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docItemCol_VOVn"><div class="docItemContainer_Djhp"><article><nav class="theme-doc-breadcrumbs breadcrumbsContainer_Z_bl" aria-label="Breadcrumbs"><ul class="breadcrumbs"><li class="breadcrumbs__item"><a aria-label="Home page" class="breadcrumbs__link" href="/"><svg viewBox="0 0 24 24" class="breadcrumbHomeIcon_YNFT"><path d="M10 19v-5h4v5c0 .55.45 1 1 1h3c.55 0 1-.45 1-1v-7h1.7c.46 0 .68-.57.33-.87L12.67 3.6c-.38-.34-.96-.34-1.34 0l-8.36 7.53c-.34.3-.13.87.33.87H5v7c0 .55.45 1 1 1h3c.55 0 1-.45 1-1z" fill="currentColor"></path></svg></a></li><li class="breadcrumbs__item"><span class="breadcrumbs__link">Compute Engine</span></li><li class="breadcrumbs__item breadcrumbs__item--active"><span class="breadcrumbs__link">Special Functions</span></li></ul></nav><div class="tocCollapsible_ETCw theme-doc-toc-mobile tocMobile_ITEo"><button type="button" class="clean-btn tocCollapsibleButton_TO0P">On this page</button></div><div class="theme-doc-markdown markdown"><header><h1>Special Functions</h1></header><p>The functions in this section take a <code>number</code> argument, the type that includes
<code class="language-math math-inline">\pm\infty</code> and <code>NaN</code> alongside the finite numbers. They are not restricted to
finite input the way the <a class="" href="/compute-engine/reference/number-theory/">number theory</a>
functions are:</p>
<ul>
<li class="">An <strong>infinite</strong> argument is accepted. Where the function has a limit there,
that limit is the value — <code class="language-math math-inline">\operatorname{erf}(\infty) = 1</code>,
<code class="language-math math-inline">\operatorname{erfc}(\infty) = 0</code>. Otherwise the expression stays symbolic,
as <code>["Gamma", "PositiveInfinity"]</code> does.</li>
<li class="">A <strong><code>NaN</code></strong> argument gives <code>NaN</code>. This happens under plain <code>evaluate()</code>, not
only under <code>N()</code>: <code>NaN</code> is not an exact value, so there is nothing to hold
symbolically.</li>
</ul>
<p>A function may still be infinite at a finite point — <code class="language-math math-inline">K(1) = \infty</code> — in which
case the result is <code>PositiveInfinity</code>.</p>
<section id="Erf" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Erf</b>(<em>z:number</em>)</p><p>Evaluate to the <strong>error function</strong> of a complex number.</p><p>The error function is an odd function ( </p><div class="language-math math-inline codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-math codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain"> \operatorname{erf} -z = -</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">\operatorname{erf} z</span><br></div></code></pre></div></div> ) that is used in statistics to calculate probabilities
of normally distributed events.<p></p><p>The formula for the error function of a complex number is:</p><p><code class="language-math math-inline"> \operatorname{erf} z = \frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} dt</code></p><p>where <code class="language-math math-inline">z</code> is a complex number.</p></section>
<section id="Erfc" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Erfc</b>(<em>z:number</em>)</p><p>Evaluate to the <strong>complementary error function</strong> of a complex number.</p><p>It is defined as <code class="language-math math-inline">\operatorname{erfc} z = 1 - \operatorname {erf} z</code>.</p></section>
<section id="ErfInv" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>ErfInv</b>(<em>x:number</em>)</p><p>Evaluate to the <strong>inverse error function</strong> of a real number <code class="language-math math-inline">-1 < x < 1</code></p><p>Outside that interval the value is not defined: <code>["ErfInv", 2]</code> and
<code>["ErfInv", "PositiveInfinity"]</code> both evaluate to <code>NaN</code>.</p><p>It is defined as </p><div class="language-math math-inline codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-math codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">\operatorname{erf} \left(\operatorname{erf} ^{-1}x\right)</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">= x</span><br></div></code></pre></div></div>.<p></p></section>
