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<div class="section" id="orthogonal-polynomials">
<h1>Orthogonal polynomials<a class="headerlink" href="#orthogonal-polynomials" title="Permalink to this headline">¶</a></h1>
<p>An orthogonal polynomial sequence is a sequence of polynomials <img class="math" src="../_images/math/d26d6755d1ba55b7c33c13a32226df5c64bc8b95.png" alt="P_0(x), P_1(x), \ldots"/> of degree <img class="math" src="../_images/math/3f5ec7bff9454a5deecb5b6340d98a6e9ab97801.png" alt="0, 1, \ldots"/>, which are mutually orthogonal in the sense that</p>
<div class="math">
<p><img src="../_images/math/45dfa8b0de5579f275740c1f1bb2f69813fe8256.png" alt="\int_S P_n(x) P_m(x) w(x) dx =
\begin{cases}
c_n \ne 0 & \text{if $m = n$} \\
0 & \text{if $m \ne n$}
\end{cases}" /></p>
</div><p>where <img class="math" src="../_images/math/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" alt="S"/> is some domain (e.g. an interval <img class="math" src="../_images/math/3b76e28a88e5404196d7722767713b9ba940020a.png" alt="[a,b] \in \mathbb{R}"/>) and <img class="math" src="../_images/math/1fbf2dc9f4233f5839c7f4334e8eedda044ea649.png" alt="w(x)"/> is a fixed <em>weight function</em>. A sequence of orthogonal polynomials is determined completely by <img class="math" src="../_images/math/9ee4b825a2e36ae093ed7be5e4851ef453b34914.png" alt="w"/>, <img class="math" src="../_images/math/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" alt="S"/>, and a normalization convention (e.g. <img class="math" src="../_images/math/5932fc6cd01c1eda6d154917ff9fd2c2b6a786a9.png" alt="c_n = 1"/>). Applications of orthogonal polynomials include function approximation and solution of differential equations.</p>
<p>Orthogonal polynomials are sometimes defined using the differential equations they satisfy (as functions of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>) or the recurrence relations they satisfy with respect to the order <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>. Other ways of defining orthogonal polynomials include differentiation formulas and generating functions. The standard orthogonal polynomials can also be represented as hypergeometric series (see <a class="reference external" href="hypergeometric.html"><em>Hypergeometric functions</em></a>), more specifically using the Gauss hypergeometric function <img class="math" src="../_images/math/88949b5db676bcd9b635abe892e514d84a12f031.png" alt="\,_2F_1"/> in most cases. The following functions are generally implemented using hypergeometric functions since this is computationally efficient and easily generalizes.</p>
<p>For more information, see the <a class="reference external" href="http://en.wikipedia.org/wiki/Orthogonal_polynomials">Wikipedia article on orthogonal polynomials</a>.</p>
<div class="section" id="legendre-functions">
<h2>Legendre functions<a class="headerlink" href="#legendre-functions" title="Permalink to this headline">¶</a></h2>
<div class="section" id="legendre">
<h3><tt class="xref docutils literal"><span class="pre">legendre()</span></tt><a class="headerlink" href="#legendre" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.legendre">
<tt class="descclassname">mpmath.</tt><tt class="descname">legendre</tt><big>(</big><em>n</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.legendre" title="Permalink to this definition">¶</a></dt>
<dd><p><tt class="docutils literal"><span class="pre">legendre(n,</span> <span class="pre">x)</span></tt> evaluates the Legendre polynomial <img class="math" src="../_images/math/9f7ba371de1369bc2066fe2ef779dd977b4cbdf6.png" alt="P_n(x)"/>.
The Legendre polynomials are given by the formula</p>
<div class="math">
<p><img src="../_images/math/38fe540809e4b52fca0bdc2ba12f34f2e40dfae1.png" alt="P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 -1)^n." /></p>
</div><p>Alternatively, they can be computed recursively using</p>
<div class="math">
<p><img src="../_images/math/cc7661d774d98d949e870523ff276e438459c663.png" alt="P_0(x) = 1
P_1(x) = x
(n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)." /></p>
</div><p>A third definition is in terms of the hypergeometric function
<img class="math" src="../_images/math/88949b5db676bcd9b635abe892e514d84a12f031.png" alt="\,_2F_1"/>, whereby they can be generalized to arbitrary <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>:</p>
<div class="math">
<p><img src="../_images/math/4a8af3820475912ebd06768ac74171c27fe276f5.png" alt="P_n(x) = \,_2F_1\left(-n, n+1, 1, \frac{1-x}{2}\right)" /></p>
</div><p><strong>Plots</strong></p>
<div class="highlight-python"><div class="highlight"><pre><span class="c"># Legendre polynomials P_n(x) on [-1,1] for n=0,1,2,3,4</span>
<span class="n">f0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">plot</span><span class="p">([</span><span class="n">f0</span><span class="p">,</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">],[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
</pre></div>
</div>
<img alt="../_images/legendre.png" src="../_images/legendre.png" />
<p><strong>Basic evaluation</strong></p>
<p>The Legendre polynomials assume fixed values at the points
<img class="math" src="../_images/math/8d4c38a419ae56f13a202216dd45a1978bc91a54.png" alt="x = -1"/> and <img class="math" src="../_images/math/34310f2f36ed6d724838edb08788ee62acb33386.png" alt="x = 1"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">([</span><span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)])</span>
<span class="go">[1.0, 1.0, 1.0, 1.0, 1.0, 1.0]</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">([</span><span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)])</span>
<span class="go">[1.0, -1.0, 1.0, -1.0, 1.0, -1.0]</span>
</pre></div>
</div>
<p>The coefficients of Legendre polynomials can be recovered
using degree-<img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/> Taylor expansion:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
<span class="gp">... </span> <span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)))</span>
<span class="gp">...</span>
<span class="go">[1.0]</span>
<span class="go">[0.0, 1.0]</span>
<span class="go">[-0.5, 0.0, 1.5]</span>
<span class="go">[0.0, -1.5, 0.0, 2.5]</span>
<span class="go">[0.375, 0.0, -3.75, 0.0, 4.375]</span>
</pre></div>
</div>
<p>The roots of Legendre polynomials are located symmetrically
on the interval <img class="math" src="../_images/math/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png" alt="[-1, 1]"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
<span class="gp">... </span> <span class="n">nprint</span><span class="p">(</span><span class="n">polyroots</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]))</span>
<span class="gp">...</span>
<span class="go">[]</span>
<span class="go">[0.0]</span>
<span class="go">[-0.57735, 0.57735]</span>
<span class="go">[-0.774597, 0.0, 0.774597]</span>
<span class="go">[-0.861136, -0.339981, 0.339981, 0.861136]</span>
</pre></div>
</div>
<p>An example of an evaluation for arbitrary <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">legendre</span><span class="p">(</span><span class="mf">0.75</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">4</span><span class="n">j</span><span class="p">)</span>
<span class="go">(1.94952805264875 + 2.1071073099422j)</span>
</pre></div>
</div>
<p><strong>Orthogonality</strong></p>
<p>The Legendre polynomials are orthogonal on <img class="math" src="../_images/math/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png" alt="[-1, 1]"/> with respect
to the trivial weight <img class="math" src="../_images/math/ddf7c7bf07793db4262aeb2e1b26a318a6e85c27.png" alt="w(x) = 1"/>. That is, <img class="math" src="../_images/math/21600e9d56166633998cdda57e4d8234553c1063.png" alt="P_m(x) P_n(x)"/>
integrates to zero if <img class="math" src="../_images/math/45e3b3079da1f5ad7cc7f003099272b9ae298584.png" alt="m \ne n"/> and to <img class="math" src="../_images/math/285bc2fbf54290282cb92b66c1b5a67a76ea1adf.png" alt="2/(2n+1)"/> if <img class="math" src="../_images/math/a39526a4fd8fa70a09be9251b26dbfb4bb423219.png" alt="m = n"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span>
