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<div class="section" id="exponential-integrals-and-error-functions">
<h1>Exponential integrals and error functions<a class="headerlink" href="#exponential-integrals-and-error-functions" title="Permalink to this headline">¶</a></h1>
<p>Exponential integrals give closed-form solutions to a large class of commonly occurring transcendental integrals that cannot be evaluated using elementary functions. Integrals of this type include those with an integrand of the form <img class="math" src="../_images/math/462d300df961dc8a9f6cdffea7a6b17f3fc6aea9.png" alt="t^a e^{t}"/> or <img class="math" src="../_images/math/1416120cf059226aaa74a003c2f78fee7310e8ed.png" alt="e^{-x^2}"/>, the latter giving rise to the Gaussian (or normal) probability distribution.</p>
<p>The most general function in this section is the incomplete gamma function, to which all others can be reduced. The incomplete gamma function, in turn, can be expressed using hypergeometric functions (see <a class="reference external" href="hypergeometric.html"><em>Hypergeometric functions</em></a>).</p>
<div class="section" id="incomplete-gamma-functions">
<h2>Incomplete gamma functions<a class="headerlink" href="#incomplete-gamma-functions" title="Permalink to this headline">¶</a></h2>
<div class="section" id="gammainc">
<h3><tt class="xref docutils literal"><span class="pre">gammainc()</span></tt><a class="headerlink" href="#gammainc" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.gammainc">
<tt class="descclassname">mpmath.</tt><tt class="descname">gammainc</tt><big>(</big><em>z</em>, <em>a=0</em>, <em>b=inf</em>, <em>regularized=False</em><big>)</big><a class="headerlink" href="#mpmath.gammainc" title="Permalink to this definition">¶</a></dt>
<dd><p><tt class="docutils literal"><span class="pre">gammainc(z,</span> <span class="pre">a=0,</span> <span class="pre">b=inf)</span></tt> computes the (generalized) incomplete
gamma function with integration limits <img class="math" src="../_images/math/da2e551d2ca2155b8d8f4935d2e9757722c9bab6.png" alt="[a, b]"/>:</p>
<div class="math">
<p><img src="../_images/math/1c35e8a24ac29184aa204a28b66fee318bab20dd.png" alt="\Gamma(z,a,b) = \int_a^b t^{z-1} e^{-t} \, dt" /></p>
</div><p>The generalized incomplete gamma function reduces to the
following special cases when one or both endpoints are fixed:</p>
<ul class="simple">
<li><img class="math" src="../_images/math/de26bc44efa7144cd4e6a60ee93f4e5f7c39d8e1.png" alt="\Gamma(z,0,\infty)"/> is the standard (“complete”)
gamma function, <img class="math" src="../_images/math/f3bdf82eb6ddbd7c4870a75d6652ffbbb8f9224b.png" alt="\Gamma(z)"/> (available directly
as the mpmath function <a title="mpmath.gamma" class="reference external" href="gamma.html#mpmath.gamma"><tt class="xref docutils literal"><span class="pre">gamma()</span></tt></a>)</li>
<li><img class="math" src="../_images/math/341baca0e80ef692894416999286ebd67b090b3b.png" alt="\Gamma(z,a,\infty)"/> is the “upper” incomplete gamma
function, <img class="math" src="../_images/math/4f0145b57fff1fcb4af69a8f4a7af00e525b11d4.png" alt="\Gamma(z,a)"/></li>
<li><img class="math" src="../_images/math/9ee8563ab8c1fe0852edb67920855aff8c2dd80b.png" alt="\Gamma(z,0,b)"/> is the “lower” incomplete gamma
function, <img class="math" src="../_images/math/78d0784efd9748c0c726d80e96aa81c72a5419e6.png" alt="\gamma(z,b)"/>.</li>
</ul>
<p>Of course, we have
<img class="math" src="../_images/math/e19701f34ac72a2affda63599f878c7d8664b9cb.png" alt="\Gamma(z,0,x) + \Gamma(z,x,\infty) = \Gamma(z)"/>
for all <img class="math" src="../_images/math/b13f21416d84e13708696f34dea81026cda583c9.png" alt="z"/> and <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>.</p>
<p>Note however that some authors reverse the order of the
arguments when defining the lower and upper incomplete
gamma function, so one should be careful to get the correct
definition.</p>
<p>If also given the keyword argument <tt class="docutils literal"><span class="pre">regularized=True</span></tt>,
<a title="mpmath.gammainc" class="reference internal" href="#mpmath.gammainc"><tt class="xref docutils literal"><span class="pre">gammainc()</span></tt></a> computes the “regularized” incomplete gamma
function</p>
<div class="math">
<p><img src="../_images/math/4e50287e68f70409610b11ce66b425056b08cf2c.png" alt="P(z,a,b) = \frac{\Gamma(z,a,b)}{\Gamma(z)}." /></p>
</div><p><strong>Examples</strong></p>
<p>We can compare with numerical quadrature to verify that
<a title="mpmath.gammainc" class="reference internal" href="#mpmath.gammainc"><tt class="xref docutils literal"><span class="pre">gammainc()</span></tt></a> computes the integral in the definition:</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">gammainc</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">4</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="go">(0.00977212668627705160602312 - 0.0770637306312989892451977j)</span>
<span class="gp">>>> </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">t</span><span class="o">**</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="o">-</span><span class="mi">1</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="p">),</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
<span class="go">(0.00977212668627705160602312 - 0.0770637306312989892451977j)</span>
</pre></div>
</div>
<p>Argument symmetries follow directly from the integral definition:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="n">gammainc</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">gammainc</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="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">gammainc</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="mi">4</span><span class="p">);</span> <span class="n">gammainc</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="mi">4</span><span class="p">)</span>
<span class="go">1.523793388892911312363331</span>
<span class="go">1.523793388892911312363331</span>
<span class="gp">>>> </span><span class="n">findroot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">gammainc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">3.0</span>
</pre></div>
</div>
<p>Evaluation for arbitrarily large arguments:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">gammainc</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">4.083660630910611272288592e-26</span>
<span class="gp">>>> </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10000000000000000</span><span class="p">)</span>
<span class="go">5.290402449901174752972486e-4342944819032375</span>
<span class="gp">>>> </span><span class="n">gammainc</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="mi">1000000</span><span class="o">+</span><span class="mi">1000000</span><span class="n">j</span><span class="p">)</span>
<span class="go">(-1.257913707524362408877881e-434284 + 2.556691003883483531962095e-434284j)</span>
</pre></div>
</div>
<p>Evaluation of a generalized incomplete gamma function automatically chooses
the representation that gives a more accurate result, depending on which
parameter is larger:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">10000000</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="n">gammainc</span><span class="p">(</span><span class="mi">10000000</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="c"># Bad</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">10000000</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c"># Good</span>
<span class="go">1.755146243738946045873491e+4771204</span>
<span class="gp">>>> </span><span class="n">gammainc</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">100000001</span><span class="p">)</span> <span class="o">-</span> <span class="n">gammainc</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">100000000</span><span class="p">)</span> <span class="c"># Bad</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">100000000</span><span class="p">,</span> <span class="mi">100000001</span><span class="p">)</span> <span class="c"># Good</span>
<span class="go">4.078258353474186729184421e-43429441</span>
</pre></div>
</div>
<p>The incomplete gamma functions satisfy simple recurrence
relations:</p>
<div class="highlight-python"><div class="highlight"><pre><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="gp">>>> </span><span class="n">z</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">3.5</span><span class="p">),</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">gammainc</span><span class="p">(</span><span class="n">z</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">);</span> <span class="n">z</span><span class="o">*</span><span class="n">gammainc</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="n">a</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span>
<span class="go">10.60130296933533459267329</span>
<span class="go">10.60130296933533459267329</span>