<section id="Factorial" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Factorial</b>(<em>n</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span>n!</span><br></div></div><div class="display_sbg0">$$$n!$$</div></div><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factorial"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// -> 120</span><br></div></code></pre></div></div></section>
<section id="Factorial2" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Factorial2</b>(<em>n</em>)</p><p>The double factorial of <code>n</code>: </p><div class="language-math math-inline codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-math codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain"> n!! = n \cdot (n-2) \cdot (n-4) \times</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">\cdots</span><br></div></code></pre></div></div>, that is the product of all the positive integers up to <code>n</code> that have
the same parity (odd or even) as <code>n</code>.<p></p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span>n!!</span><br></div></div><div class="display_sbg0">$$$n!!$$</div></div><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Factorial2"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// -> 15</span><br></div></code></pre></div></div><p>It can also be written in terms of the <code class="language-math math-inline">\Gamma</code> function:</p><div class="language-math math-display codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-math codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">n!! = 2^{\frac{n}{2}+\frac{1}{4}(1-\cos(\pi n))}\pi^{\frac{1}{4}(\cos(\pi</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">n)-1)}\Gamma\left(\frac{n}{2}+1\right)</span><br></div></code></pre></div></div><p>This is not the same as the factorial of the factorial of <code>n</code> (i.e.
<code class="language-math math-inline">((n!)!)</code>).</p><p><strong>Reference</strong></p><ul>
<li class="">WikiPedia: <a href="https://en.wikipedia.org/wiki/Double_factorial" target="_blank" rel="noopener noreferrer" class="">Double Factorial</a></li>
</ul></section>
<section id="Gamma" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Gamma</b>(<em>z</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\Gamma</span><span>(n) = (n-1)!</span><br></div></div><div class="display_sbg0">$$$\Gamma(n) = (n-1)!$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Gamma_function" target="_blank" rel="noopener noreferrer" class="">Gamma Function</a> is an
extension of the factorial function, with its argument shifted by 1, to real and
complex numbers.</p><div class="language-math math-display codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-math codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">\operatorname{\Gamma}\left(z\right) = \int\limits_{0}^{\infty} t^{z-1}</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">\mathrm{e}^{-t} \, \mathrm{d}t</span><br></div></code></pre></div></div><ul>
<li class="">Wikidata: <a href="https://www.wikidata.org/wiki/Q190573" target="_blank" rel="noopener noreferrer" class="">Q190573</a></li>
<li class="">NIST: <a href="http://dlmf.nist.gov/5.2.E1" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/5.2.E1</a></li>
</ul><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Gamma"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// 24</span><br></div></code></pre></div></div><p class="signature_CWyf"><b>Gamma</b>(<em>s</em>, <em>z</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\Gamma</span><span>(s, z)</span><br></div></div><div class="display_sbg0">$$$\Gamma(s, z)$$</div></div><p>With two arguments, <code>Gamma</code> is the <strong>upper incomplete Gamma function</strong>. The lower
limit of the integral is the second argument <code>z</code> instead of <code>0</code>:</p><div class="language-math math-display codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-math codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">\operatorname{\Gamma}\left(s, z\right) = \int\limits_{z}^{\infty} t^{s-1}</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">\mathrm{e}^{-t} \, \mathrm{d}t</span><br></div></code></pre></div></div><p>The order <code>s</code> and the lower limit <code>z</code> may be real or complex, including negative
and fractional orders. The two-argument form is evaluated numerically with
<code>.N()</code> and otherwise stays symbolic; <code class="language-math math-inline">\Gamma(s, 0)</code> reduces to <code class="language-math math-inline">\Gamma(s)</code>.