<span class="gp">>>> </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span>
<span class="gp">>>> </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">0.222222222222222</span>
</pre></div>
</div>
<p><strong>Differential equation</strong></p>
<p>The Legendre polynomials satisfy the differential equation</p>
<div class="math">
<p><img src="../_images/math/5e711cfea764d0ab83c9d3936a63da07b53633dc.png" alt="((1-x^2) y')' + n(n+1) y' = 0." /></p>
</div><p>We can verify this numerically:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">n</span> <span class="o">=</span> <span class="mf">3.6</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="mf">0.73</span>
<span class="gp">>>> </span><span class="n">P</span> <span class="o">=</span> <span class="n">legendre</span>
<span class="gp">>>> </span><span class="n">A</span> <span class="o">=</span> <span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">u</span><span class="p">:</span> <span class="n">P</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">u</span><span class="p">),</span> <span class="n">t</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">B</span> <span class="o">=</span> <span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">P</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">A</span><span class="o">+</span><span class="n">B</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">9.0e-16</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="legenp">
<h3><tt class="xref docutils literal"><span class="pre">legenp()</span></tt><a class="headerlink" href="#legenp" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.legenp">
<tt class="descclassname">mpmath.</tt><tt class="descname">legenp</tt><big>(</big><em>n</em>, <em>m</em>, <em>z</em>, <em>type=2</em><big>)</big><a class="headerlink" href="#mpmath.legenp" title="Permalink to this definition">¶</a></dt>
<dd><p>Calculates the (associated) Legendre function of the first kind of
degree <em>n</em> and order <em>m</em>, <img class="math" src="../_images/math/df26d26618564cc9d53e98ffad13b8db3cd84f5a.png" alt="P_n^m(z)"/>. Taking <img class="math" src="../_images/math/f5507214a312b4e8009e11bba68cd7453d4ba0cf.png" alt="m = 0"/> gives the ordinary
Legendre function of the first kind, <img class="math" src="../_images/math/31519e614892fb7e7f8b443fa15ac68149839215.png" alt="P_n(z)"/>. The parameters may be
complex numbers.</p>
<p>In terms of the Gauss hypergeometric function, the (associated) Legendre
function is defined as</p>
<div class="math">
<p><img src="../_images/math/1c8015385f0b412abc35c3fcc6d4de4dbc475a6b.png" alt="P_n^m(z) = \frac{1}{\Gamma(1-m)} \frac{(1+z)^{m/2}}{(1-z)^{m/2}}
\,_2F_1\left(-n, n+1, 1-m, \frac{1-z}{2}\right)." /></p>
</div><p>With <em>type=3</em> instead of <em>type=2</em>, the alternative
definition</p>
<div class="math">
<p><img src="../_images/math/ad598da7927137371ac6f2ce7733cce6ba539339.png" alt="\hat{P}_n^m(z) = \frac{1}{\Gamma(1-m)} \frac{(z+1)^{m/2}}{(z-1)^{m/2}}
\,_2F_1\left(-n, n+1, 1-m, \frac{1-z}{2}\right)." /></p>
</div><p>is used. These functions correspond respectively to <tt class="docutils literal"><span class="pre">LegendreP[n,m,2,z]</span></tt>
and <tt class="docutils literal"><span class="pre">LegendreP[n,m,3,z]</span></tt> in Mathematica.</p>
<p>The general solution of the (associated) Legendre differential equation</p>
<div class="math">
<p><img src="../_images/math/7e4a9c802eebd1e8eede8a5e520bd6b5480c9996.png" alt="(1-z^2) f''(z) - 2zf'(z) + \left(n(n+1)-\frac{m^2}{1-z^2}\right)f(z) = 0" /></p>
</div><p>is given by <img class="math" src="../_images/math/e1acb1603031612de2f742eaa69a8f86e2cd9229.png" alt="C_1 P_n^m(z) + C_2 Q_n^m(z)"/> for arbitrary constants
<img class="math" src="../_images/math/70b97eb4ad6e4e7a5a97e2547d9ed6e965960d0c.png" alt="C_1"/>, <img class="math" src="../_images/math/b65e6facfe122c182fe14d503aeace2c52a9739d.png" alt="C_2"/>, where <img class="math" src="../_images/math/459b2f4f0c552216e54b0c076a6064fed7120049.png" alt="Q_n^m(z)"/> is a Legendre function of the
second kind as implemented by <a title="mpmath.legenq" class="reference internal" href="#mpmath.legenq"><tt class="xref docutils literal"><span class="pre">legenq()</span></tt></a>.</p>
<p><strong>Examples</strong></p>
<p>Evaluation for arbitrary parameters and arguments:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">legenp</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">);</span> <span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="go">149.5</span>
<span class="go">149.5</span>
<span class="gp">>>> </span><span class="n">legenp</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span>
<span class="go">(1.972260393822275434196053 - 1.972260393822275434196053j)</span>
<span class="gp">>>> </span><span class="n">legenp</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="o">+</span><span class="mi">4</span><span class="n">j</span><span class="p">)</span>
<span class="go">(-3.335677248386698208736542 - 5.663270217461022307645625j)</span>
<span class="gp">>>> </span><span class="n">chop</span><span class="p">(</span><span class="n">legenp</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="nb">type</span><span class="o">=</span><span class="mi">2</span><span class="p">))</span>
<span class="go">28.125</span>
<span class="gp">>>> </span><span class="n">chop</span><span class="p">(</span><span class="n">legenp</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="nb">type</span><span class="o">=</span><span class="mi">3</span><span class="p">))</span>
<span class="go">-28.125</span>
</pre></div>
</div>
<p>Verifying the associated Legendre differential equation:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">C1</span><span class="p">,</span> <span class="n">C2</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span>
<span class="gp">>>> </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">C1</span><span class="o">*</span><span class="n">legenp</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">C2</span><span class="o">*</span><span class="n">legenq</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">deq</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> \
<span class="gp">... </span> <span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">-</span><span class="n">m</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
<span class="gp">>>> </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="o">+</span><span class="mi">2</span><span class="n">j</span><span class="p">]:</span>
<span class="gp">... </span> <span class="n">chop</span><span class="p">(</span><span class="n">deq</span><span class="p">(</span><span class="n">mpmathify</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
<span class="gp">...</span>
<span class="go">0.0</span>
<span class="go">0.0</span>
<span class="go">0.0</span>
<span class="go">0.0</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="legenq">
<h3><tt class="xref docutils literal"><span class="pre">legenq()</span></tt><a class="headerlink" href="#legenq" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.legenq">
<tt class="descclassname">mpmath.</tt><tt class="descname">legenq</tt><big>(</big><em>n</em>, <em>m</em>, <em>z</em>, <em>type=2</em><big>)</big><a class="headerlink" href="#mpmath.legenq" title="Permalink to this definition">¶</a></dt>
<dd><p>Calculates the (associated) Legendre function of the second kind of
degree <em>n</em> and order <em>m</em>, <img class="math" src="../_images/math/459b2f4f0c552216e54b0c076a6064fed7120049.png" alt="Q_n^m(z)"/>. Taking <img class="math" src="../_images/math/f5507214a312b4e8009e11bba68cd7453d4ba0cf.png" alt="m = 0"/> gives the ordinary
Legendre function of the second kind, <img class="math" src="../_images/math/f182d870432fbf0f8135a048b1b5ff6b4f5f1ad1.png" alt="Q_n(z)"/>. The parameters may
complex numbers.</p>
<p>The Legendre functions of the second kind give a second set of
solutions to the (associated) Legendre differential equation.
