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<h1>Mathematical functions<a class="headerlink" href="#mathematical-functions" title="Permalink to this headline">ΒΆ</a></h1>
<p>Mpmath implements the standard functions from Python’s <tt class="docutils literal"><span class="pre">math</span></tt> and <tt class="docutils literal"><span class="pre">cmath</span></tt> modules, for both real and complex numbers and with arbitrary precision. Many other functions are also available in mpmath, including commonly-used variants of standard functions (such as the alternative trigonometric functions sec, csc, cot), but also a large number of “special functions” such as the gamma function, the Riemann zeta function, error functions, Bessel functions, etc.</p>
<ul>
<li class="toctree-l1"><a class="reference external" href="constants.html">Mathematical constants</a><ul>
<li class="toctree-l2"><a class="reference external" href="constants.html#exact-constants">Exact constants</a></li>
<li class="toctree-l2"><a class="reference external" href="constants.html#pi-pi">Pi (<tt class="docutils literal"><span class="pre">pi</span></tt>)</a></li>
<li class="toctree-l2"><a class="reference external" href="constants.html#degree-degree">Degree (<tt class="docutils literal"><span class="pre">degree</span></tt>)</a></li>
<li class="toctree-l2"><a class="reference external" href="constants.html#base-of-the-natural-logarithm-e">Base of the natural logarithm (<tt class="docutils literal"><span class="pre">e</span></tt>)</a></li>
<li class="toctree-l2"><a class="reference external" href="constants.html#golden-ratio-phi">Golden ratio (<tt class="docutils literal"><span class="pre">phi</span></tt>)</a></li>
<li class="toctree-l2"><a class="reference external" href="constants.html#euler-s-constant-euler">Euler’s constant (<tt class="docutils literal"><span class="pre">euler</span></tt>)</a></li>
<li class="toctree-l2"><a class="reference external" href="constants.html#catalan-s-constant-catalan">Catalan’s constant (<tt class="docutils literal"><span class="pre">catalan</span></tt>)</a></li>
<li class="toctree-l2"><a class="reference external" href="constants.html#apery-s-constant-apery">Apery’s constant (<tt class="docutils literal"><span class="pre">apery</span></tt>)</a></li>
<li class="toctree-l2"><a class="reference external" href="constants.html#khinchin-s-constant-khinchin">Khinchin’s constant (<tt class="docutils literal"><span class="pre">khinchin</span></tt>)</a></li>
<li class="toctree-l2"><a class="reference external" href="constants.html#glaisher-s-constant-glaisher">Glaisher’s constant (<tt class="docutils literal"><span class="pre">glaisher</span></tt>)</a></li>
<li class="toctree-l2"><a class="reference external" href="constants.html#mertens-constant-mertens">Mertens constant (<tt class="docutils literal"><span class="pre">mertens</span></tt>)</a></li>
<li class="toctree-l2"><a class="reference external" href="constants.html#twin-prime-constant-twinprime">Twin prime constant (<tt class="docutils literal"><span class="pre">twinprime</span></tt>)</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference external" href="powers.html">Powers and logarithms</a><ul>
<li class="toctree-l2"><a class="reference external" href="powers.html#nth-roots">Nth roots</a></li>
<li class="toctree-l2"><a class="reference external" href="powers.html#exponentiation">Exponentiation</a></li>
<li class="toctree-l2"><a class="reference external" href="powers.html#logarithms">Logarithms</a></li>
<li class="toctree-l2"><a class="reference external" href="powers.html#lambert-w-function">Lambert W function</a></li>
<li class="toctree-l2"><a class="reference external" href="powers.html#arithmetic-geometric-mean">Arithmetic-geometric mean</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference external" href="trigonometric.html">Trigonometric functions</a><ul>
<li class="toctree-l2"><a class="reference external" href="trigonometric.html#degree-radian-conversion">Degree-radian conversion</a></li>
<li class="toctree-l2"><a class="reference external" href="trigonometric.html#id1">Trigonometric functions</a></li>
<li class="toctree-l2"><a class="reference external" href="trigonometric.html#trigonometric-functions-with-modified-argument">Trigonometric functions with modified argument</a></li>
<li class="toctree-l2"><a class="reference external" href="trigonometric.html#inverse-trigonometric-functions">Inverse trigonometric functions</a></li>
<li class="toctree-l2"><a class="reference external" href="trigonometric.html#sinc-function">Sinc function</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference external" href="hyperbolic.html">Hyperbolic functions</a><ul>