(This matches the <code>Gamma[s, z]</code> convention of Mathematica.)</p><ul>
<li class="">NIST: <a href="http://dlmf.nist.gov/8.2.E2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/8.2.E2</a></li>
</ul><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"N"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Gamma"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// 0.7357588823428849</span><br></div></code></pre></div></div></section>
<section id="GammaLn" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>GammaLn</b>(<em>z</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\ln</span><span>(</span><span class="command_yVKk">\Gamma</span><span>(z))</span><br></div></div><div class="display_sbg0">$$$\ln(\Gamma(z))$$</div></div><p>This function is called <code>gammaln</code> in MatLab and SciPy and <code>LogGamma</code> in
Mathematica.</p></section>
<section id="Zeta" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Zeta</b>(<em>s</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\zeta</span><span>(s)</span><br></div></div><div class="display_sbg0">$$$\zeta(s)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Riemann_zeta_function" target="_blank" rel="noopener noreferrer" class="">Riemann zeta function</a>,
defined for complex numbers with real part greater than 1 as:</p><code class="language-math math-display">\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}</code><p>and extended to other values by analytic continuation.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Zeta"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ π²/6</span><br></div></code></pre></div></div><ul>
<li class="">Wikidata: <a href="https://www.wikidata.org/wiki/Q187235" target="_blank" rel="noopener noreferrer" class="">Q187235</a></li>
<li class="">NIST: <a href="http://dlmf.nist.gov/25.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/25.2</a></li>
</ul></section>
<section id="Beta" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Beta</b>(<em>a</em>, <em>b</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\Beta</span><span>(a, b)</span><br></div></div><div class="display_sbg0">$$$\Beta(a, b)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Beta_function" target="_blank" rel="noopener noreferrer" class="">Euler beta function</a>, defined as:</p><code class="language-math math-display">\operatorname{B}(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}</code><p>It can also be expressed as an integral:</p><code class="language-math math-display">\operatorname{B}(a, b) = \int_0^1 t^{a-1}(1-t)^{b-1} \, dt</code><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Beta"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 1/12</span><br></div></code></pre></div></div><ul>
<li class="">Wikidata: <a href="https://www.wikidata.org/wiki/Q192828" target="_blank" rel="noopener noreferrer" class="">Q192828</a></li>
<li class="">NIST: <a href="http://dlmf.nist.gov/5.12" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/5.12</a></li>
</ul></section>
<section id="GammaRegularized" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>GammaRegularized</b>(<em>a</em>, <em>z</em>)</p><p>The <strong>upper regularized incomplete gamma function</strong>, defined as:</p><code class="language-math math-display">Q(a, z) = \frac{\Gamma(a, z)}{\Gamma(a)} = \frac{1}{\Gamma(a)}\int_z^\infty t^{a-1}e^{-t} \, dt</code><p>Exact arguments stay symbolic, and special values fold:
<code>GammaRegularized(a, 0)</code> is <code class="language-math math-inline">1</code> and <code>GammaRegularized(1, z)</code> is <code class="language-math math-inline">e^{-z}</code>.
Use <code>N()</code> for a numeric value, at machine or arbitrary precision.</p><p>The cumulative distribution function of a Poisson distribution evaluates to
this function: </p><div class="language-math math-inline codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-math codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">\operatorname{CDF}(\operatorname{Poisson}(\lambda), k) =</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">Q(k+1, \lambda)</span><br></div></code></pre></div></div>.<p></p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"N"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"GammaRegularized"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 0.12465201948308115</span><br></div></code></pre></div></div><ul>
<li class="">NIST: <a href="http://dlmf.nist.gov/8.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/8.2</a></li>
</ul></section>
<section id="BetaRegularized" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>BetaRegularized</b>(<em>x</em>, <em>a</em>, <em>b</em>)</p><p>The <strong>regularized incomplete beta function</strong>, defined as:</p><code class="language-math math-display">I_x(a, b) = \frac{1}{\operatorname{B}(a, b)}\int_0^x t^{a-1}(1-t)^{b-1} \, dt</code><p>Exact arguments stay symbolic, and the endpoints fold:
<code>BetaRegularized(0, a, b)</code> is <code class="language-math math-inline">0</code> and <code>BetaRegularized(1, a, b)</code> is <code class="language-math math-inline">1</code>.