<span class="gp">>>> </span><span class="n">gammainc</span><span class="p">(</span><span class="n">z</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">a</span><span class="p">);</span> <span class="n">z</span><span class="o">*</span><span class="n">gammainc</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">a</span><span class="p">)</span> <span class="o">-</span> <span class="n">a</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span>
<span class="go">1.030425427232114336470932</span>
<span class="go">1.030425427232114336470932</span>
</pre></div>
</div>
<p>Evaluation at integers and poles:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">gammainc</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">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">)</span>
<span class="go">(-0.2214577048967798566234192 + 0.0j)</span>
<span class="gp">>>> </span><span class="n">gammainc</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
<span class="go">+inf</span>
</pre></div>
</div>
<p>If <img class="math" src="../_images/math/b13f21416d84e13708696f34dea81026cda583c9.png" alt="z"/> is an integer, the recurrence reduces the incomplete gamma
function to <img class="math" src="../_images/math/f78411aea3227e4ea059cf9814f1cb899d74bdb3.png" alt="P(a) \exp(-a) + Q(b) \exp(-b)"/> where <img class="math" src="../_images/math/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png" alt="P"/> and
<img class="math" src="../_images/math/9866e3a998d628ba0941eb4fea0666ac391d149a.png" alt="Q"/> are polynomials:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">gammainc</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">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
<span class="go">0.1353352832366126918939995</span>
<span class="go">0.1353352832366126918939995</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">50</span>
<span class="gp">>>> </span><span class="n">identify</span><span class="p">(</span><span class="n">gammainc</span><span class="p">(</span><span class="mi">6</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="p">[</span><span class="s">'exp(-1)'</span><span class="p">,</span> <span class="s">'exp(-2)'</span><span class="p">])</span>
<span class="go">'(326*exp(-1) + (-872)*exp(-2))'</span>
</pre></div>
</div>
<p>The incomplete gamma functions reduce to functions such as
the exponential integral Ei and the error function for special
arguments:</p>
<div class="highlight-python"><div class="highlight"><pre><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="gp">>>> </span><span class="n">gammainc</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="o">-</span><span class="n">ei</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">)</span>
<span class="go">0.00377935240984890647887486</span>
<span class="go">0.00377935240984890647887486</span>
<span class="gp">>>> </span><span class="n">gammainc</span><span class="p">(</span><span class="mf">0.5</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="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
<span class="go">1.691806732945198336509541</span>
<span class="go">1.691806732945198336509541</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="exponential-integrals">
<h2>Exponential integrals<a class="headerlink" href="#exponential-integrals" title="Permalink to this headline">¶</a></h2>
<div class="section" id="ei">
<h3><tt class="xref docutils literal"><span class="pre">ei()</span></tt><a class="headerlink" href="#ei" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.ei">
<tt class="descclassname">mpmath.</tt><tt class="descname">ei</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.ei" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the exponential integral or Ei-function, <img class="math" src="../_images/math/22547b8d0ed094d330ea27f6fb07e82a2bd29881.png" alt="\mathrm{Ei}(x)"/>.
The exponential integral is defined as</p>
<div class="math">
<p><img src="../_images/math/b6f50d1fcdf735206d82775af178e1e7db15f202.png" alt="\mathrm{Ei}(x) = \int_{-\infty\,}^x \frac{e^t}{t} \, dt." /></p>
</div><p>When the integration range includes <img class="math" src="../_images/math/288e2a39b58d673182c57dc6214b702a448341ea.png" alt="t = 0"/>, the exponential
integral is interpreted as providing the Cauchy principal value.</p>
<p>For real <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, the Ei-function behaves roughly like
<img class="math" src="../_images/math/bfa32ad8fa62d527d51b3594a987c41477327f46.png" alt="\mathrm{Ei}(x) \approx \exp(x) + \log(|x|)"/>.</p>
<p>The Ei-function is related to the more general family of exponential
integral functions denoted by <img class="math" src="../_images/math/0d3cbf9ab1e976e485e5a8b8459491ffa0b561dc.png" alt="E_n"/>, which are available as <a title="mpmath.expint" class="reference internal" href="#mpmath.expint"><tt class="xref docutils literal"><span class="pre">expint()</span></tt></a>.</p>
<p><strong>Basic examples</strong></p>
<p>Some basic values and limits are:</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">ei</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">-inf</span>
<span class="gp">>>> </span><span class="n">ei</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">1.89511781635594</span>
<span class="gp">>>> </span><span class="n">ei</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">+inf</span>
<span class="gp">>>> </span><span class="n">ei</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
<span class="go">0.0</span>
</pre></div>
</div>
<p>For <img class="math" src="../_images/math/36a68e8aa6ec6f48231ad80a4851f84bacf641b3.png" alt="x < 0"/>, the defining integral can be evaluated
numerically as a reference:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">ei</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">)</span>
<span class="go">-0.00377935240984891</span>
<span class="gp">>>> </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="n">t</span><span class="p">)</span><span class="o">/</span><span class="n">t</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">])</span>
<span class="go">-0.00377935240984891</span>
</pre></div>
</div>
<p><a title="mpmath.ei" class="reference internal" href="#mpmath.ei"><tt class="xref docutils literal"><span class="pre">ei()</span></tt></a> supports complex arguments and arbitrary
precision evaluation:</p>
<div class="highlight-python"><div class="highlight"><pre><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">50</span>
<span class="gp">>>> </span><span class="n">ei</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
<span class="go">10.928374389331410348638445906907535171566338835056</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="gp">>>> </span><span class="n">ei</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="go">(-4.154091651642689822535359 + 4.294418620024357476985535j)</span>
</pre></div>
</div>
<p><strong>Related functions</strong></p>
<p>The exponential integral is closely related to the logarithmic
integral. See <a title="mpmath.li" class="reference internal" href="#mpmath.li"><tt class="xref docutils literal"><span class="pre">li()</span></tt></a> for additional information.</p>
<p>The exponential integral is related to the hyperbolic
and trigonometric integrals (see <a title="mpmath.chi" class="reference internal" href="#mpmath.chi"><tt class="xref docutils literal"><span class="pre">chi()</span></tt></a>, <a title="mpmath.shi" class="reference internal" href="#mpmath.shi"><tt class="xref docutils literal"><span class="pre">shi()</span></tt></a>,
<a title="mpmath.ci" class="reference internal" href="#mpmath.ci"><tt class="xref docutils literal"><span class="pre">ci()</span></tt></a>, <a title="mpmath.si" class="reference internal" href="#mpmath.si"><tt class="xref docutils literal"><span class="pre">si()</span></tt></a>) similarly to how the ordinary
exponential function is related to the hyperbolic and
trigonometric functions:</p>
<div class="highlight-python"><div class="highlight"><pre><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">ei</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">9.93383257062542</span>
<span class="gp">>>> </span><span class="n">chi</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="n">shi</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">9.93383257062542</span>
<span class="gp">>>> </span><span class="n">chop</span><span class="p">(</span><span class="n">ci</span><span class="p">(</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span> <span class="o">-</span> <span class="n">j</span><span class="o">*</span><span class="n">si</span><span class="p">(</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span> <span class="o">-</span> <span class="n">pi</span><span class="o">*</span><span class="n">j</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
<span class="go">9.93383257062542</span>
</pre></div>
</div>
<p>Beware that logarithmic corrections, as in the last example
above, are required to obtain the correct branch in general.