(See <a title="mpmath.legenp" class="reference internal" href="#mpmath.legenp"><tt class="xref docutils literal"><span class="pre">legenp()</span></tt></a>.)
Unlike the Legendre functions of the first kind, they are not
polynomials of <img class="math" src="../_images/math/b13f21416d84e13708696f34dea81026cda583c9.png" alt="z"/> for integer <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>, <img class="math" src="../_images/math/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" alt="m"/> but rational or logarithmic
functions with poles at <img class="math" src="../_images/math/0d1d3efda3e96b7763cf2660767f837c3c1c5f52.png" alt="z = \pm 1"/>.</p>
<p>There are various ways to define Legendre functions of
the second kind, giving rise to different complex structure.
A version can be selected using the <em>type</em> keyword argument.
The <em>type=2</em> and <em>type=3</em> functions are given respectively by</p>
<div class="math">
<p><img src="../_images/math/07b7374b512011a09c636ce6543bc5d0b30caaa6.png" alt="Q_n^m(z) = \frac{\pi}{2 \sin(\pi m)}
\left( \cos(\pi m) P_n^m(z) -
\frac{\Gamma(1+m+n)}{\Gamma(1-m+n)} P_n^{-m}(z)\right)
\hat{Q}_n^m(z) = \frac{\pi}{2 \sin(\pi m)} e^{\pi i m}
\left( \hat{P}_n^m(z) -
\frac{\Gamma(1+m+n)}{\Gamma(1-m+n)} \hat{P}_n^{-m}(z)\right)" /></p>
</div><p>where <img class="math" src="../_images/math/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png" alt="P"/> and <img class="math" src="../_images/math/4d88c8da55420edccc79578e17c0525c08689c46.png" alt="\hat{P}"/> are the <em>type=2</em> and <em>type=3</em> Legendre functions
of the first kind. The formulas above should be understood as limits
when <img class="math" src="../_images/math/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" alt="m"/> is an integer.</p>
<p>These functions correspond to <tt class="docutils literal"><span class="pre">LegendreQ[n,m,2,z]</span></tt> (or <tt class="docutils literal"><span class="pre">LegendreQ[n,m,z]</span></tt>)
and <tt class="docutils literal"><span class="pre">LegendreQ[n,m,3,z]</span></tt> in Mathematica. The <em>type=3</em> function
is essentially the same as the function defined in
Abramowitz & Stegun (eq. 8.1.3) but with <img class="math" src="../_images/math/8c9d691ae30bde236b7785fef9cc3f9c351fad6a.png" alt="(z+1)^{m/2}(z-1)^{m/2}"/> instead
of <img class="math" src="../_images/math/a6bc271d565ef85d91da5d032be01e04d25253eb.png" alt="(z^2-1)^{m/2}"/>, giving slightly different branches.</p>
<p><strong>Examples</strong></p>
<p>Evaluation for arbitrary parameters and arguments:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">legenq</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="go">-0.8186632680417568557122028</span>
<span class="gp">>>> </span><span class="n">legenq</span><span class="p">(</span><span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span>
<span class="go">(0.6655964618250228714288277 + 0.3937692045497259717762649j)</span>
<span class="gp">>>> </span><span class="n">legenq</span><span class="p">(</span><span class="mi">2</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4</span><span class="n">j</span><span class="p">,</span> <span class="o">-</span><span class="mi">6</span><span class="o">+</span><span class="mi">5</span><span class="n">j</span><span class="p">)</span>
<span class="go">(-10001.95256487468541686564 - 6011.691337610097577791134j)</span>
</pre></div>
</div>
<p>Different versions of the function:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">legenq</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="go">0.7298060598018049369381857</span>
<span class="gp">>>> </span><span class="n">legenq</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">)</span>
<span class="go">(-7.902916572420817192300921 + 0.1998650072605976600724502j)</span>
<span class="gp">>>> </span><span class="n">legenq</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="nb">type</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="go">(2.040524284763495081918338 - 0.7298060598018049369381857j)</span>
<span class="gp">>>> </span><span class="n">chop</span><span class="p">(</span><span class="n">legenq</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="nb">type</span><span class="o">=</span><span class="mi">3</span><span class="p">))</span>
<span class="go">-0.1998650072605976600724502</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="chebyshev-polynomials">
<h2>Chebyshev polynomials<a class="headerlink" href="#chebyshev-polynomials" title="Permalink to this headline">¶</a></h2>
<div class="section" id="chebyt">
<h3><tt class="xref docutils literal"><span class="pre">chebyt()</span></tt><a class="headerlink" href="#chebyt" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.chebyt">
<tt class="descclassname">mpmath.</tt><tt class="descname">chebyt</tt><big>(</big><em>n</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.chebyt" title="Permalink to this definition">¶</a></dt>
<dd><p><tt class="docutils literal"><span class="pre">chebyt(n,</span> <span class="pre">x)</span></tt> evaluates the Chebyshev polynomial of the first
kind <img class="math" src="../_images/math/e4fe51661da2eabbd80cb6c7080652f96263b19d.png" alt="T_n(x)"/>, defined by the identity</p>
<div class="math">
<p><img src="../_images/math/1add736efb518e69c365c9a11b7d48fb62e17754.png" alt="T_n(\cos x) = \cos(n x)." /></p>
</div><p>The Chebyshev polynomials of the first kind are a special
case of the Jacobi polynomials, and by extension of the
hypergeometric function <img class="math" src="../_images/math/88949b5db676bcd9b635abe892e514d84a12f031.png" alt="\,_2F_1"/>. They can thus also be
evaluated for nonintegral <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>.</p>
<p><strong>Plots</strong></p>
<div class="highlight-python"><div class="highlight"><pre><span class="c"># Chebyshev polynomials T_n(x) on [-1,1] for n=0,1,2,3,4</span>
<span class="n">f0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">plot</span><span class="p">([</span><span class="n">f0</span><span class="p">,</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">],[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
</pre></div>
</div>
<img alt="../_images/chebyt.png" src="../_images/chebyt.png" />
<p><strong>Basic evaluation</strong></p>
<p>The coefficients of the <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>-th polynomial can be recovered
using using degree-<img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/> Taylor expansion:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
<span class="gp">... </span> <span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)))</span>
<span class="gp">...</span>
<span class="go">[1.0]</span>
<span class="go">[0.0, 1.0]</span>
<span class="go">[-1.0, 0.0, 2.0]</span>
<span class="go">[0.0, -3.0, 0.0, 4.0]</span>
<span class="go">[1.0, 0.0, -8.0, 0.0, 8.0]</span>
</pre></div>
</div>
<p><strong>Orthogonality</strong></p>
<p>The Chebyshev polynomials of the first kind are orthogonal
on the interval <img class="math" src="../_images/math/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png" alt="[-1, 1]"/> with respect to the weight
function <img class="math" src="../_images/math/d76ea6713fb4ffe020fafb88f2098ea16f38f842.png" alt="w(x) = 1/\sqrt{1-x^2}"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">chebyt</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span>
<span class="gp">>>> </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">1.57079632596448</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="chebyu">