Use <code>N()</code> for a numeric value, at machine or arbitrary precision.</p><p>The cumulative distribution function of a binomial distribution evaluates to
this function: </p><div class="language-math math-inline codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-math codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">\operatorname{CDF}(\operatorname{Binomial}(n, p), k) =</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">I_{1-p}(n-k, k+1)</span><br></div></code></pre></div></div>.<p></p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"N"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"BetaRegularized"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0.5</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 0.6875</span><br></div></code></pre></div></div><ul>
<li class="">NIST: <a href="http://dlmf.nist.gov/8.17" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/8.17</a></li>
</ul></section>
<section id="LambertW" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>LambertW</b>(<em>x</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\operatorname</span><span class="punctuation_YpjP">{</span><span>W</span><span class="punctuation_YpjP">}</span><span>(x)</span><br></div></div><div class="display_sbg0">$$$\operatorname{W}(x)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Lambert_W_function" target="_blank" rel="noopener noreferrer" class="">Lambert W function</a>,
also called the product logarithm. It is the inverse function of
<code class="language-math math-inline">f(w) = w e^w</code>.</p><p>For a given value <code class="language-math math-inline">x</code>, <code class="language-math math-inline">W(x)</code> is the value <code class="language-math math-inline">w</code> such that <code class="language-math math-inline">w e^w = x</code>.</p><p>The derivative of the Lambert W function is:</p><code class="language-math math-display">\frac{d}{dx} W(x) = \frac{W(x)}{x(1 + W(x))}</code><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"LambertW"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ Ω ≈ 0.5671 (the Omega constant)</span><br></div></code></pre></div></div><ul>
<li class="">Wikidata: <a href="https://www.wikidata.org/wiki/Q429963" target="_blank" rel="noopener noreferrer" class="">Q429963</a></li>
<li class="">NIST: <a href="http://dlmf.nist.gov/4.13" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/4.13</a></li>
</ul></section>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="fresnel-integrals">Fresnel Integrals<a href="#fresnel-integrals" class="hash-link" aria-label="Direct link to Fresnel Integrals" title="Direct link to Fresnel Integrals" translate="no"></a></h2>
<p>The <a href="https://en.wikipedia.org/wiki/Fresnel_integral" target="_blank" rel="noopener noreferrer" class="">Fresnel integrals</a> arise in
the description of near-field diffraction and in the geometry of the Cornu
spiral (Euler spiral).</p>
<section id="FresnelS" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>FresnelS</b>(<em>x</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\operatorname</span><span class="punctuation_YpjP">{</span><span>FresnelS</span><span class="punctuation_YpjP">}</span><span>(x)</span><br></div></div><div class="display_sbg0">$$$\operatorname{FresnelS}(x)$$</div></div><p>The Fresnel S integral:</p><code class="language-math math-display">S(x) = \int_0^x \sin\!\left(\frac{\pi t^2}{2}\right) dt</code><p>It is an odd function (<code class="language-math math-inline">S(-x) = -S(x)</code>) with asymptotic value
<code class="language-math math-inline">S(\infty) = \tfrac{1}{2}</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"FresnelS"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 0.4383</span><br></div></code></pre></div></div><ul>
<li class="">Wikipedia: <a href="https://en.wikipedia.org/wiki/Fresnel_integral" target="_blank" rel="noopener noreferrer" class="">Fresnel integral</a></li>
<li class="">NIST: <a href="http://dlmf.nist.gov/7.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/7.2</a></li>
</ul></section>
<section id="FresnelC" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>FresnelC</b>(<em>x</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\operatorname</span><span class="punctuation_YpjP">{</span><span>FresnelC</span><span class="punctuation_YpjP">}</span><span>(x)</span><br></div></div><div class="display_sbg0">$$$\operatorname{FresnelC}(x)$$</div></div><p>The Fresnel C integral:</p><code class="language-math math-display">C(x) = \int_0^x \cos\!\left(\frac{\pi t^2}{2}\right) dt</code><p>It is an odd function (<code class="language-math math-inline">C(-x) = -C(x)</code>) with asymptotic value
<code class="language-math math-inline">C(\infty) = \tfrac{1}{2}</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"FresnelC"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 0.7799</span><br></div></code></pre></div></div><ul>