For details, see [1].</p>
<p>The exponential integral is also a special case of the
hypergeometric function <img class="math" src="../_images/math/1f6fb6dd6d16bd0ea3f9258e3b552ae516743eed.png" alt="\,_2F_2"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">z</span> <span class="o">=</span> <span class="mf">0.6</span>
<span class="gp">>>> </span><span class="n">z</span><span class="o">*</span><span class="n">hyper</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="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">ln</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">-</span><span class="n">ln</span><span class="p">(</span><span class="mi">1</span><span class="o">/</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">euler</span>
<span class="go">0.769881289937359</span>
<span class="gp">>>> </span><span class="n">ei</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
<span class="go">0.769881289937359</span>
</pre></div>
</div>
<p><strong>References</strong></p>
<ol class="arabic simple">
<li>Relations between Ei and other functions:
<a class="reference external" href="http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/27/01/">http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/27/01/</a></li>
<li>Abramowitz & Stegun, section 5:
<a class="reference external" href="http://www.math.sfu.ca/~cbm/aands/page_228.htm">http://www.math.sfu.ca/~cbm/aands/page_228.htm</a></li>
<li>Asymptotic expansion for Ei:
<a class="reference external" href="http://mathworld.wolfram.com/En-Function.html">http://mathworld.wolfram.com/En-Function.html</a></li>
</ol>
</dd></dl>
</div>
<div class="section" id="e1">
<h3><tt class="xref docutils literal"><span class="pre">e1()</span></tt><a class="headerlink" href="#e1" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.e1">
<tt class="descclassname">mpmath.</tt><tt class="descname">e1</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.e1" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the exponential integral <img class="math" src="../_images/math/2cca7336acceec044052743bc80eb0c775702420.png" alt="\mathrm{E}_1(z)"/>, given by</p>
<div class="math">
<p><img src="../_images/math/29a66a0fefdb5f734d89dd12e5bd3e1428de70bf.png" alt="\mathrm{E}_1(z) = \int_z^{\infty} \frac{e^{-t}}{t} dt." /></p>
</div><p>This is equivalent to <a title="mpmath.expint" class="reference internal" href="#mpmath.expint"><tt class="xref docutils literal"><span class="pre">expint()</span></tt></a> with <img class="math" src="../_images/math/c766f9fce6070981e62f805a7cf7294baed4498e.png" alt="n = 1"/>.</p>
<p><strong>Examples</strong></p>
<p>Two ways to evaluate this function:</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">e1</span><span class="p">(</span><span class="mf">6.25</span><span class="p">)</span>
<span class="go">0.0002704758872637179088496194</span>
<span class="gp">>>> </span><span class="n">expint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mf">6.25</span><span class="p">)</span>
<span class="go">0.0002704758872637179088496194</span>
</pre></div>
</div>
<p>The E1-function is essentially the same as the Ei-function (<a title="mpmath.ei" class="reference internal" href="#mpmath.ei"><tt class="xref docutils literal"><span class="pre">ei()</span></tt></a>)
with negated argument, except for an imaginary branch cut term:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">e1</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
<span class="go">0.02491491787026973549562801</span>
<span class="gp">>>> </span><span class="o">-</span><span class="n">ei</span><span class="p">(</span><span class="o">-</span><span class="mf">2.5</span><span class="p">)</span>
<span class="go">0.02491491787026973549562801</span>
<span class="gp">>>> </span><span class="n">e1</span><span class="p">(</span><span class="o">-</span><span class="mf">2.5</span><span class="p">)</span>
<span class="go">(-7.073765894578600711923552 - 3.141592653589793238462643j)</span>
<span class="gp">>>> </span><span class="o">-</span><span class="n">ei</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
<span class="go">-7.073765894578600711923552</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="expint">
<h3><tt class="xref docutils literal"><span class="pre">expint()</span></tt><a class="headerlink" href="#expint" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.expint">
<tt class="descclassname">mpmath.</tt><tt class="descname">expint</tt><big>(</big><em>*args</em><big>)</big><a class="headerlink" href="#mpmath.expint" title="Permalink to this definition">¶</a></dt>
<dd><p><tt class="xref docutils literal"><span class="pre">expint(n,z)()</span></tt> gives the generalized exponential integral
or En-function,</p>
<div class="math">
<p><img src="../_images/math/b1fcac81169f1d6a27095d4eb2fa5070cd52be0b.png" alt="\mathrm{E}_n(z) = \int_1^{\infty} \frac{e^{-zt}}{t^n} dt," /></p>
</div><p>where <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/> and <img class="math" src="../_images/math/b13f21416d84e13708696f34dea81026cda583c9.png" alt="z"/> may both be complex numbers. The case with <img class="math" src="../_images/math/c766f9fce6070981e62f805a7cf7294baed4498e.png" alt="n = 1"/> is
also given by <a title="mpmath.e1" class="reference internal" href="#mpmath.e1"><tt class="xref docutils literal"><span class="pre">e1()</span></tt></a>.</p>
<p><strong>Examples</strong></p>
<p>Evaluation at real and complex 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">expint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">6.25</span><span class="p">)</span>
<span class="go">0.0002704758872637179088496194</span>
<span class="gp">>>> </span><span class="n">expint</span><span class="p">(</span><span class="o">-</span><span class="mi">3</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">(0.00299658467335472929656159 + 0.06100816202125885450319632j)</span>
<span class="gp">>>> </span><span class="n">expint</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">4</span><span class="o">-</span><span class="mi">5</span><span class="n">j</span><span class="p">)</span>
<span class="go">(0.001803529474663565056945248 - 0.002235061547756185403349091j)</span>
</pre></div>
</div>
<p>At negative integer values of <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/>, <img class="math" src="../_images/math/293ac7f9abdef50b197d8a3aee3acdbd1476db34.png" alt="E_n(z)"/> reduces to a
rational-exponential function:</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">n</span><span class="p">,</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="nb">sum</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="n">k</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">n</span><span class="o">+</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span>\
<span class="gp">... </span> <span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">z</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="p">)</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">z</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">pi</span>
<span class="gp">>>> </span><span class="n">expint</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
<span class="go">584.2604820613019908668219</span>
<span class="gp">>>> </span><span class="n">f</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="go">584.2604820613019908668219</span>
<span class="gp">>>> </span><span class="n">n</span> <span class="o">=</span> <span class="mi">5</span>
<span class="gp">>>> </span><span class="n">expint</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
<span class="go">115366.5762594725451811138</span>
<span class="gp">>>> </span><span class="n">f</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="go">115366.5762594725451811138</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="logarithmic-integral">
<h2>Logarithmic integral<a class="headerlink" href="#logarithmic-integral" title="Permalink to this headline">¶</a></h2>
<div class="section" id="li">
<h3><tt class="xref docutils literal"><span class="pre">li()</span></tt><a class="headerlink" href="#li" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.li">
<tt class="descclassname">mpmath.</tt><tt class="descname">li</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.li" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the logarithmic integral or li-function
<img class="math" src="../_images/math/0d6e907c00d0dad55d41518059061920eb2e90c8.png" alt="\mathrm{li}(x)"/>, defined by</p>
<div class="math">
<p><img src="../_images/math/f75ae63efadf7c64db115111b33de06ad147df96.png" alt="\mathrm{li}(x) = \int_0^x \frac{1}{\log t} \, dt" /></p>