<h3><tt class="xref docutils literal"><span class="pre">chebyu()</span></tt><a class="headerlink" href="#chebyu" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.chebyu">
<tt class="descclassname">mpmath.</tt><tt class="descname">chebyu</tt><big>(</big><em>n</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.chebyu" title="Permalink to this definition">¶</a></dt>
<dd><p><tt class="docutils literal"><span class="pre">chebyu(n,</span> <span class="pre">x)</span></tt> evaluates the Chebyshev polynomial of the second
kind <img class="math" src="../_images/math/bc224bf0d5fd957095a282940921bd63fc68a468.png" alt="U_n(x)"/>, defined by the identity</p>
<div class="math">
<p><img src="../_images/math/a1ae609ead051c4e0471f855c65795d622d76dbd.png" alt="U_n(\cos x) = \frac{\sin((n+1)x)}{\sin(x)}." /></p>
</div><p>The Chebyshev polynomials of the second kind are a special
case of the Jacobi polynomials, and by extension of the
hypergeometric function <img class="math" src="../_images/math/88949b5db676bcd9b635abe892e514d84a12f031.png" alt="\,_2F_1"/>. They can thus also be
evaluated for nonintegral <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>.</p>
<p><strong>Plots</strong></p>
<div class="highlight-python"><div class="highlight"><pre><span class="c"># Chebyshev polynomials U_n(x) on [-1,1] for n=0,1,2,3,4</span>
<span class="n">f0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">plot</span><span class="p">([</span><span class="n">f0</span><span class="p">,</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">],[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
</pre></div>
</div>
<img alt="../_images/chebyu.png" src="../_images/chebyu.png" />
<p><strong>Basic evaluation</strong></p>
<p>The coefficients of the <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>-th polynomial can be recovered
using using degree-<img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/> Taylor expansion:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
<span class="gp">... </span> <span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)))</span>
<span class="gp">...</span>
<span class="go">[1.0]</span>
<span class="go">[0.0, 2.0]</span>
<span class="go">[-1.0, 0.0, 4.0]</span>
<span class="go">[0.0, -4.0, 0.0, 8.0]</span>
<span class="go">[1.0, 0.0, -12.0, 0.0, 16.0]</span>
</pre></div>
</div>
<p><strong>Orthogonality</strong></p>
<p>The Chebyshev polynomials of the second kind are orthogonal
on the interval <img class="math" src="../_images/math/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png" alt="[-1, 1]"/> with respect to the weight
function <img class="math" src="../_images/math/60470bac7b7be5ec9204ca80aa90ea82150a475d.png" alt="w(x) = \sqrt{1-x^2}"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">chebyu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span>
<span class="gp">>>> </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span>
<span class="gp">>>> </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">1.5707963267949</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="jacobi-polynomials">
<h2>Jacobi polynomials<a class="headerlink" href="#jacobi-polynomials" title="Permalink to this headline">¶</a></h2>
<div class="section" id="jacobi">
<h3><tt class="xref docutils literal"><span class="pre">jacobi()</span></tt><a class="headerlink" href="#jacobi" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.jacobi">
<tt class="descclassname">mpmath.</tt><tt class="descname">jacobi</tt><big>(</big><em>n</em>, <em>a</em>, <em>b</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.jacobi" title="Permalink to this definition">¶</a></dt>
<dd><p><tt class="docutils literal"><span class="pre">jacobi(n,</span> <span class="pre">a,</span> <span class="pre">b,</span> <span class="pre">x)</span></tt> evaluates the Jacobi polynomial
<img class="math" src="../_images/math/ca97c3d1353d63ad511d603d70a7143fc1239c03.png" alt="P_n^{(a,b)}(x)"/>. The Jacobi polynomials are a special
case of the hypergeometric function <img class="math" src="../_images/math/88949b5db676bcd9b635abe892e514d84a12f031.png" alt="\,_2F_1"/> given by:</p>
<div class="math">
<p><img src="../_images/math/ac4d80ba2147ce445beaf36dd549824cc68970bb.png" alt="P_n^{(a,b)}(x) = {n+a \choose n}
\,_2F_1\left(-n,1+a+b+n,a+1,\frac{1-x}{2}\right)." /></p>
</div><p>Note that this definition generalizes to nonintegral values
of <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>. When <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/> is an integer, the hypergeometric series
terminates after a finite number of terms, giving
a polynomial in <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>.</p>
<p><strong>Evaluation of Jacobi polynomials</strong></p>
<p>A special evaluation is <img class="math" src="../_images/math/a26de8ee7daa76d6e686b74f5fdab8944be6cc66.png" alt="P_n^{(a,b)}(1) = {n+a \choose n}"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">jacobi</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">2.4609375</span>
<span class="gp">>>> </span><span class="n">binomial</span><span class="p">(</span><span class="mi">4</span><span class="o">+</span><span class="mf">0.5</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="go">2.4609375</span>
</pre></div>
</div>
<p>A Jacobi polynomial of degree <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/> is equal to its
Taylor polynomial of degree <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>. The explicit
coefficients of Jacobi polynomials can therefore
be recovered easily using <a title="mpmath.taylor" class="reference external" href="../calculus/approximation.html#mpmath.taylor"><tt class="xref docutils literal"><span class="pre">taylor()</span></tt></a>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
<span class="gp">... </span> <span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
<span class="gp">...</span>
<span class="go">[1.0]</span>
<span class="go">[-0.5, 2.5]</span>
<span class="go">[-0.75, -1.5, 5.25]</span>
<span class="go">[0.5, -3.5, -3.5, 10.5]</span>
<span class="go">[0.625, 2.5, -11.25, -7.5, 20.625]</span>
</pre></div>
</div>
<p>For nonintegral <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>, the Jacobi “polynomial” is no longer
a polynomial:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">jacobi</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="go">[0.309983, 1.84119, -1.26933, 1.26699, -1.34808]</span>
</pre></div>
</div>
<p><strong>Orthogonality</strong></p>
<p>The Jacobi polynomials are orthogonal on the interval
<img class="math" src="../_images/math/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png" alt="[-1, 1]"/> with respect to the weight function
<img class="math" src="../_images/math/b13f14bd4cb8445a3401f1ac1bb66910fd16cc58.png" alt="w(x) = (1-x)^a (1+x)^b"/>. That is,
<img class="math" src="../_images/math/01b58ee5325433b90eefd8e8f723f93b30684874.png" alt="w(x) P_n^{(a,b)}(x) P_m^{(a,b)}(x)"/> integrates to
zero if <img class="math" src="../_images/math/45e3b3079da1f5ad7cc7f003099272b9ae298584.png" alt="m \ne n"/> and to a nonzero number if <img class="math" src="../_images/math/a39526a4fd8fa70a09be9251b26dbfb4bb423219.png" alt="m = n"/>.</p>
<p>The orthogonality is easy to verify using numerical
quadrature:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">P</span> <span class="o">=</span> <span class="n">jacobi</span>
<span class="gp">>>> </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">a</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">b</span> <span class="o">*</span> <span class="n">P</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">P</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="mi">2</span>