<li class="">Wikipedia: <a href="https://en.wikipedia.org/wiki/Fresnel_integral" target="_blank" rel="noopener noreferrer" class="">Fresnel integral</a></li>
<li class="">NIST: <a href="http://dlmf.nist.gov/7.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/7.2</a></li>
</ul></section>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="bessel-functions">Bessel Functions<a href="#bessel-functions" class="hash-link" aria-label="Direct link to Bessel Functions" title="Direct link to Bessel Functions" translate="no"></a></h2>
<p>Bessel functions are solutions to Bessel's differential equation:</p>
<code class="language-math math-display">x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0</code>
<p>They arise in problems with cylindrical or spherical symmetry.</p>
<section id="BesselJ" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>BesselJ</b>(<em>n</em>, <em>x</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span>J_n(x)</span><br></div></div><div class="display_sbg0">$$$J_n(x)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Bessel_function#Bessel_functions_of_the_first_kind" target="_blank" rel="noopener noreferrer" class="">Bessel function of the first kind</a>
of order <code class="language-math math-inline">n</code>.</p><p>The derivative with respect to <code class="language-math math-inline">x</code> is:</p><code class="language-math math-display">\frac{d}{dx} J_n(x) = \frac{1}{2}(J_{n-1}(x) - J_{n+1}(x))</code><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"BesselJ"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ J₀(1) ≈ 0.7652</span><br></div></code></pre></div></div><ul>
<li class="">NIST: <a href="http://dlmf.nist.gov/10.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/10.2</a></li>
</ul></section>
<section id="BesselY" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>BesselY</b>(<em>n</em>, <em>x</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span>Y_n(x)</span><br></div></div><div class="display_sbg0">$$$Y_n(x)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Bessel_function#Bessel_functions_of_the_second_kind" target="_blank" rel="noopener noreferrer" class="">Bessel function of the second kind</a>
of order <code class="language-math math-inline">n</code>, also called the Neumann function.</p><p>The derivative with respect to <code class="language-math math-inline">x</code> is:</p><code class="language-math math-display">\frac{d}{dx} Y_n(x) = \frac{1}{2}(Y_{n-1}(x) - Y_{n+1}(x))</code><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"BesselY"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ Y₀(1) ≈ 0.0883</span><br></div></code></pre></div></div><ul>
<li class="">NIST: <a href="http://dlmf.nist.gov/10.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/10.2</a></li>
</ul></section>
<section id="BesselI" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>BesselI</b>(<em>n</em>, <em>x</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span>I_n(x)</span><br></div></div><div class="display_sbg0">$$$I_n(x)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions" target="_blank" rel="noopener noreferrer" class="">modified Bessel function of the first kind</a>
of order <code class="language-math math-inline">n</code>.</p><p>The derivative with respect to <code class="language-math math-inline">x</code> is:</p><code class="language-math math-display">\frac{d}{dx} I_n(x) = \frac{1}{2}(I_{n-1}(x) + I_{n+1}(x))</code><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"BesselI"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ I₀(1) ≈ 1.2661</span><br></div></code></pre></div></div><ul>
<li class="">NIST: <a href="http://dlmf.nist.gov/10.25" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/10.25</a></li>
</ul></section>
<section id="BesselK" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>BesselK</b>(<em>n</em>, <em>x</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span>K_n(x)</span><br></div></div><div class="display_sbg0">$$$K_n(x)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions" target="_blank" rel="noopener noreferrer" class="">modified Bessel function of the second kind</a>
of order <code class="language-math math-inline">n</code>, also called the MacDonald function.</p><p>The derivative with respect to <code class="language-math math-inline">x</code> is:</p><code class="language-math math-display">\frac{d}{dx} K_n(x) = -\frac{1}{2}(K_{n-1}(x) + K_{n+1}(x))</code><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"BesselK"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ K₀(1) ≈ 0.4210</span><br></div></code></pre></div></div><ul>
<li class="">NIST: <a href="http://dlmf.nist.gov/10.25" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/10.25</a></li>