</div><p>The logarithmic integral has a singularity at <img class="math" src="../_images/math/34310f2f36ed6d724838edb08788ee62acb33386.png" alt="x = 1"/>.</p>
<p>Alternatively, <tt class="docutils literal"><span class="pre">li(x,</span> <span class="pre">offset=True)</span></tt> computes the offset
logarithmic integral (used in number theory)</p>
<div class="math">
<p><img src="../_images/math/b968979126cad6e37986a4f71553ab559a04388d.png" alt="\mathrm{Li}(x) = \int_2^x \frac{1}{\log t} \, dt." /></p>
</div><p>These two functions are related via the simple identity
<img class="math" src="../_images/math/afd2720cc15b5e1df0fc0dbcdd26f48aa1cf6aca.png" alt="\mathrm{Li}(x) = \mathrm{li}(x) - \mathrm{li}(2)"/>.</p>
<p>The logarithmic integral should also not be confused with
the polylogarithm (also denoted by Li), which is implemented
as <a title="mpmath.polylog" class="reference external" href="zeta.html#mpmath.polylog"><tt class="xref docutils literal"><span class="pre">polylog()</span></tt></a>.</p>
<p><strong>Examples</strong></p>
<p>Some basic values and limits:</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">30</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">li</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">li</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">-inf</span>
<span class="gp">>>> </span><span class="n">li</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">-inf</span>
<span class="gp">>>> </span><span class="n">li</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">1.04516378011749278484458888919</span>
<span class="gp">>>> </span><span class="n">findroot</span><span class="p">(</span><span class="n">li</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="go">1.45136923488338105028396848589</span>
<span class="gp">>>> </span><span class="n">li</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">+inf</span>
<span class="gp">>>> </span><span class="n">li</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">offset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">li</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">offset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="go">-inf</span>
<span class="gp">>>> </span><span class="n">li</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">offset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="go">-1.04516378011749278484458888919</span>
<span class="gp">>>> </span><span class="n">li</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="n">offset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="go">5.12043572466980515267839286347</span>
</pre></div>
</div>
<p>The logarithmic integral can be evaluated for arbitrary
complex arguments:</p>
<div class="highlight-python"><div class="highlight"><pre><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">20</span>
<span class="gp">>>> </span><span class="n">li</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="go">(3.1343755504645775265 + 2.6769247817778742392j)</span>
</pre></div>
</div>
<p>The logarithmic integral is related to the exponential integral:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">ei</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
<span class="go">2.1635885946671919729</span>
<span class="gp">>>> </span><span class="n">li</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">2.1635885946671919729</span>
</pre></div>
</div>
<p>The logarithmic integral grows like <img class="math" src="../_images/math/c43ced27cffc61a02cf7bdd2dc5761e7f1c3d379.png" alt="O(x/\log(x))"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><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">x</span> <span class="o">=</span> <span class="mi">10</span><span class="o">**</span><span class="mi">100</span>
<span class="gp">>>> </span><span class="n">x</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">4.34294481903252e+97</span>
<span class="gp">>>> </span><span class="n">li</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">4.3619719871407e+97</span>
</pre></div>
</div>
<p>The prime number theorem states that the number of primes less
than <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/> is asymptotic to <img class="math" src="../_images/math/581063678c8b649d703e950eda717730c7be682b.png" alt="\mathrm{Li}(x)"/> (equivalently
<img class="math" src="../_images/math/0d6e907c00d0dad55d41518059061920eb2e90c8.png" alt="\mathrm{li}(x)"/>). For example, it is known that there are
exactly 1,925,320,391,606,803,968,923 prime numbers less than
<img class="math" src="../_images/math/97e2bfde528cba61dcb20b160c0f3919e7ada9b1.png" alt="10^{23}"/> [1]. The logarithmic integral provides a very
accurate estimate:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">li</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">23</span><span class="p">,</span> <span class="n">offset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="go">1.92532039161405e+21</span>
</pre></div>
</div>
<p>A definite integral is:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">quad</span><span class="p">(</span><span class="n">li</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">-0.693147180559945</span>
<span class="gp">>>> </span><span class="o">-</span><span class="n">ln</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">-0.693147180559945</span>
</pre></div>
</div>
<p><strong>References</strong></p>
<ol class="arabic simple">
<li><a class="reference external" href="http://mathworld.wolfram.com/PrimeCountingFunction.html">http://mathworld.wolfram.com/PrimeCountingFunction.html</a></li>
<li><a class="reference external" href="http://mathworld.wolfram.com/LogarithmicIntegral.html">http://mathworld.wolfram.com/LogarithmicIntegral.html</a></li>
</ol>
</dd></dl>
</div>
</div>
<div class="section" id="trigonometric-integrals">
<h2>Trigonometric integrals<a class="headerlink" href="#trigonometric-integrals" title="Permalink to this headline">¶</a></h2>
<div class="section" id="ci">
<h3><tt class="xref docutils literal"><span class="pre">ci()</span></tt><a class="headerlink" href="#ci" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.ci">
<tt class="descclassname">mpmath.</tt><tt class="descname">ci</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.ci" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the cosine integral,</p>
<div class="math">
<p><img src="../_images/math/88a47638ad4c93c10e6b3625519e81bc9371a542.png" alt="\mathrm{Ci}(x) = -\int_x^{\infty} \frac{\cos t}{t}\,dt
= \gamma + \log x + \int_0^x \frac{\cos t - 1}{t}\,dt" /></p>
</div><p><strong>Examples</strong></p>
<p>Some values and limits:</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">ci</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">-inf</span>
<span class="gp">>>> </span><span class="n">ci</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0.3374039229009681346626462</span>
<span class="gp">>>> </span><span class="n">ci</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
<span class="go">0.07366791204642548599010096</span>
<span class="gp">>>> </span><span class="n">ci</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">ci</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
<span class="go">(0.0 + 3.141592653589793238462643j)</span>
<span class="gp">>>> </span><span class="n">ci</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">(1.408292501520849518759125 - 2.983617742029605093121118j)</span>
</pre></div>
</div>
<p>The cosine integral behaves roughly like the sinc function
(see <a title="mpmath.sinc" class="reference external" href="trigonometric.html#mpmath.sinc"><tt class="xref docutils literal"><span class="pre">sinc()</span></tt></a>) for large real <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">ci</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
<span class="go">-4.875060251748226537857298e-11</span>
<span class="gp">>>> </span><span class="n">sinc</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
<span class="go">-4.875060250875106915277943e-11</span>
<span class="gp">>>> </span><span class="n">chop</span><span class="p">(</span><span class="n">limit</span><span class="p">(</span><span class="n">ci</span><span class="p">,</span> <span class="n">inf</span><span class="p">))</span>
<span class="go">0.0</span>
</pre></div>
</div>
<p>It has infinitely many roots on the positive real axis:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">findroot</span><span class="p">(</span><span class="n">ci</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">0.6165054856207162337971104</span>
<span class="gp">>>> </span><span class="n">findroot</span><span class="p">(</span><span class="n">ci</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="go">3.384180422551186426397851</span>