<span class="gp">>>> </span><span class="n">b</span> <span class="o">=</span> <span class="mi">3</span>
<span class="gp">>>> </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span>
<span class="gp">>>> </span><span class="n">chop</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span>
<span class="gp">>>> </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">1.9047619047619</span>
</pre></div>
</div>
<p><strong>Differential equation</strong></p>
<p>The Jacobi polynomials are solutions of the differential
equation</p>
<div class="math">
<p><img src="../_images/math/a96ce75376e38b9ab152e89cae52bd733c918c79.png" alt="(1-x^2) y'' + (b-a-(a+b+2)x) y' + n (n+a+b+1) y = 0." /></p>
</div><p>We can verify that <a title="mpmath.jacobi" class="reference internal" href="#mpmath.jacobi"><tt class="xref docutils literal"><span class="pre">jacobi()</span></tt></a> approximately satisfies
this equation:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="mf">2.5</span>
<span class="gp">>>> </span><span class="n">b</span> <span class="o">=</span> <span class="mi">4</span>
<span class="gp">>>> </span><span class="n">n</span> <span class="o">=</span> <span class="mi">3</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">pi</span>
<span class="gp">>>> </span><span class="n">A0</span> <span class="o">=</span> <span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">A1</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="o">-</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">A2</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">A2</span> <span class="o">+</span> <span class="n">A1</span> <span class="o">+</span> <span class="n">A0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">4.0e-12</span>
</pre></div>
</div>
<p>The difference of order <img class="math" src="../_images/math/6336cd9b79f2499e10784c0214c316dc2641adf9.png" alt="10^{-12}"/> is as close to zero as
it could be at 15-digit working precision, since the terms
are large:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">A0</span><span class="p">,</span> <span class="n">A1</span><span class="p">,</span> <span class="n">A2</span>
<span class="go">(26560.2328981879, -21503.7641037294, -5056.46879445852)</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="gegenbauer-polynomials">
<h2>Gegenbauer polynomials<a class="headerlink" href="#gegenbauer-polynomials" title="Permalink to this headline">¶</a></h2>
<div class="section" id="gegenbauer">
<h3><tt class="xref docutils literal"><span class="pre">gegenbauer()</span></tt><a class="headerlink" href="#gegenbauer" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.gegenbauer">
<tt class="descclassname">mpmath.</tt><tt class="descname">gegenbauer</tt><big>(</big><em>n</em>, <em>a</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.gegenbauer" title="Permalink to this definition">¶</a></dt>
<dd><p>Evaluates the Gegenbauer polynomial, or ultraspherical polynomial,</p>
<div class="math">
<p><img src="../_images/math/e8d3dd33caf8cf4590357b0aa2e7aa24b5698d24.png" alt="C_n^{(a)}(z) = {n+2a-1 \choose n} \,_2F_1\left(-n, n+2a;
a+\frac{1}{2}; \frac{1}{2}(1-z)\right)." /></p>
</div><p>When <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/> is a nonnegative integer, this formula gives a polynomial
in <img class="math" src="../_images/math/b13f21416d84e13708696f34dea81026cda583c9.png" alt="z"/> of degree <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>, but all parameters are permitted to be
complex numbers. With <img class="math" src="../_images/math/b5c1efcd8e4680a8a20c6037b71494a6c427b7dc.png" alt="a = 1/2"/>, the Gegenbauer polynomial
reduces to a Legendre polynomial.</p>
<p><strong>Examples</strong></p>
<p>Evaluation for arbitrary arguments:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">gegenbauer</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">)</span>
<span class="go">-2485.0</span>
<span class="gp">>>> </span><span class="n">gegenbauer</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="go">3.012757178975667428359374e+2322</span>
<span class="gp">>>> </span><span class="n">gegenbauer</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.75</span><span class="p">,</span> <span class="o">-</span><span class="mi">1000</span><span class="n">j</span><span class="p">)</span>
<span class="go">(-5038991.358609026523401901 + 9414549.285447104177860806j)</span>
</pre></div>
</div>
<p>Evaluation at negative integer orders:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">gegenbauer</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">)</span>
<span class="go">-1.0</span>
<span class="gp">>>> </span><span class="n">gegenbauer</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">gegenbauer</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="n">j</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">gegenbauer</span><span class="p">(</span><span class="o">-</span><span class="mi">7</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="go">8989.0</span>
</pre></div>
</div>
<p>The Gegenbauer polynomials solve the differential equation:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">n</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="mf">4.5</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2</span><span class="n">j</span>
<span class="gp">>>> </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
<span class="gp">>>> </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="n">j</span><span class="p">]:</span>
<span class="gp">... </span> <span class="n">chop</span><span class="p">((</span><span class="mi">1</span><span class="o">-</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
<span class="gp">...</span>
<span class="go">0.0</span>
<span class="go">0.0</span>
<span class="go">0.0</span>
</pre></div>
</div>
<p>The Gegenbauer polynomials have generating function
<img class="math" src="../_images/math/7bf34994b503447d946212e98e25d8985e49f5a7.png" alt="(1-2zt+t^2)^{-a}"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">a</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">1</span>
<span class="gp">>>> </span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">*</span><span class="n">t</span><span class="o">+</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="go">[1.0, 5.0, 15.0, 35.0]</span>
<span class="gp">>>> </span><span class="p">[</span><span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">)]</span>
<span class="go">[1.0, 5.0, 15.0, 35.0]</span>
</pre></div>
</div>
<p>The Gegenbauer polynomials are orthogonal on <img class="math" src="../_images/math/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png" alt="[-1, 1]"/> with respect
to the weight <img class="math" src="../_images/math/b342b6a42721d088aec75e20cba163cd53b2e759.png" alt="(1-z^2)^{a-\frac{1}{2}}"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span>
<span class="gp">>>> </span><span class="n">Cn</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">zeroprec</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">Cm</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">gegenbauer</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">zeroprec</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">chop</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">Cn</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">Cm</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="mf">0.5</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]))</span>
<span class="go">0.0</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="hermite-polynomials">
