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<div class="section" id="hyperbolic-functions">
<h1>Hyperbolic functions<a class="headerlink" href="#hyperbolic-functions" title="Permalink to this headline">¶</a></h1>
<div class="section" id="id1">
<h2>Hyperbolic functions<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h2>
<div class="section" id="cosh">
<h3><tt class="xref docutils literal"><span class="pre">cosh()</span></tt><a class="headerlink" href="#cosh" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.cosh">
<tt class="descclassname">mpmath.</tt><tt class="descname">cosh</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.cosh" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the hyperbolic cosine of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/3732ad83966f9b85d1b83c01f0ea37f4ec467ec8.png" alt="\cosh(x) = (e^x + e^{-x})/2"/>. 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">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">cosh</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">1.0</span>
<span class="gp">>>> </span><span class="n">cosh</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">1.543080634815243778477906</span>
<span class="gp">>>> </span><span class="n">cosh</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">),</span> <span class="n">cosh</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">)</span>
<span class="go">(+inf, +inf)</span>
</pre></div>
</div>
<p>The hyperbolic cosine is an even, convex function with
a global minimum at <img class="math" src="../_images/math/2d348bde3e15456e71734dc2c56fc7425c95927f.png" alt="x = 0"/>, having a Maclaurin series
that starts:</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">cosh</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">[1.0, 0.0, 0.5, 0.0, 0.0416667, 0.0]</span>
</pre></div>
</div>
<p>Generalized to complex numbers, the hyperbolic cosine is
equivalent to a cosine with the argument rotated
in the imaginary direction, or <img class="math" src="../_images/math/f0a0b4d06f22d8bc20f6f1592b18f65d243765ea.png" alt="\cosh x = \cos ix"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">cosh</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">(-3.724545504915322565473971 + 0.5118225699873846088344638j)</span>
<span class="gp">>>> </span><span class="n">cos</span><span class="p">(</span><span class="mi">3</span><span class="o">-</span><span class="mi">2</span><span class="n">j</span><span class="p">)</span>
<span class="go">(-3.724545504915322565473971 + 0.5118225699873846088344638j)</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="sinh">
<h3><tt class="xref docutils literal"><span class="pre">sinh()</span></tt><a class="headerlink" href="#sinh" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.sinh">
<tt class="descclassname">mpmath.</tt><tt class="descname">sinh</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.sinh" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the hyperbolic sine of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/9c750e3bd89537a51a94d226b330d08b3368703e.png" alt="\sinh(x) = (e^x - e^{-x})/2"/>. 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">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">sinh</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">sinh</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">1.175201193643801456882382</span>
<span class="gp">>>> </span><span class="n">sinh</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">),</span> <span class="n">sinh</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">)</span>
<span class="go">(-inf, +inf)</span>
</pre></div>
</div>
<p>The hyperbolic sine is an odd function, with a Maclaurin
series that starts:</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">sinh</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, 1.0, 0.0, 0.166667, 0.0, 0.00833333]</span>
</pre></div>
</div>
<p>Generalized to complex numbers, the hyperbolic sine is
essentially a sine with a rotation <img class="math" src="../_images/math/34857b3ba74ce5cd8607f3ebd23e9015908ada71.png" alt="i"/> applied to
the argument; more precisely, <img class="math" src="../_images/math/f02d498569463dab4b4f46ed42789fac5dbb0963.png" alt="\sinh x = -i \sin ix"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">sinh</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">(-3.590564589985779952012565 + 0.5309210862485198052670401j)</span>
<span class="gp">>>> </span><span class="n">j</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="mi">3</span><span class="o">-</span><span class="mi">2</span><span class="n">j</span><span class="p">)</span>