</pre></div>
</div>
<p>Evaluation is supported for <img class="math" src="../_images/math/b13f21416d84e13708696f34dea81026cda583c9.png" alt="z"/> anywhere in the complex plane:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">ci</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">6</span><span class="o">*</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="go">(4.449410587611035724984376e+434287 + 9.75744874290013526417059e+434287j)</span>
</pre></div>
</div>
<p>We can evaluate the defining integral as a reference:</p>
<div class="highlight-python"><div class="highlight"><pre><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="o">-</span><span class="n">quadosc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">/</span><span class="n">t</span><span class="p">,</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">omega</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="go">-0.190029749656644</span>
<span class="gp">>>> </span><span class="n">ci</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="go">-0.190029749656644</span>
</pre></div>
</div>
<p>Some infinite series can be evaluated using the
cosine integral:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="o">/</span><span class="p">(</span><span class="n">fac</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</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">k</span><span class="p">)),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
<span class="go">-0.239811742000565</span>
<span class="gp">>>> </span><span class="n">ci</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">euler</span>
<span class="go">-0.239811742000565</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="si">
<h3><tt class="xref docutils literal"><span class="pre">si()</span></tt><a class="headerlink" href="#si" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.si">
<tt class="descclassname">mpmath.</tt><tt class="descname">si</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.si" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the sine integral,</p>
<div class="math">
<p><img src="../_images/math/c6f4de1fa90d03021c350326a1de82372fcc8486.png" alt="\mathrm{Si}(x) = \int_0^x \frac{\sin t}{t}\,dt." /></p>
</div><p>The sine integral is thus the antiderivative of the sinc
function (see <a title="mpmath.sinc" class="reference external" href="trigonometric.html#mpmath.sinc"><tt class="xref docutils literal"><span class="pre">sinc()</span></tt></a>).</p>
<p><strong>Examples</strong></p>
<p>Some values and limits:</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">si</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">si</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0.9460830703671830149413533</span>
<span class="gp">>>> </span><span class="n">si</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">-0.9460830703671830149413533</span>
<span class="gp">>>> </span><span class="n">si</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
<span class="go">1.851937051982466170361053</span>
<span class="gp">>>> </span><span class="n">si</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">1.570796326794896619231322</span>
<span class="gp">>>> </span><span class="n">si</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
<span class="go">-1.570796326794896619231322</span>
<span class="gp">>>> </span><span class="n">si</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">(4.547513889562289219853204 + 1.399196580646054789459839j)</span>
</pre></div>
</div>
<p>The sine integral approaches <img class="math" src="../_images/math/039bf945c704ff98e6f81626bbad32550cc37193.png" alt="\pi/2"/> for large real <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">si</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
<span class="go">1.570796326707584656968511</span>
<span class="gp">>>> </span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span>
<span class="go">1.570796326794896619231322</span>
</pre></div>
</div>
<p>Evaluation is supported for <img class="math" src="../_images/math/b13f21416d84e13708696f34dea81026cda583c9.png" alt="z"/> anywhere in the complex plane:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">si</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">6</span><span class="o">*</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="go">(-9.75744874290013526417059e+434287 + 4.449410587611035724984376e+434287j)</span>
</pre></div>
</div>
<p>We can evaluate the defining integral as a reference:</p>
<div class="highlight-python"><div class="highlight"><pre><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">quad</span><span class="p">(</span><span class="n">sinc</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
<span class="go">1.54993124494467</span>
<span class="gp">>>> </span><span class="n">si</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="go">1.54993124494467</span>
</pre></div>
</div>
<p>Some infinite series can be evaluated using the
sine integral:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="o">/</span><span class="p">(</span><span class="n">fac</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="o">+</span><span class="mi">1</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">k</span><span class="o">+</span><span class="mi">1</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">0.946083070367183</span>
<span class="gp">>>> </span><span class="n">si</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0.946083070367183</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="hyperbolic-integrals">
<h2>Hyperbolic integrals<a class="headerlink" href="#hyperbolic-integrals" title="Permalink to this headline">¶</a></h2>
<div class="section" id="chi">
<h3><tt class="xref docutils literal"><span class="pre">chi()</span></tt><a class="headerlink" href="#chi" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.chi">
<tt class="descclassname">mpmath.</tt><tt class="descname">chi</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.chi" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the hyperbolic cosine integral, defined
in analogy with the cosine integral (see <a title="mpmath.ci" class="reference internal" href="#mpmath.ci"><tt class="xref docutils literal"><span class="pre">ci()</span></tt></a>) as</p>
<div class="math">
<p><img src="../_images/math/a1ba2d9e8191139ad1ef2a63a2601774b544321f.png" alt="\mathrm{Chi}(x) = -\int_x^{\infty} \frac{\cosh t}{t}\,dt
= \gamma + \log x + \int_0^x \frac{\cosh t - 1}{t}\,dt" /></p>
</div><p>Some values and limits:</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">chi</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">-inf</span>
<span class="gp">>>> </span><span class="n">chi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0.8378669409802082408946786</span>
<span class="gp">>>> </span><span class="n">chi</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">+inf</span>
<span class="gp">>>> </span><span class="n">findroot</span><span class="p">(</span><span class="n">chi</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="go">0.5238225713898644064509583</span>
<span class="gp">>>> </span><span class="n">chi</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">(-0.1683628683277204662429321 + 2.625115880451325002151688j)</span>
</pre></div>
</div>
<p>Evaluation is supported for <img class="math" src="../_images/math/b13f21416d84e13708696f34dea81026cda583c9.png" alt="z"/> anywhere in the complex plane:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">chi</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">6</span><span class="o">*</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="go">(4.449410587611035724984376e+434287 - 9.75744874290013526417059e+434287j)</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="shi">
<h3><tt class="xref docutils literal"><span class="pre">shi()</span></tt><a class="headerlink" href="#shi" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.shi">
<tt class="descclassname">mpmath.</tt><tt class="descname">shi</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.shi" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the hyperbolic sine integral, defined
in analogy with the sine integral (see <a title="mpmath.si" class="reference internal" href="#mpmath.si"><tt class="xref docutils literal"><span class="pre">si()</span></tt></a>) as</p>
<div class="math">
<p><img src="../_images/math/71770811bdc4686450bc654f03f550e9dcac3f18.png" alt="\mathrm{Shi}(x) = \int_0^x \frac{\sinh t}{t}\,dt." /></p>