<h2>Hermite polynomials<a class="headerlink" href="#hermite-polynomials" title="Permalink to this headline">¶</a></h2>
<div class="section" id="hermite">
<h3><tt class="xref docutils literal"><span class="pre">hermite()</span></tt><a class="headerlink" href="#hermite" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.hermite">
<tt class="descclassname">mpmath.</tt><tt class="descname">hermite</tt><big>(</big><em>n</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.hermite" title="Permalink to this definition">¶</a></dt>
<dd><p>Evaluates the Hermite polynomial <img class="math" src="../_images/math/5033e2e22e5ed9caf21225ee4090314dfbf58500.png" alt="H_n(z)"/>, which may be defined using
the recurrence</p>
<div class="math">
<p><img src="../_images/math/75b5fc753a430e7389262811a8a9b9319948cc27.png" alt="H_0(z) = 1
H_1(z) = 2z
H_{n+1} = 2z H_n(z) - 2n H_{n-1}(z)." /></p>
</div><p>The Hermite polynomials are orthogonal on <img class="math" src="../_images/math/57f240143a85a508de7721a7548b3b3c545547bc.png" alt="(-\infty, \infty)"/> with
respect to the weight <img class="math" src="../_images/math/bf5dce5a26b0584f66e28b1c50a5973c7d82e6e0.png" alt="e^{-z^2}"/>. More generally, allowing arbitrary complex
values of <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>, the Hermite function <img class="math" src="../_images/math/5033e2e22e5ed9caf21225ee4090314dfbf58500.png" alt="H_n(z)"/> is defined as</p>
<div class="math">
<p><img src="../_images/math/b31ec18c24de864ebf69620a0f263bc87b542886.png" alt="H_n(z) = (2z)^n \,_2F_0\left(-\frac{n}{2}, \frac{1-n}{2},
-\frac{1}{z^2}\right)" /></p>
</div><p>for <img class="math" src="../_images/math/bf01ae35b63816c0899ff913fa79b3f03b894160.png" alt="\Re{z} > 0"/>, or generally</p>
<div class="math">
<p><img src="../_images/math/44d4342a440ed69b0cafabe81984ad73470d4a98.png" alt="H_n(z) = 2^n \sqrt{\pi} \left(
\frac{1}{\Gamma\left(\frac{1-n}{2}\right)}
\,_1F_1\left(-\frac{n}{2}, \frac{1}{2}, z^2\right) -
\frac{2z}{\Gamma\left(-\frac{n}{2}\right)}
\,_1F_1\left(\frac{1-n}{2}, \frac{3}{2}, z^2\right)
\right)." /></p>
</div><p><strong>Plots</strong></p>
<div class="highlight-python"><div class="highlight"><pre><span class="c"># Hermite polynomials H_n(x) on the real line for n=0,1,2,3,4</span>
<span class="n">f0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">plot</span><span class="p">([</span><span class="n">f0</span><span class="p">,</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">],[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">],[</span><span class="o">-</span><span class="mi">25</span><span class="p">,</span><span class="mi">25</span><span class="p">])</span>
</pre></div>
</div>
<img alt="../_images/hermite.png" src="../_images/hermite.png" />
<p><strong>Examples</strong></p>
<p>Evaluation for arbitrary arguments:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">hermite</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="go">1.0</span>
<span class="gp">>>> </span><span class="n">hermite</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">);</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="go">20.0</span>
<span class="go">398.0</span>
<span class="gp">>>> </span><span class="n">hermite</span><span class="p">(</span><span class="mi">10000</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="go">4.950440066552087387515653e+19334</span>
<span class="gp">>>> </span><span class="n">hermite</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">8</span><span class="p">)</span>
<span class="go">-7999999999999998800000000.0</span>
<span class="gp">>>> </span><span class="n">hermite</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">8</span><span class="p">)</span>
<span class="go">1.675159751729877682920301e+4342944819032534</span>
<span class="gp">>>> </span><span class="n">hermite</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="mi">2</span><span class="n">j</span><span class="p">)</span>
<span class="go">(-0.07652130602993513389421901 - 0.1084662449961914580276007j)</span>
</pre></div>
</div>
<p>Coefficients of the first few Hermite polynomials are:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">7</span><span class="p">):</span>
<span class="gp">... </span> <span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
<span class="gp">...</span>
<span class="go">[1.0]</span>
<span class="go">[0.0, 2.0]</span>
<span class="go">[-2.0, 0.0, 4.0]</span>
<span class="go">[0.0, -12.0, 0.0, 8.0]</span>
<span class="go">[12.0, 0.0, -48.0, 0.0, 16.0]</span>
<span class="go">[0.0, 120.0, 0.0, -160.0, 0.0, 32.0]</span>
<span class="go">[-120.0, 0.0, 720.0, 0.0, -480.0, 0.0, 64.0]</span>
</pre></div>
</div>
<p>Values at <img class="math" src="../_images/math/aa9f0d97d2f39f78f05e05da40bf04f5a7c0726c.png" alt="z = 0"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">9</span><span class="p">):</span>
<span class="gp">... </span> <span class="n">hermite</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="gp">...</span>
<span class="go">0.02769459142039868792653387</span>
<span class="go">0.08333333333333333333333333</span>
<span class="go">0.2215567313631895034122709</span>
<span class="go">0.5</span>
<span class="go">0.8862269254527580136490837</span>
<span class="go">1.0</span>
<span class="go">0.0</span>
<span class="go">-2.0</span>
<span class="go">0.0</span>
<span class="go">12.0</span>
<span class="go">0.0</span>
<span class="go">-120.0</span>
<span class="go">0.0</span>
<span class="go">1680.0</span>
</pre></div>
</div>
<p>Hermite functions satisfy the differential equation:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">n</span> <span class="o">=</span> <span class="mi">4</span>
<span class="gp">>>> </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">z</span> <span class="o">=</span> <span class="mf">1.5</span>
<span class="gp">>>> </span><span class="n">chop</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
<span class="go">0.0</span>
</pre></div>
</div>
<p>Verifying orthogonality:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">chop</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">hermite</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="n">inf</span><span class="p">]))</span>
<span class="go">0.0</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="laguerre-polynomials">
<h2>Laguerre polynomials<a class="headerlink" href="#laguerre-polynomials" title="Permalink to this headline">¶</a></h2>
<div class="section" id="laguerre">
<h3><tt class="xref docutils literal"><span class="pre">laguerre()</span></tt><a class="headerlink" href="#laguerre" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.laguerre">
<tt class="descclassname">mpmath.</tt><tt class="descname">laguerre</tt><big>(</big><em>n</em>, <em>a</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.laguerre" title="Permalink to this definition">¶</a></dt>
<dd><p>Gives the generalized (associated) Laguerre polynomial, defined by</p>
<div class="math">
<p><img src="../_images/math/25665af88af510a7dd4a3464709efdc9fa8db92c.png" alt="L_n^a(z) = \frac{\Gamma(n+b+1)}{\Gamma(b+1) \Gamma(n+1)}
\,_1F_1(-n, a+1, z)." /></p>
</div><p>With <img class="math" src="../_images/math/b5969c2cfb1ffadf981fbbddcf796c8eb0037546.png" alt="a = 0"/> and <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/> a nonnegative integer, this reduces to an ordinary
Laguerre polynomial, the sequence of which begins
<img class="math" src="../_images/math/08424294ec8b0ce0f433a848518546edc44844f0.png" alt="L_0(z) = 1, L_1(z) = 1-z, L_2(z) = z^2-2z+1, \ldots"/>.</p>
<p>The Laguerre polynomials are orthogonal with respect to the weight