<span class="go">(-3.590564589985779952012565 + 0.5309210862485198052670401j)</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="tanh">
<h3><tt class="xref docutils literal"><span class="pre">tanh()</span></tt><a class="headerlink" href="#tanh" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.tanh">
<tt class="descclassname">mpmath.</tt><tt class="descname">tanh</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.tanh" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the hyperbolic tangent of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/950b32d9b0df97b3df6b9437f246638314046cb8.png" alt="\tanh(x) = \sinh(x)/\cosh(x)"/>. 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">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">tanh</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">tanh</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0.7615941559557648881194583</span>
<span class="gp">>>> </span><span class="n">tanh</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">),</span> <span class="n">tanh</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">(-1.0, 1.0)</span>
</pre></div>
</div>
<p>The hyperbolic tangent is an odd, sigmoidal function, similar
to the inverse tangent and error function. Its Maclaurin
series is:</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">tanh</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, 1.0, 0.0, -0.333333, 0.0, 0.133333]</span>
</pre></div>
</div>
<p>Generalized to complex numbers, the hyperbolic tangent is
essentially a tangent with a rotation <img class="math" src="../_images/math/34857b3ba74ce5cd8607f3ebd23e9015908ada71.png" alt="i"/> applied to
the argument; more precisely, <img class="math" src="../_images/math/3b086acaefd15f1d02fd7afb36aaaf5718b579ab.png" alt="\tanh x = -i \tan ix"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">tanh</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">(0.9653858790221331242784803 - 0.009884375038322493720314034j)</span>
<span class="gp">>>> </span><span class="n">j</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="mi">3</span><span class="o">-</span><span class="mi">2</span><span class="n">j</span><span class="p">)</span>
<span class="go">(0.9653858790221331242784803 - 0.009884375038322493720314034j)</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="sech">
<h3><tt class="xref docutils literal"><span class="pre">sech()</span></tt><a class="headerlink" href="#sech" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.sech">
<tt class="descclassname">mpmath.</tt><tt class="descname">sech</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.sech" title="Permalink to this definition">¶</a></dt>
<dd>Computes the hyperbolic secant of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/dc326752061626d1054325ef79ba61e23bb6b44d.png" alt="\mathrm{sech}(x) = \frac{1}{\cosh(x)}"/>.</dd></dl>
</div>
<div class="section" id="csch">
<h3><tt class="xref docutils literal"><span class="pre">csch()</span></tt><a class="headerlink" href="#csch" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.csch">
<tt class="descclassname">mpmath.</tt><tt class="descname">csch</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.csch" title="Permalink to this definition">¶</a></dt>
<dd>Computes the hyperbolic cosecant of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/1dcdba16431ffa1be6270c9a1b867e11851ab870.png" alt="\mathrm{csch}(x) = \frac{1}{\sinh(x)}"/>.</dd></dl>
</div>
<div class="section" id="coth">
<h3><tt class="xref docutils literal"><span class="pre">coth()</span></tt><a class="headerlink" href="#coth" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.coth">
<tt class="descclassname">mpmath.</tt><tt class="descname">coth</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.coth" title="Permalink to this definition">¶</a></dt>
<dd>Computes the hyperbolic cotangent of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/2e7af4a78169a6991408505a6a3ff7c082672897.png" alt="\mathrm{coth}(x) = \frac{\cosh(x)}{\sinh(x)}"/>.</dd></dl>
</div>
</div>
<div class="section" id="inverse-hyperbolic-functions">
<h2>Inverse hyperbolic functions<a class="headerlink" href="#inverse-hyperbolic-functions" title="Permalink to this headline">¶</a></h2>
<div class="section" id="acosh">
<h3><tt class="xref docutils literal"><span class="pre">acosh()</span></tt><a class="headerlink" href="#acosh" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.acosh">
<tt class="descclassname">mpmath.</tt><tt class="descname">acosh</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.acosh" title="Permalink to this definition">¶</a></dt>