<li class="toctree-l2"><a class="reference external" href="hyperbolic.html#id1">Hyperbolic functions</a></li>
<li class="toctree-l2"><a class="reference external" href="hyperbolic.html#inverse-hyperbolic-functions">Inverse hyperbolic functions</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference external" href="gamma.html">Factorials and gamma functions</a><ul>
<li class="toctree-l2"><a class="reference external" href="gamma.html#factorials">Factorials</a></li>
<li class="toctree-l2"><a class="reference external" href="gamma.html#binomial-coefficients">Binomial coefficients</a></li>
<li class="toctree-l2"><a class="reference external" href="gamma.html#gamma-function">Gamma function</a></li>
<li class="toctree-l2"><a class="reference external" href="gamma.html#rising-and-falling-factorials">Rising and falling factorials</a></li>
<li class="toctree-l2"><a class="reference external" href="gamma.html#beta-function">Beta function</a></li>
<li class="toctree-l2"><a class="reference external" href="gamma.html#super-and-hyperfactorials">Super- and hyperfactorials</a></li>
<li class="toctree-l2"><a class="reference external" href="gamma.html#polygamma-functions-and-harmonic-numbers">Polygamma functions and harmonic numbers</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference external" href="expintegrals.html">Exponential integrals and error functions</a><ul>
<li class="toctree-l2"><a class="reference external" href="expintegrals.html#incomplete-gamma-functions">Incomplete gamma functions</a></li>
<li class="toctree-l2"><a class="reference external" href="expintegrals.html#exponential-integrals">Exponential integrals</a></li>
<li class="toctree-l2"><a class="reference external" href="expintegrals.html#logarithmic-integral">Logarithmic integral</a></li>
<li class="toctree-l2"><a class="reference external" href="expintegrals.html#trigonometric-integrals">Trigonometric integrals</a></li>
<li class="toctree-l2"><a class="reference external" href="expintegrals.html#hyperbolic-integrals">Hyperbolic integrals</a></li>
<li class="toctree-l2"><a class="reference external" href="expintegrals.html#error-functions">Error functions</a></li>
<li class="toctree-l2"><a class="reference external" href="expintegrals.html#the-normal-distribution">The normal distribution</a></li>
<li class="toctree-l2"><a class="reference external" href="expintegrals.html#fresnel-integrals">Fresnel integrals</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference external" href="bessel.html">Bessel functions and related functions</a><ul>
<li class="toctree-l2"><a class="reference external" href="bessel.html#bessel-functions">Bessel functions</a></li>
<li class="toctree-l2"><a class="reference external" href="bessel.html#bessel-function-zeros">Bessel function zeros</a></li>
<li class="toctree-l2"><a class="reference external" href="bessel.html#hankel-functions">Hankel functions</a></li>
<li class="toctree-l2"><a class="reference external" href="bessel.html#kelvin-functions">Kelvin functions</a></li>
<li class="toctree-l2"><a class="reference external" href="bessel.html#struve-functions">Struve functions</a></li>
<li class="toctree-l2"><a class="reference external" href="bessel.html#anger-weber-functions">Anger-Weber functions</a></li>
<li class="toctree-l2"><a class="reference external" href="bessel.html#lommel-functions">Lommel functions</a></li>
<li class="toctree-l2"><a class="reference external" href="bessel.html#airy-and-scorer-functions">Airy and Scorer functions</a></li>
<li class="toctree-l2"><a class="reference external" href="bessel.html#coulomb-wave-functions">Coulomb wave functions</a></li>
<li class="toctree-l2"><a class="reference external" href="bessel.html#confluent-u-and-whittaker-functions">Confluent U and Whittaker functions</a></li>
<li class="toctree-l2"><a class="reference external" href="bessel.html#parabolic-cylinder-functions">Parabolic cylinder functions</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference external" href="orthogonal.html">Orthogonal polynomials</a><ul>
<li class="toctree-l2"><a class="reference external" href="orthogonal.html#legendre-functions">Legendre functions</a></li>
<li class="toctree-l2"><a class="reference external" href="orthogonal.html#chebyshev-polynomials">Chebyshev polynomials</a></li>
<li class="toctree-l2"><a class="reference external" href="orthogonal.html#jacobi-polynomials">Jacobi polynomials</a></li>
<li class="toctree-l2"><a class="reference external" href="orthogonal.html#gegenbauer-polynomials">Gegenbauer polynomials</a></li>
<li class="toctree-l2"><a class="reference external" href="orthogonal.html#hermite-polynomials">Hermite polynomials</a></li>