<img class="math" src="../_images/math/ced4c2d88cf637d4b56a416bdf8c2faca7bd73f4.png" alt="z^a e^{-z}"/> on <img class="math" src="../_images/math/a86f243b13e6ed9b3b2934d2dd9024dc39cdc6b8.png" alt="[0, \infty)"/>.</p>
<p><strong>Plots</strong></p>
<div class="highlight-python"><div class="highlight"><pre><span class="c"># Hermite polynomials L_n(x) on the real line for n=0,1,2,3,4</span>
<span class="n">f0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
<span class="n">plot</span><span class="p">([</span><span class="n">f0</span><span class="p">,</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">],[</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">])</span>
</pre></div>
</div>
<img alt="../_images/laguerre.png" src="../_images/laguerre.png" />
<p><strong>Examples</strong></p>
<p>Evaluation for arbitrary arguments:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">laguerre</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">)</span>
<span class="go">0.03726399739583333333333333</span>
<span class="gp">>>> </span><span class="n">laguerre</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span>
<span class="go">(4.474921610704496808379097 - 11.02058050372068958069241j)</span>
<span class="gp">>>> </span><span class="n">laguerre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10000</span><span class="p">)</span>
<span class="go">49980001.0</span>
<span class="gp">>>> </span><span class="n">laguerre</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10000</span><span class="p">)</span>
<span class="go">-9.327764910194842158583189e+4328</span>
</pre></div>
</div>
<p>The first few Laguerre polynomials, normalized to have integer
coefficients:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">7</span><span class="p">):</span>
<span class="gp">... </span> <span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">laguerre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
<span class="gp">...</span>
<span class="go">[1.0]</span>
<span class="go">[1.0, -1.0]</span>
<span class="go">[2.0, -4.0, 1.0]</span>
<span class="go">[6.0, -18.0, 9.0, -1.0]</span>
<span class="go">[24.0, -96.0, 72.0, -16.0, 1.0]</span>
<span class="go">[120.0, -600.0, 600.0, -200.0, 25.0, -1.0]</span>
<span class="go">[720.0, -4320.0, 5400.0, -2400.0, 450.0, -36.0, 1.0]</span>
</pre></div>
</div>
<p>Verifying orthogonality:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">Lm</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">Ln</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span>
<span class="gp">>>> </span><span class="n">chop</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">t</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="n">Lm</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">Ln</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">]))</span>
<span class="go">0.0</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="spherical-harmonics">
<h2>Spherical harmonics<a class="headerlink" href="#spherical-harmonics" title="Permalink to this headline">¶</a></h2>
<div class="section" id="spherharm">
<h3><tt class="xref docutils literal"><span class="pre">spherharm()</span></tt><a class="headerlink" href="#spherharm" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.spherharm">
<tt class="descclassname">mpmath.</tt><tt class="descname">spherharm</tt><big>(</big><em>l</em>, <em>m</em>, <em>theta</em>, <em>phi</em><big>)</big><a class="headerlink" href="#mpmath.spherharm" title="Permalink to this definition">¶</a></dt>
<dd><p>Evaluates the spherical harmonic <img class="math" src="../_images/math/2fabe01ef8bc01c79cb116512dca79311c870b95.png" alt="Y_l^m(\theta,\phi)"/>,</p>
<div class="math">
<p><img src="../_images/math/e368cb6772c14753dd2f73ae503399c0455595d0.png" alt="Y_l^m(\theta,\phi) = \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}
P_l^m(\cos \theta) e^{i m \phi}" /></p>
</div><p>where <img class="math" src="../_images/math/d8a3014951eb2ea04abd44b40541cac5a0fa4925.png" alt="P_l^m"/> is an associated Legendre function (see <a title="mpmath.legenp" class="reference internal" href="#mpmath.legenp"><tt class="xref docutils literal"><span class="pre">legenp()</span></tt></a>).</p>
<p>Here <img class="math" src="../_images/math/2881d0a92b53eb5568e81d2c74ea8ade94faa9fe.png" alt="\theta \in [0, \pi]"/> denotes the polar coordinate (ranging
from the north pole to the south pole) and <img class="math" src="../_images/math/c7371cd9a7326bd7e49ea7b4e35c8a84f40c76da.png" alt="\phi \in [0, 2 \pi]"/> denotes the
azimuthal coordinate on a sphere. Care should be used since many different
conventions for spherical coordinate variables are used.</p>
<p>Usually spherical harmonics are considered for <img class="math" src="../_images/math/237dcc5205c860f4c8e784f3ee130f9d9d991c94.png" alt="l \in \mathbb{N}"/>,
<img class="math" src="../_images/math/66f3365ae75a053eb41dbdff1a4d79415c1e8b22.png" alt="m \in \mathbb{Z}"/>, <img class="math" src="../_images/math/4e092c1d72ab491868c4fb6a9f469439ee524565.png" alt="|m| \le l"/>. More generally, <img class="math" src="../_images/math/c315ce0762c031bf8d8c1f4f45daaaab50cd46cc.png" alt="l,m,\theta,\phi"/>
are permitted to be complex numbers.</p>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last"><a title="mpmath.spherharm" class="reference internal" href="#mpmath.spherharm"><tt class="xref docutils literal"><span class="pre">spherharm()</span></tt></a> returns a complex number, even the value is
purely real.</p>
</div>
<p><strong>Plots</strong></p>
<div class="highlight-python"><div class="highlight"><pre><span class="c"># Real part of spherical harmonic Y_(4,0)(theta,phi)</span>
<span class="k">def</span> <span class="nf">Y</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">m</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">g</span><span class="p">(</span><span class="n">theta</span><span class="p">,</span><span class="n">phi</span><span class="p">):</span>
<span class="n">R</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">re</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">spherharm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">theta</span><span class="p">,</span><span class="n">phi</span><span class="p">)))</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">R</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">R</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">R</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="k">return</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">]</span>
<span class="k">return</span> <span class="n">g</span>
<span class="n">fp</span><span class="o">.</span><span class="n">splot</span><span class="p">(</span><span class="n">Y</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">fp</span><span class="o">.</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">pi</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">300</span><span class="p">)</span>
<span class="c"># fp.splot(Y(4,0), [0,fp.pi], [0,2*fp.pi], points=300)</span>
<span class="c"># fp.splot(Y(4,1), [0,fp.pi], [0,2*fp.pi], points=300)</span>
<span class="c"># fp.splot(Y(4,2), [0,fp.pi], [0,2*fp.pi], points=300)</span>
<span class="c"># fp.splot(Y(4,3), [0,fp.pi], [0,2*fp.pi], points=300)</span>
</pre></div>
</div>
<p><img class="math" src="../_images/math/67786de490601eb00f46371c76a516dd9aef5671.png" alt="Y_{4,0}"/>:</p>
<img alt="../_images/spherharm40.png" src="../_images/spherharm40.png" />
<p><img class="math" src="../_images/math/265d32f4a740077e5033b0db988576a66a033cff.png" alt="Y_{4,1}"/>:</p>
<img alt="../_images/spherharm41.png" src="../_images/spherharm41.png" />