</ul></section>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="airy-functions">Airy Functions<a href="#airy-functions" class="hash-link" aria-label="Direct link to Airy Functions" title="Direct link to Airy Functions" translate="no"></a></h2>
<p>Airy functions are solutions to the Airy differential equation:</p>
<code class="language-math math-display">\frac{d^2 y}{dx^2} - xy = 0</code>
<p>They arise in physics, particularly in quantum mechanics and optics.</p>
<section id="AiryAi" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>AiryAi</b>(<em>x</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\operatorname</span><span class="punctuation_YpjP">{</span><span>Ai</span><span class="punctuation_YpjP">}</span><span>(x)</span><br></div></div><div class="display_sbg0">$$$\operatorname{Ai}(x)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Airy_function" target="_blank" rel="noopener noreferrer" class="">Airy function of the first kind</a>.</p><p>It is the solution to the Airy equation that decays exponentially for
positive <code class="language-math math-inline">x</code> and oscillates for negative <code class="language-math math-inline">x</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"AiryAi"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 1/(3^(2/3) Γ(2/3)) ≈ 0.3550</span><br></div></code></pre></div></div><ul>
<li class="">NIST: <a href="http://dlmf.nist.gov/9.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/9.2</a></li>
</ul></section>
<section id="AiryBi" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>AiryBi</b>(<em>x</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\operatorname</span><span class="punctuation_YpjP">{</span><span>Bi</span><span class="punctuation_YpjP">}</span><span>(x)</span><br></div></div><div class="display_sbg0">$$$\operatorname{Bi}(x)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Airy_function" target="_blank" rel="noopener noreferrer" class="">Airy function of the second kind</a>.</p><p>It is the solution to the Airy equation that grows exponentially for
positive <code class="language-math math-inline">x</code> and oscillates for negative <code class="language-math math-inline">x</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"AiryBi"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 1/(3^(1/6) Γ(2/3)) ≈ 0.6149</span><br></div></code></pre></div></div><ul>
<li class="">NIST: <a href="http://dlmf.nist.gov/9.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/9.2</a></li>
</ul></section>
<section id="AiryAiPrime" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>AiryAiPrime</b>(<em>x</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\operatorname</span><span class="punctuation_YpjP">{</span><span>Ai</span><span class="punctuation_YpjP">}</span><span>'(x)</span><br></div></div><div class="display_sbg0">$$$\operatorname{Ai}'(x)$$</div></div><p>The derivative of the <a href="https://en.wikipedia.org/wiki/Airy_function" target="_blank" rel="noopener noreferrer" class="">Airy function of the first kind</a>.</p><p>It satisfies <code class="language-math math-inline">\operatorname{Ai}''(x) = x\operatorname{Ai}(x)</code>, so
differentiating an Airy expression stays within the Airy family.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"AiryAiPrime"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ −1/(3^(1/3) Γ(1/3)) ≈ −0.2588</span><br></div></code></pre></div></div><ul>
<li class="">NIST: <a href="http://dlmf.nist.gov/9.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/9.2</a></li>
</ul></section>
<section id="AiryBiPrime" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>AiryBiPrime</b>(<em>x</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\operatorname</span><span class="punctuation_YpjP">{</span><span>Bi</span><span class="punctuation_YpjP">}</span><span>'(x)</span><br></div></div><div class="display_sbg0">$$$\operatorname{Bi}'(x)$$</div></div><p>The derivative of the <a href="https://en.wikipedia.org/wiki/Airy_function" target="_blank" rel="noopener noreferrer" class="">Airy function of the second kind</a>.</p><p>It satisfies <code class="language-math math-inline">\operatorname{Bi}''(x) = x\operatorname{Bi}(x)</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"AiryBiPrime"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 3^(1/6)/Γ(1/3) ≈ 0.4483</span><br></div></code></pre></div></div><ul>
<li class="">NIST: <a href="http://dlmf.nist.gov/9.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/9.2</a></li>
</ul></section>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="elliptic-integrals">Elliptic Integrals<a href="#elliptic-integrals" class="hash-link" aria-label="Direct link to Elliptic Integrals" title="Direct link to Elliptic Integrals" translate="no"></a></h2>