<li class="toctree-l2"><a class="reference external" href="orthogonal.html#laguerre-polynomials">Laguerre polynomials</a></li>
<li class="toctree-l2"><a class="reference external" href="orthogonal.html#spherical-harmonics">Spherical harmonics</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference external" href="hypergeometric.html">Hypergeometric functions</a><ul>
<li class="toctree-l2"><a class="reference external" href="hypergeometric.html#common-hypergeometric-series">Common hypergeometric series</a></li>
<li class="toctree-l2"><a class="reference external" href="hypergeometric.html#generalized-hypergeometric-functions">Generalized hypergeometric functions</a></li>
<li class="toctree-l2"><a class="reference external" href="hypergeometric.html#meijer-g-function">Meijer G-function</a></li>
<li class="toctree-l2"><a class="reference external" href="hypergeometric.html#bilateral-hypergeometric-series">Bilateral hypergeometric series</a></li>
<li class="toctree-l2"><a class="reference external" href="hypergeometric.html#hypergeometric-functions-of-two-variables">Hypergeometric functions of two variables</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference external" href="elliptic.html">Elliptic functions</a><ul>
<li class="toctree-l2"><a class="reference external" href="elliptic.html#elliptic-arguments">Elliptic arguments</a></li>
<li class="toctree-l2"><a class="reference external" href="elliptic.html#legendre-elliptic-integrals">Legendre elliptic integrals</a></li>
<li class="toctree-l2"><a class="reference external" href="elliptic.html#carlson-symmetric-elliptic-integrals">Carlson symmetric elliptic integrals</a></li>
<li class="toctree-l2"><a class="reference external" href="elliptic.html#jacobi-theta-functions">Jacobi theta functions</a></li>
<li class="toctree-l2"><a class="reference external" href="elliptic.html#jacobi-elliptic-functions">Jacobi elliptic functions</a></li>
<li class="toctree-l2"><a class="reference external" href="elliptic.html#modular-functions">Modular functions</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference external" href="zeta.html">Zeta functions, L-series and polylogarithms</a><ul>
<li class="toctree-l2"><a class="reference external" href="zeta.html#riemann-and-hurwitz-zeta-functions">Riemann and Hurwitz zeta functions</a></li>
<li class="toctree-l2"><a class="reference external" href="zeta.html#dirichlet-l-series">Dirichlet L-series</a></li>
<li class="toctree-l2"><a class="reference external" href="zeta.html#stieltjes-constants">Stieltjes constants</a></li>
<li class="toctree-l2"><a class="reference external" href="zeta.html#zeta-function-zeros">Zeta function zeros</a></li>
<li class="toctree-l2"><a class="reference external" href="zeta.html#lerch-transcendent">Lerch transcendent</a></li>
<li class="toctree-l2"><a class="reference external" href="zeta.html#polylogarithms-and-clausen-functions">Polylogarithms and Clausen functions</a></li>
<li class="toctree-l2"><a class="reference external" href="zeta.html#zeta-function-variants">Zeta function variants</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference external" href="numtheory.html">Number-theoretical, combinatorial and integer functions</a><ul>
<li class="toctree-l2"><a class="reference external" href="numtheory.html#fibonacci-numbers">Fibonacci numbers</a></li>
<li class="toctree-l2"><a class="reference external" href="numtheory.html#bernoulli-numbers-and-polynomials">Bernoulli numbers and polynomials</a></li>
<li class="toctree-l2"><a class="reference external" href="numtheory.html#euler-numbers-and-polynomials">Euler numbers and polynomials</a></li>
<li class="toctree-l2"><a class="reference external" href="numtheory.html#bell-numbers-and-polynomials">Bell numbers and polynomials</a></li>
<li class="toctree-l2"><a class="reference external" href="numtheory.html#prime-counting-functions">Prime counting functions</a></li>
<li class="toctree-l2"><a class="reference external" href="numtheory.html#cyclotomic-polynomials">Cyclotomic polynomials</a></li>
<li class="toctree-l2"><a class="reference external" href="numtheory.html#arithmetic-functions">Arithmetic functions</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference external" href="qfunctions.html">q-functions</a><ul>
<li class="toctree-l2"><a class="reference external" href="qfunctions.html#q-pochhammer-symbol">q-Pochhammer symbol</a></li>
<li class="toctree-l2"><a class="reference external" href="qfunctions.html#q-gamma-and-factorial">q-gamma and factorial</a></li>
<li class="toctree-l2"><a class="reference external" href="qfunctions.html#hypergeometric-q-series">Hypergeometric q-series</a></li>
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