</div><p>Some values and limits:</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">shi</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">shi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">1.057250875375728514571842</span>
<span class="gp">>>> </span><span class="n">shi</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">-1.057250875375728514571842</span>
<span class="gp">>>> </span><span class="n">shi</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">+inf</span>
<span class="gp">>>> </span><span class="n">shi</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">(-0.1931890762719198291678095 + 2.645432555362369624818525j)</span>
</pre></div>
</div>
<p>Evaluation is supported for <img class="math" src="../_images/math/b13f21416d84e13708696f34dea81026cda583c9.png" alt="z"/> anywhere in the complex plane:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">shi</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">6</span><span class="o">*</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="go">(4.449410587611035724984376e+434287 - 9.75744874290013526417059e+434287j)</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="error-functions">
<h2>Error functions<a class="headerlink" href="#error-functions" title="Permalink to this headline">¶</a></h2>
<div class="section" id="erf">
<h3><tt class="xref docutils literal"><span class="pre">erf()</span></tt><a class="headerlink" href="#erf" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.erf">
<tt class="descclassname">mpmath.</tt><tt class="descname">erf</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.erf" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the error function, <img class="math" src="../_images/math/24a55f83d76fde51296370fe18ea839a557f53b4.png" alt="\mathrm{erf}(x)"/>. The error
function is the normalized antiderivative of the Gaussian function
<img class="math" src="../_images/math/fb60aca3a6fcb6082e294c04c9cca57f49916065.png" alt="\exp(-t^2)"/>. More precisely,</p>
<div class="math">
<p><img src="../_images/math/ea21142041f1b312ff67d54e1b750820675ff290.png" alt="\mathrm{erf}(x) = \frac{2}{\sqrt \pi} \int_0^x \exp(-t^2) \,dt" /></p>
</div><p><strong>Basic examples</strong></p>
<p>Simple values and limits include:</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">erf</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0.842700792949715</span>
<span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">-0.842700792949715</span>
<span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">1.0</span>
<span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
<span class="go">-1.0</span>
</pre></div>
</div>
<p>For large real <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, <img class="math" src="../_images/math/24a55f83d76fde51296370fe18ea839a557f53b4.png" alt="\mathrm{erf}(x)"/> approaches 1 very
rapidly:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">0.999977909503001</span>
<span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="go">0.999999999998463</span>
</pre></div>
</div>
<p>The error function is an odd function:</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">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">erf</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)))</span>
<span class="go">[0.0, 1.12838, 0.0, -0.376126, 0.0, 0.112838]</span>
</pre></div>
</div>
<p><a title="mpmath.erf" class="reference internal" href="#mpmath.erf"><tt class="xref docutils literal"><span class="pre">erf()</span></tt></a> implements arbitrary-precision evaluation and
supports complex numbers:</p>
<div class="highlight-python"><div class="highlight"><pre><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">50</span>
<span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
<span class="go">0.52049987781304653768274665389196452873645157575796</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="gp">>>> </span><span class="n">erf</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="go">(1.316151281697947644880271 + 0.1904534692378346862841089j)</span>
</pre></div>
</div>
<p>Evaluation is supported for large arguments:</p>
<div class="highlight-python"><div class="highlight"><pre><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="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="s">'1e1000'</span><span class="p">)</span>
<span class="go">1.0</span>
<span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="s">'-1e1000'</span><span class="p">)</span>
<span class="go">-1.0</span>
<span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="s">'1e-1000'</span><span class="p">)</span>
<span class="go">1.128379167095512573896159e-1000</span>
<span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="s">'1e7j'</span><span class="p">)</span>
<span class="go">(0.0 + 8.593897639029319267398803e+43429448190317j)</span>
<span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="s">'1e7+1e7j'</span><span class="p">)</span>
<span class="go">(0.9999999858172446172631323 + 3.728805278735270407053139e-8j)</span>
</pre></div>
</div>
<p><strong>Related functions</strong></p>
<p>See also <a title="mpmath.erfc" class="reference internal" href="#mpmath.erfc"><tt class="xref docutils literal"><span class="pre">erfc()</span></tt></a>, which is more accurate for large <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
and <a title="mpmath.erfi" class="reference internal" href="#mpmath.erfi"><tt class="xref docutils literal"><span class="pre">erfi()</span></tt></a> which gives the antiderivative of
<img class="math" src="../_images/math/b133c76827decf65dd75b6f1a1a38dcf7c9476bf.png" alt="\exp(t^2)"/>.</p>
<p>The Fresnel integrals <a title="mpmath.fresnels" class="reference internal" href="#mpmath.fresnels"><tt class="xref docutils literal"><span class="pre">fresnels()</span></tt></a> and <a title="mpmath.fresnelc" class="reference internal" href="#mpmath.fresnelc"><tt class="xref docutils literal"><span class="pre">fresnelc()</span></tt></a>
are also related to the error function.</p>
</dd></dl>
</div>
<div class="section" id="erfc">
<h3><tt class="xref docutils literal"><span class="pre">erfc()</span></tt><a class="headerlink" href="#erfc" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.erfc">
<tt class="descclassname">mpmath.</tt><tt class="descname">erfc</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.erfc" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the complementary error function,
<img class="math" src="../_images/math/a8e29942bdfda249887cb34ecc6bdda95a1c7159.png" alt="\mathrm{erfc}(x) = 1-\mathrm{erf}(x)"/>.
This function avoids cancellation that occurs when naively
computing the complementary error function as <tt class="docutils literal"><span class="pre">1-erf(x)</span></tt>:</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="mi">1</span> <span class="o">-</span> <span class="n">erf</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">erfc</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="go">2.08848758376254e-45</span>
</pre></div>
</div>
<p><a title="mpmath.erfc" class="reference internal" href="#mpmath.erfc"><tt class="xref docutils literal"><span class="pre">erfc()</span></tt></a> works accurately even for ludicrously large
arguments:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">erfc</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
<span class="go">4.3504398860243e-43429448190325182776</span>
</pre></div>
</div>
<p>Complex arguments are supported:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">erfc</span><span class="p">(</span><span class="mi">500</span><span class="o">+</span><span class="mi">50</span><span class="n">j</span><span class="p">)</span>
<span class="go">(1.19739830969552e-107492 + 1.46072418957528e-107491j)</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="erfi">
<h3><tt class="xref docutils literal"><span class="pre">erfi()</span></tt><a class="headerlink" href="#erfi" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.erfi">
<tt class="descclassname">mpmath.</tt><tt class="descname">erfi</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.erfi" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the imaginary error function, <img class="math" src="../_images/math/a89580e826af5a48058d0bdf9f4312452fe8d889.png" alt="\mathrm{erfi}(x)"/>.
The imaginary error function is defined in analogy with the
error function, but with a positive sign in the integrand:</p>
<div class="math">
<p><img src="../_images/math/61318b2c70c69603e7e71d9cb72ea7235ecf7b1f.png" alt="\mathrm{erfi}(x) = \frac{2}{\sqrt \pi} \int_0^x \exp(t^2) \,dt" /></p>
</div><p>Whereas the error function rapidly converges to 1 as <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/> grows,
the imaginary error function rapidly diverges to infinity.
