2tau</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">tau</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&);</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h5>
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<a name="math_toolkit.jacobi_theta.jacobi_theta2.h1"></a>
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<span class="phrase"><a name="math_toolkit.jacobi_theta.jacobi_theta2.description"></a></span><a class="link" href="jacobi_theta2.html#math_toolkit.jacobi_theta.jacobi_theta2.description">Description</a>
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</h5>
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<p>
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The functions calculate the value of second <a class="link" href="jacobi_theta_overview.html" title="Overview of the Jacobi Theta Functions">Jacobi
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Theta function</a>, parameterized either in terms of the nome <span class="emphasis"><em>q</em></span>:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/jacobi_theta2.svg"></span>
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</p></blockquote></div>
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<p>
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Or in terms of an imaginary τ:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/jacobi_theta2tau.svg"></span>
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</p></blockquote></div>
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<p>
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The nome <span class="emphasis"><em>q</em></span> is restricted to the domain (0, 1), returning
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the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
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otherwise. The following graph shows the theta function at various values
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of <span class="emphasis"><em>q</em></span>:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/jacobi_theta2.svg" align="middle"></span>
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</p></blockquote></div>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
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documentation for more details</a>.
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</p>
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<h5>
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<a name="math_toolkit.jacobi_theta.jacobi_theta2.h2"></a>
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<span class="phrase"><a name="math_toolkit.jacobi_theta.jacobi_theta2.accuracy"></a></span><a class="link" href="jacobi_theta2.html#math_toolkit.jacobi_theta.jacobi_theta2.accuracy">Accuracy</a>
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</h5>
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<p>
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The following <a class="link" href="../ulps_plots.html" title="ULPs Plots">ULPs plot</a> is
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representative, fixing <span class="emphasis"><em>q</em></span>=0.5 and varying <span class="emphasis"><em>x</em></span>
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from 0 to 2π:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/jacobi_theta2_float.svg" align="middle"></span>
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</p></blockquote></div>
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<p>
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The envelope represents the function's <a href="https://en.wikipedia.org/wiki/Condition_number#One_variable" target="_top">condition
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number</a>. Note that relative accuracy degenerates periodically near
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θ<sub>2</sub>=0.
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</p>
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<p>
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Fixing <span class="emphasis"><em>x</em></span>=0.4 and varying <span class="emphasis"><em>q</em></span>, the
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ULPs plot looks like:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/jacobi_theta2q_float.svg" align="middle"></span>
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</p></blockquote></div>
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<p>
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Accuracy tends to degenerate near <span class="emphasis"><em>q</em></span>=1 (small τ).
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</p>
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<h5>
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<a name="math_toolkit.jacobi_theta.jacobi_theta2.h3"></a>
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<span class="phrase"><a name="math_toolkit.jacobi_theta.jacobi_theta2.implementation"></a></span><a class="link" href="jacobi_theta2.html#math_toolkit.jacobi_theta.jacobi_theta2.implementation">Implementation</a>
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</h5>
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<p>
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The <span class="emphasis"><em>q</em></span> parameterization is implemented using the τ parameterization,
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where τ=-log(<span class="emphasis"><em>q</em></span>)/π.
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</p>
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<p>
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If τ is greater than or equal to 1, the summation above is used as-is. However
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if τ < 1, the following identity <a href="https://dlmf.nist.gov/20.7#viii" target="_top">DLMF
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20.7.31</a> is used, defining τ'=-1/τ:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/jacobi_theta2_imaginary.svg"></span>
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</p></blockquote></div>
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<p>
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This transformation of variables ensures that the function will always converge
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in a small number of iterations.
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</p>
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</div>
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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</p>
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</div>
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