<p><img class="math" src="../_images/math/4e63d2bc0094880bdce17dfc76bb01e0eb1471aa.png" alt="Y_{4,2}"/>:</p>
<img alt="../_images/spherharm42.png" src="../_images/spherharm42.png" />
<p><img class="math" src="../_images/math/5c345066c4d6201db89c4fd497f5071be9e58d3b.png" alt="Y_{4,3}"/>:</p>
<img alt="../_images/spherharm43.png" src="../_images/spherharm43.png" />
<p><img class="math" src="../_images/math/2104cca93a9078baa3699486688f344bd0e68f95.png" alt="Y_{4,4}"/>:</p>
<img alt="../_images/spherharm44.png" src="../_images/spherharm44.png" />
<p><strong>Examples</strong></p>
<p>Some low-order spherical harmonics with reference values:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">theta</span> <span class="o">=</span> <span class="n">pi</span><span class="o">/</span><span class="mi">4</span>
<span class="gp">>>> </span><span class="n">phi</span> <span class="o">=</span> <span class="n">pi</span><span class="o">/</span><span class="mi">3</span>
<span class="gp">>>> </span><span class="n">spherharm</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">theta</span><span class="p">,</span><span class="n">phi</span><span class="p">);</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">expj</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">(0.2820947917738781434740397 + 0.0j)</span>
<span class="go">(0.2820947917738781434740397 + 0.0j)</span>
<span class="gp">>>> </span><span class="n">spherharm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">theta</span><span class="p">,</span><span class="n">phi</span><span class="p">);</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">*</span><span class="n">expj</span><span class="p">(</span><span class="o">-</span><span class="n">phi</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="go">(0.1221506279757299803965962 - 0.2115710938304086076055298j)</span>
<span class="go">(0.1221506279757299803965962 - 0.2115710938304086076055298j)</span>
<span class="gp">>>> </span><span class="n">spherharm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">theta</span><span class="p">,</span><span class="n">phi</span><span class="p">);</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span><span class="o">*</span><span class="n">expj</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">(0.3454941494713354792652446 + 0.0j)</span>
<span class="go">(0.3454941494713354792652446 + 0.0j)</span>
<span class="gp">>>> </span><span class="n">spherharm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">theta</span><span class="p">,</span><span class="n">phi</span><span class="p">);</span> <span class="o">-</span><span class="mf">0.5</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">*</span><span class="n">expj</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="go">(-0.1221506279757299803965962 - 0.2115710938304086076055298j)</span>
<span class="go">(-0.1221506279757299803965962 - 0.2115710938304086076055298j)</span>
</pre></div>
</div>
<p>With the normalization convention used, the spherical harmonics are orthonormal
on the unit sphere:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">sphere</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">dS</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">:</span> <span class="n">fp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="c"># differential element</span>
<span class="gp">>>> </span><span class="n">Y1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">:</span> <span class="n">fp</span><span class="o">.</span><span class="n">spherharm</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span><span class="n">m1</span><span class="p">,</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">Y2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">:</span> <span class="n">fp</span><span class="o">.</span><span class="n">conj</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">spherharm</span><span class="p">(</span><span class="n">l2</span><span class="p">,</span><span class="n">m2</span><span class="p">,</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">l1</span> <span class="o">=</span> <span class="n">l2</span> <span class="o">=</span> <span class="mi">3</span><span class="p">;</span> <span class="n">m1</span> <span class="o">=</span> <span class="n">m2</span> <span class="o">=</span> <span class="mi">2</span>
<span class="gp">>>> </span><span class="k">print</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">:</span> <span class="n">Y1</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">Y2</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">dS</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">),</span> <span class="o">*</span><span class="n">sphere</span><span class="p">))</span>
<span class="go">(1+0j)</span>
<span class="gp">>>> </span><span class="n">m2</span> <span class="o">=</span> <span class="mi">1</span> <span class="c"># m1 != m2</span>
<span class="gp">>>> </span><span class="k">print</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">chop</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">:</span> <span class="n">Y1</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">Y2</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">dS</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">),</span> <span class="o">*</span><span class="n">sphere</span><span class="p">)))</span>
<span class="go">0.0</span>
</pre></div>
</div>
<p>Evaluation is accurate for large orders:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">spherharm</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="mi">750</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">)</span>
<span class="go">(3.776445785304252879026585e-102 - 5.82441278771834794493484e-102j)</span>
</pre></div>
</div>
<p>Evaluation works with complex parameter values:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">spherharm</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">2</span><span class="n">j</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="n">j</span><span class="p">)</span>
<span class="go">(64.44922331113759992154992 + 1981.693919841408089681743j)</span>
</pre></div>
</div>
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<ul>
<li><a class="reference external" href="#">Orthogonal polynomials</a><ul>
<li><a class="reference external" href="#legendre-functions">Legendre functions</a><ul>
<li><a class="reference external" href="#legendre"><tt class="docutils literal"><span class="pre">legendre()</span></tt></a></li>
<li><a class="reference external" href="#legenp"><tt class="docutils literal"><span class="pre">legenp()</span></tt></a></li>
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<li><a class="reference external" href="#chebyshev-polynomials">Chebyshev polynomials</a><ul>
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<li><a class="reference external" href="#jacobi-polynomials">Jacobi polynomials</a><ul>
<li><a class="reference external" href="#jacobi"><tt class="docutils literal"><span class="pre">jacobi()</span></tt></a></li>
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<li><a class="reference external" href="#gegenbauer-polynomials">Gegenbauer polynomials</a><ul>
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<li><a class="reference external" href="#hermite-polynomials">Hermite polynomials</a><ul>
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<li><a class="reference external" href="#laguerre-polynomials">Laguerre polynomials</a><ul>
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<li><a class="reference external" href="#spherical-harmonics">Spherical harmonics</a><ul>
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