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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.ellint.jacobi_zeta"></a><a class="link" href="jacobi_zeta.html" title="Jacobi Zeta Function">Jacobi Zeta Function</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.ellint.jacobi_zeta.h0"></a>
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<span class="phrase"><a name="math_toolkit.ellint.jacobi_zeta.synopsis"></a></span><a class="link" href="jacobi_zeta.html#math_toolkit.ellint.jacobi_zeta.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_zeta</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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_zeta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><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_zeta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><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.ellint.jacobi_zeta.h1"></a>
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<span class="phrase"><a name="math_toolkit.ellint.jacobi_zeta.description"></a></span><a class="link" href="jacobi_zeta.html#math_toolkit.ellint.jacobi_zeta.description">Description</a>
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</h5>
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<p>
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This function evaluates the Jacobi Zeta Function <span class="emphasis"><em>Z(φ, k)</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_zeta.svg"></span>
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</p></blockquote></div>
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<p>
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Please note the use of φ, and <span class="emphasis"><em>k</em></span> as the parameters, the
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function is often defined as <span class="emphasis"><em>Z(φ, m)</em></span> with <span class="emphasis"><em>m
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= k<sup>2</sup></em></span>, see for example <a href="http://mathworld.wolfram.com/JacobiZetaFunction.html" target="_top">Weisstein,
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Eric W. "Jacobi Zeta Function." From MathWorld--A Wolfram Web Resource.</a>
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Or else as <a href="https://dlmf.nist.gov/22.16#E32" target="_top"><span class="emphasis"><em>Z(x, k)</em></span></a>
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with <span class="emphasis"><em>φ = am(x, k)</em></span>, where <span class="emphasis"><em>am</em></span> is the
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<a href="https://dlmf.nist.gov/22.16#E1" target="_top">Jacobi amplitude function</a>
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which is equivalent to <span class="emphasis"><em>asin(jacobi_elliptic(k, x))</em></span>.
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</p>
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<p>
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The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
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type calculation rules</em></span></a> when the arguments are of different
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types: when they are the same type then the result is the same type as the
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arguments.
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</p>
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<p>
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Requires <span class="emphasis"><em>-1 <= k <= 1</em></span>, otherwise returns the result
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of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
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(outside this range the result would be complex).
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 21. 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 21. 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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<p>
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Note that there is no complete analogue of this function (where φ = π / 2) as
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this takes the value 0 for all <span class="emphasis"><em>k</em></span>.
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</p>
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<h5>
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<a name="math_toolkit.ellint.jacobi_zeta.h2"></a>
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<span class="phrase"><a name="math_toolkit.ellint.jacobi_zeta.accuracy"></a></span><a class="link" href="jacobi_zeta.html#math_toolkit.ellint.jacobi_zeta.accuracy">Accuracy</a>
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</h5>
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<p>
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These functions are trivially computed in terms of other elliptic integrals
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and generally have very low error rates (a few epsilon) unless parameter
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φ
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is very large, in which case the usual trigonometric function argument-reduction
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issues apply.
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</p>
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<div class="table">
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<a name="math_toolkit.ellint.jacobi_zeta.table_jacobi_zeta"></a><p class="title"><b>Table 8.68. Error rates for jacobi_zeta</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for jacobi_zeta">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5150<br> Sun Solaris<br> long double
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</p>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 14.1<br> Win32<br> double
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Elliptic Integral Jacobi Zeta: Mathworld Data
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.66ε (Mean = 0.48ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.66ε (Mean = 0.48ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.52ε (Mean = 0.357ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Elliptic Integral Jacobi Zeta: Random Data
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2.99ε (Mean = 0.824ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 3.96ε (Mean = 1.06ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 3.89ε (Mean = 0.824ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Elliptic Integral Jacobi Zeta: Large Phi Values
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2.92ε (Mean = 0.951ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 3.05ε (Mean = 1.13ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2.52ε (Mean = 0.977ε)</span>
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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</div>
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<br class="table-break"><h5>
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<a name="math_toolkit.ellint.jacobi_zeta.h3"></a>
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<span class="phrase"><a name="math_toolkit.ellint.jacobi_zeta.testing"></a></span><a class="link" href="jacobi_zeta.html#math_toolkit.ellint.jacobi_zeta.testing">Testing</a>
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</h5>
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<p>
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The tests use a mixture of spot test values calculated using values calculated
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at <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, and random
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test data generated using MPFR at 1000-bit precision and a deliberately naive
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implementation in terms of the Legendre integrals.
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</p>
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<h5>
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<a name="math_toolkit.ellint.jacobi_zeta.h4"></a>
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<span class="phrase"><a name="math_toolkit.ellint.jacobi_zeta.implementation"></a></span><a class="link" href="jacobi_zeta.html#math_toolkit.ellint.jacobi_zeta.implementation">Implementation</a>
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</h5>
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<p>
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The implementation for Z(φ, k) first makes the argument φ positive using:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic"><span class="emphasis"><em>Z(-φ, k) = -Z(φ, k)</em></span></span>
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</p></blockquote></div>
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<p>
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The function is then implemented in terms of Carlson's integral R<sub>J</sub>
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using the
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relation:
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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_zeta.svg"></span>
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</p></blockquote></div>
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<p>
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There is one special case where the above relation fails: when <span class="emphasis"><em>k
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= 1</em></span>, in that case the function simplifies to
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic"><span class="emphasis"><em>Z(φ, 1) = sign(cos(φ)) sin(φ)</em></span></span>
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</p></blockquote></div>
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<h6>
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<a name="math_toolkit.ellint.jacobi_zeta.h5"></a>
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<span class="phrase"><a name="math_toolkit.ellint.jacobi_zeta.jacobi_zeta_example"></a></span><a class="link" href="jacobi_zeta.html#math_toolkit.ellint.jacobi_zeta.jacobi_zeta_example">Example</a>
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</h6>
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<p>
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A simple example comparing use of <a href="http://www.wolframalpha.com/" target="_top">Wolfram
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Alpha</a> with Boost.Math (including much higher precision using Boost.Multiprecision)
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is <a href="../../../../example/jacobi_zeta_example.cpp" target="_top">jacobi_zeta_example.cpp</a>.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><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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