<dd>Computes the inverse hyperbolic cosine of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/773f0ac59b8ae2656106a8ea9676367501c2f427.png" alt="\mathrm{cosh}^{-1}(x) = \log(x+\sqrt{x+1}\sqrt{x-1})"/>.</dd></dl>
</div>
<div class="section" id="asinh">
<h3><tt class="xref docutils literal"><span class="pre">asinh()</span></tt><a class="headerlink" href="#asinh" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.asinh">
<tt class="descclassname">mpmath.</tt><tt class="descname">asinh</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.asinh" title="Permalink to this definition">¶</a></dt>
<dd>Computes the inverse hyperbolic sine of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/00d552bbd073717999f5417100ca51aaec3b1321.png" alt="\mathrm{sinh}^{-1}(x) = \log(x+\sqrt{1+x^2})"/>.</dd></dl>
</div>
<div class="section" id="atanh">
<h3><tt class="xref docutils literal"><span class="pre">atanh()</span></tt><a class="headerlink" href="#atanh" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.atanh">
<tt class="descclassname">mpmath.</tt><tt class="descname">atanh</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.atanh" title="Permalink to this definition">¶</a></dt>
<dd>Computes the inverse hyperbolic tangent of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/48186734cfa7ab7c7db57ad6d732d5d2edb6688d.png" alt="\mathrm{tanh}^{-1}(x) = \frac{1}{2}\left(\log(1+x)-\log(1-x)\right)"/>.</dd></dl>
</div>
<div class="section" id="asech">
<h3><tt class="xref docutils literal"><span class="pre">asech()</span></tt><a class="headerlink" href="#asech" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.asech">
<tt class="descclassname">mpmath.</tt><tt class="descname">asech</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.asech" title="Permalink to this definition">¶</a></dt>
<dd>Computes the inverse hyperbolic secant of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/252da35eb0895e001fc56b61962b5f49257ddd53.png" alt="\mathrm{sech}^{-1}(x) = \cosh^{-1}(1/x)"/>.</dd></dl>
</div>
<div class="section" id="acsch">
<h3><tt class="xref docutils literal"><span class="pre">acsch()</span></tt><a class="headerlink" href="#acsch" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.acsch">
<tt class="descclassname">mpmath.</tt><tt class="descname">acsch</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.acsch" title="Permalink to this definition">¶</a></dt>
<dd>Computes the inverse hyperbolic cosecant of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/cab4520534b205f35806fbd1ac0142fc8520d1e5.png" alt="\mathrm{csch}^{-1}(x) = \sinh^{-1}(1/x)"/>.</dd></dl>
</div>
<div class="section" id="acoth">
<h3><tt class="xref docutils literal"><span class="pre">acoth()</span></tt><a class="headerlink" href="#acoth" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.acoth">
<tt class="descclassname">mpmath.</tt><tt class="descname">acoth</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.acoth" title="Permalink to this definition">¶</a></dt>
<dd>Computes the inverse hyperbolic cotangent of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/cec881617442e790e2604ae1acc19b9bf11252e3.png" alt="\mathrm{coth}^{-1}(x) = \tanh^{-1}(1/x)"/>.</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="#">Hyperbolic functions</a><ul>
<li><a class="reference external" href="#id1">Hyperbolic functions</a><ul>
<li><a class="reference external" href="#cosh"><tt class="docutils literal"><span class="pre">cosh()</span></tt></a></li>
<li><a class="reference external" href="#sinh"><tt class="docutils literal"><span class="pre">sinh()</span></tt></a></li>
<li><a class="reference external" href="#tanh"><tt class="docutils literal"><span class="pre">tanh()</span></tt></a></li>
<li><a class="reference external" href="#sech"><tt class="docutils literal"><span class="pre">sech()</span></tt></a></li>
<li><a class="reference external" href="#csch"><tt class="docutils literal"><span class="pre">csch()</span></tt></a></li>
<li><a class="reference external" href="#coth"><tt class="docutils literal"><span class="pre">coth()</span></tt></a></li>
</ul>
</li>
<li><a class="reference external" href="#inverse-hyperbolic-functions">Inverse hyperbolic functions</a><ul>
<li><a class="reference external" href="#acosh"><tt class="docutils literal"><span class="pre">acosh()</span></tt></a></li>
<li><a class="reference external" href="#asinh"><tt class="docutils literal"><span class="pre">asinh()</span></tt></a></li>
<li><a class="reference external" href="#atanh"><tt class="docutils literal"><span class="pre">atanh()</span></tt></a></li>
<li><a class="reference external" href="#asech"><tt class="docutils literal"><span class="pre">asech()</span></tt></a></li>
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