The functions are related as
<img class="math" src="../_images/math/3018e8ed0c527e8a994940c6df66b75d9b9522e9.png" alt="\mathrm{erfi}(x) = -i\,\mathrm{erf}(ix)"/> for all complex
numbers <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>.</p>
<p><strong>Examples</strong></p>
<p>Basic values and limits:</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">erfi</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">erfi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">1.65042575879754</span>
<span class="gp">>>> </span><span class="n">erfi</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">-1.65042575879754</span>
<span class="gp">>>> </span><span class="n">erfi</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">+inf</span>
<span class="gp">>>> </span><span class="n">erfi</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
<span class="go">-inf</span>
</pre></div>
</div>
<p>Note the symmetry between erf and erfi:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">erfi</span><span class="p">(</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span>
<span class="go">(0.0 + 0.999977909503001j)</span>
<span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">0.999977909503001</span>
<span class="gp">>>> </span><span class="n">erf</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="p">)</span>
<span class="go">(-0.536643565778565 - 5.04914370344703j)</span>
<span class="gp">>>> </span><span class="n">erfi</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">1</span><span class="n">j</span><span class="p">)</span>
<span class="go">(-5.04914370344703 - 0.536643565778565j)</span>
</pre></div>
</div>
<p>Large arguments are supported:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">erfi</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
<span class="go">1.71130938718796e+434291</span>
<span class="gp">>>> </span><span class="n">erfi</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
<span class="go">7.3167287567024e+43429448190325182754</span>
<span class="gp">>>> </span><span class="n">erfi</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
<span class="go">-7.3167287567024e+43429448190325182754</span>
<span class="gp">>>> </span><span class="n">erfi</span><span class="p">(</span><span class="mi">1000</span><span class="o">-</span><span class="mi">500</span><span class="n">j</span><span class="p">)</span>
<span class="go">(2.49895233563961e+325717 + 2.6846779342253e+325717j)</span>
<span class="gp">>>> </span><span class="n">erfi</span><span class="p">(</span><span class="mi">100000</span><span class="n">j</span><span class="p">)</span>
<span class="go">(0.0 + 1.0j)</span>
<span class="gp">>>> </span><span class="n">erfi</span><span class="p">(</span><span class="o">-</span><span class="mi">100000</span><span class="n">j</span><span class="p">)</span>
<span class="go">(0.0 - 1.0j)</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="erfinv">
<h3><tt class="xref docutils literal"><span class="pre">erfinv()</span></tt><a class="headerlink" href="#erfinv" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.erfinv">
<tt class="descclassname">mpmath.</tt><tt class="descname">erfinv</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.erfinv" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the inverse error function, satisfying</p>
<div class="math">
<p><img src="../_images/math/a7d28a51d8aa10dd203b36eded7a6b7707eba7d4.png" alt="\mathrm{erf}(\mathrm{erfinv}(x)) =
\mathrm{erfinv}(\mathrm{erf}(x)) = x." /></p>
</div><p>This function is defined only for <img class="math" src="../_images/math/f260c78fb4c935f83297e9c2cb2cf7c3c3e3adfc.png" alt="-1 \le x \le 1"/>.</p>
<p><strong>Examples</strong></p>
<p>Special values include:</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">erfinv</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">erfinv</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">+inf</span>
<span class="gp">>>> </span><span class="n">erfinv</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">-inf</span>
</pre></div>
</div>
<p>The domain is limited to the standard interval:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">erfinv</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="nc">ValueError</span>: <span class="n-Identifier">erfinv(x) is defined only for -1 <= x <= 1</span>
</pre></div>
</div>
<p>It is simple to check that <a title="mpmath.erfinv" class="reference internal" href="#mpmath.erfinv"><tt class="xref docutils literal"><span class="pre">erfinv()</span></tt></a> computes inverse values of
<a title="mpmath.erf" class="reference internal" href="#mpmath.erf"><tt class="xref docutils literal"><span class="pre">erf()</span></tt></a> as promised:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="n">erfinv</span><span class="p">(</span><span class="mf">0.75</span><span class="p">))</span>
<span class="go">0.75</span>
<span class="gp">>>> </span><span class="n">erf</span><span class="p">(</span><span class="n">erfinv</span><span class="p">(</span><span class="o">-</span><span class="mf">0.995</span><span class="p">))</span>
<span class="go">-0.995</span>
</pre></div>
</div>
<p><a title="mpmath.erfinv" class="reference internal" href="#mpmath.erfinv"><tt class="xref docutils literal"><span class="pre">erfinv()</span></tt></a> supports arbitrary-precision evaluation:</p>
<div class="highlight-python"><div class="highlight"><pre><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">50</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">erf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">x</span>
<span class="go">0.99532226501895273416206925636725292861089179704006</span>
<span class="gp">>>> </span><span class="n">erfinv</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">2.0</span>
</pre></div>
</div>
<p>A definite integral involving the inverse error function:</p>
<div class="highlight-python"><div class="highlight"><pre><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">quad</span><span class="p">(</span><span class="n">erfinv</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">0.564189583547756</span>
<span class="gp">>>> </span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
<span class="go">0.564189583547756</span>
</pre></div>
</div>
<p>The inverse error function can be used to generate random numbers
with a Gaussian distribution (although this is a relatively
inefficient algorithm):</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">erfinv</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">rand</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="c"># doctest: +SKIP</span>
<span class="go">[-0.586747, 1.10233, -0.376796, 0.926037, -0.708142, -0.732012]</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="the-normal-distribution">
<h2>The normal distribution<a class="headerlink" href="#the-normal-distribution" title="Permalink to this headline">¶</a></h2>
<div class="section" id="npdf">
<h3><tt class="xref docutils literal"><span class="pre">npdf()</span></tt><a class="headerlink" href="#npdf" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.npdf">
<tt class="descclassname">mpmath.</tt><tt class="descname">npdf</tt><big>(</big><em>x</em>, <em>mu=0</em>, <em>sigma=1</em><big>)</big><a class="headerlink" href="#mpmath.npdf" title="Permalink to this definition">¶</a></dt>
<dd><p><tt class="docutils literal"><span class="pre">npdf(x,</span> <span class="pre">mu=0,</span> <span class="pre">sigma=1)</span></tt> evaluates the probability density
function of a normal distribution with mean value <img class="math" src="../_images/math/2d8c833ed800824727cd7bd2fb9de1a12ad7e674.png" alt="\mu"/>
and variance <img class="math" src="../_images/math/741fb9098efcb98055f467f87630a5d0ca599b6b.png" alt="\sigma^2"/>.</p>
<p>Elementary properties of the probability distribution can
be verified using numerical integration:</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">quad</span><span class="p">(</span><span class="n">npdf</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">1.0</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">npdf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
<span class="go">0.5</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">npdf</span><span class="p">(</span><span class="n">x</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="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
<span class="go">0.5</span>
</pre></div>
</div>
<p>See also <a title="mpmath.ncdf" class="reference internal" href="#mpmath.ncdf"><tt class="xref docutils literal"><span class="pre">ncdf()</span></tt></a>, which gives the cumulative
distribution.</p>
</dd></dl>
</div>
<div class="section" id="ncdf">
<h3><tt class="xref docutils literal"><span class="pre">ncdf()</span></tt><a class="headerlink" href="#ncdf" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.ncdf">
<tt class="descclassname">mpmath.</tt><tt class="descname">ncdf</tt><big>(</big><em>x</em>, <em>mu=0</em>, <em>sigma=1</em><big>)</big><a class="headerlink" href="#mpmath.ncdf" title="Permalink to this definition">¶</a></dt>
<dd><p><tt class="docutils literal"><span class="pre">ncdf(x,</span> <span class="pre">mu=0,</span> <span class="pre">sigma=1)</span></tt> evaluates the cumulative distribution
function of a normal distribution with mean value <img class="math" src="../_images/math/2d8c833ed800824727cd7bd2fb9de1a12ad7e674.png" alt="\mu"/>
and variance <img class="math" src="../_images/math/741fb9098efcb98055f467f87630a5d0ca599b6b.png" alt="\sigma^2"/>.</p>
<p>See also <a title="mpmath.npdf" class="reference internal" href="#mpmath.npdf"><tt class="xref docutils literal"><span class="pre">npdf()</span></tt></a>, which gives the probability density.</p>
<p>Elementary properties include:</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">ncdf</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">mu</span><span class="o">=</span><span class="n">pi</span><span class="p">)</span>
<span class="go">0.5</span>
<span class="gp">>>> </span><span class="n">ncdf</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">ncdf</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">)</span>
<span class="go">1.0</span>
</pre></div>
</div>
<p>The cumulative distribution is the integral of the density
function having identical mu and sigma:</p>
<div class="highlight-python"><div class="highlight"><pre><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">diff</span><span class="p">(</span><span class="n">ncdf</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="go">0.053990966513188</span>
<span class="gp">>>> </span><span class="n">npdf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">0.053990966513188</span>
<span class="gp">>>> </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">ncdf</span><span class="p">(</span><span class="n">x</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="mi">0</span><span class="p">)</span>
<span class="go">0.107981933026376</span>
<span class="gp">>>> </span><span class="n">npdf</span><span class="p">(</span><span class="mi">0</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">0.107981933026376</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="fresnel-integrals">
<h2>Fresnel integrals<a class="headerlink" href="#fresnel-integrals" title="Permalink to this headline">¶</a></h2>
<div class="section" id="fresnels">
<h3><tt class="xref docutils literal"><span class="pre">fresnels()</span></tt><a class="headerlink" href="#fresnels" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.fresnels">
<tt class="descclassname">mpmath.</tt><tt class="descname">fresnels</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.fresnels" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the Fresnel sine integral</p>
<div class="math">
<p><img src="../_images/math/9733ffe71bb6a0806cce5f481093b7126f3eb2c5.png" alt="S(x) = \int_0^x \sin\left(\frac{\pi t^2}{2}\right) \,dt" /></p>
</div><p>Note that some sources define this function
without the normalization factor <img class="math" src="../_images/math/039bf945c704ff98e6f81626bbad32550cc37193.png" alt="\pi/2"/>.</p>
<p><strong>Examples</strong></p>
<p>Some basic values and limits:</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">fresnels</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">fresnels</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">0.5</span>
<span class="gp">>>> </span><span class="n">fresnels</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
<span class="go">-0.5</span>
<span class="gp">>>> </span><span class="n">fresnels</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0.4382591473903547660767567</span>
<span class="gp">>>> </span><span class="n">fresnels</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="p">)</span>
<span class="go">(36.72546488399143842838788 + 15.58775110440458732748279j)</span>
</pre></div>
</div>
<p>Comparing with the definition:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">fresnels</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">0.4963129989673750360976123</span>
<span class="gp">>>> </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">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">),</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">0.4963129989673750360976123</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="fresnelc">
<h3><tt class="xref docutils literal"><span class="pre">fresnelc()</span></tt><a class="headerlink" href="#fresnelc" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.fresnelc">
<tt class="descclassname">mpmath.</tt><tt class="descname">fresnelc</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.fresnelc" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the Fresnel cosine integral</p>
<div class="math">
<p><img src="../_images/math/b120f1ef25e8a877e5b748b59645ef709e763a04.png" alt="C(x) = \int_0^x \cos\left(\frac{\pi t^2}{2}\right) \,dt" /></p>
</div><p>Note that some sources define this function
without the normalization factor <img class="math" src="../_images/math/039bf945c704ff98e6f81626bbad32550cc37193.png" alt="\pi/2"/>.</p>
<p><strong>Examples</strong></p>
<p>Some basic values and limits:</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">fresnelc</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">fresnelc</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">0.5</span>
<span class="gp">>>> </span><span class="n">fresnelc</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
<span class="go">-0.5</span>
<span class="gp">>>> </span><span class="n">fresnelc</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0.7798934003768228294742064</span>
<span class="gp">>>> </span><span class="n">fresnelc</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="p">)</span>
<span class="go">(16.08787137412548041729489 - 36.22568799288165021578758j)</span>
</pre></div>
</div>
<p>Comparing with the definition:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">fresnelc</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">0.6057207892976856295561611</span>
<span class="gp">>>> </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">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">),</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">0.6057207892976856295561611</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
</div>
</div>
</div>
</div>
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<h3><a href="../index.html">Table Of Contents</a></h3>
<ul>
<li><a class="reference external" href="#">Exponential integrals and error functions</a><ul>
<li><a class="reference external" href="#incomplete-gamma-functions">Incomplete gamma functions</a><ul>
<li><a class="reference external" href="#gammainc"><tt class="docutils literal"><span class="pre">gammainc()</span></tt></a></li>
</ul>
</li>
<li><a class="reference external" href="#exponential-integrals">Exponential integrals</a><ul>
<li><a class="reference external" href="#ei"><tt class="docutils literal"><span class="pre">ei()</span></tt></a></li>
<li><a class="reference external" href="#e1"><tt class="docutils literal"><span class="pre">e1()</span></tt></a></li>
<li><a class="reference external" href="#expint"><tt class="docutils literal"><span class="pre">expint()</span></tt></a></li>
</ul>
</li>
<li><a class="reference external" href="#logarithmic-integral">Logarithmic integral</a><ul>
<li><a class="reference external" href="#li"><tt class="docutils literal"><span class="pre">li()</span></tt></a></li>
</ul>
</li>
<li><a class="reference external" href="#trigonometric-integrals">Trigonometric integrals</a><ul>
<li><a class="reference external" href="#ci"><tt class="docutils literal"><span class="pre">ci()</span></tt></a></li>
<li><a class="reference external" href="#si"><tt class="docutils literal"><span class="pre">si()</span></tt></a></li>
</ul>
</li>
<li><a class="reference external" href="#hyperbolic-integrals">Hyperbolic integrals</a><ul>
<li><a class="reference external" href="#chi"><tt class="docutils literal"><span class="pre">chi()</span></tt></a></li>
<li><a class="reference external" href="#shi"><tt class="docutils literal"><span class="pre">shi()</span></tt></a></li>
</ul>
</li>
<li><a class="reference external" href="#error-functions">Error functions</a><ul>
<li><a class="reference external" href="#erf"><tt class="docutils literal"><span class="pre">erf()</span></tt></a></li>
<li><a class="reference external" href="#erfc"><tt class="docutils literal"><span class="pre">erfc()</span></tt></a></li>
<li><a class="reference external" href="#erfi"><tt class="docutils literal"><span class="pre">erfi()</span></tt></a></li>
<li><a class="reference external" href="#erfinv"><tt class="docutils literal"><span class="pre">erfinv()</span></tt></a></li>
</ul>
</li>
<li><a class="reference external" href="#the-normal-distribution">The normal distribution</a><ul>
<li><a class="reference external" href="#npdf"><tt class="docutils literal"><span class="pre">npdf()</span></tt></a></li>
<li><a class="reference external" href="#ncdf"><tt class="docutils literal"><span class="pre">ncdf()</span></tt></a></li>
</ul>
</li>
<li><a class="reference external" href="#fresnel-integrals">Fresnel integrals</a><ul>
<li><a class="reference external" href="#fresnels"><tt class="docutils literal"><span class="pre">fresnels()</span></tt></a></li>
<li><a class="reference external" href="#fresnelc"><tt class="docutils literal"><span class="pre">fresnelc()</span></tt></a></li>
</ul>
</li>
</ul>
</li>
</ul>
<h4>Previous topic</h4>
<p class="topless"><a href="gamma.html"
title="previous chapter">Factorials and gamma functions</a></p>
<h4>Next topic</h4>
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title="next chapter">Bessel functions and related functions</a></p>
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