<p>The complete elliptic integrals arise in computing the arc length of an
ellipse, the period of a pendulum, and throughout number theory.</p>
<p><strong>Convention:</strong> these functions use the <strong>parameter</strong> <code class="language-math math-inline">m = k^2</code>, where <code class="language-math math-inline">k</code>
is the modulus. This matches Mathematica, mpmath and
<a href="https://fungrim.org" target="_blank" rel="noopener noreferrer" class="">Fungrim</a>. To evaluate in terms of the modulus <code class="language-math math-inline">k</code>,
pass <code class="language-math math-inline">k^2</code>.</p>
<section id="EllipticK" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>EllipticK</b>(<em>m</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span>K(m)</span><br></div></div><div class="display_sbg0">$$$K(m)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind" target="_blank" rel="noopener noreferrer" class="">complete elliptic integral of the first kind</a>:</p><code class="language-math math-display">K(m) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1 - m \sin^2\theta}}</code><p>It is computed via the arithmetic-geometric mean: </p><div class="language-math math-inline codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-math codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">K(m) =</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">\dfrac{\pi}{2\operatorname{agm}(1, \sqrt{1-m})}</span><br></div></code></pre></div></div>.<p></p><p><code class="language-math math-inline">K(1) = \infty</code>. For <code class="language-math math-inline">m > 1</code> the value is complex.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"EllipticK"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0.5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 1.85407467730137</span><br></div></code></pre></div></div><ul>
<li class="">Wikidata: <a href="https://www.wikidata.org/wiki/Q1080993" target="_blank" rel="noopener noreferrer" class="">Q1080993</a></li>
<li class="">NIST: <a href="http://dlmf.nist.gov/19.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/19.2</a></li>
</ul></section>
<section id="EllipticE" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>EllipticE</b>(<em>m</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span>E(m)</span><br></div></div><div class="display_sbg0">$$$E(m)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_second_kind" target="_blank" rel="noopener noreferrer" class="">complete elliptic integral of the second kind</a>:</p><code class="language-math math-display">E(m) = \int_0^{\pi/2} \sqrt{1 - m \sin^2\theta} \, d\theta</code><p>The perimeter of an ellipse with semi-major axis <code class="language-math math-inline">a</code> and eccentricity <code class="language-math math-inline">e</code>
is <code class="language-math math-inline">4aE(e^2)</code>.</p><p><code class="language-math math-inline">E(1) = 1</code>. For <code class="language-math math-inline">m > 1</code> the value is complex.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"EllipticE"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0.5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 1.35064388104768</span><br></div></code></pre></div></div><ul>
<li class="">Wikidata: <a href="https://www.wikidata.org/wiki/Q1375529" target="_blank" rel="noopener noreferrer" class="">Q1375529</a></li>
<li class="">NIST: <a href="http://dlmf.nist.gov/19.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/19.2</a></li>
</ul></section>
<section id="AGM" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>AGM</b>(<em>a</em>, <em>b</em>)</p><p class="signature_CWyf"><b>AGM</b>(<em>z</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\operatorname</span><span class="punctuation_YpjP">{</span><span>agm</span><span class="punctuation_YpjP">}</span><span>(a, b)</span><br></div></div><div class="display_sbg0">$$$\operatorname{agm}(a, b)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean" target="_blank" rel="noopener noreferrer" class="">arithmetic-geometric mean</a>
of two numbers: the common limit of the sequences </p><div class="language-math math-inline codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-math codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">a_{n+1} = \frac{a_n +</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">b_n}{2}</span><br></div></code></pre></div></div> and <code class="language-math math-inline">b_{n+1} = \sqrt{a_n b_n}</code>.<p></p><p>With a single argument, <code class="language-math math-inline">\operatorname{agm}(z)</code> is shorthand for
<code class="language-math math-inline">\operatorname{agm}(1, z)</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"AGM"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 1.45679103104691</span><br></div></code></pre></div></div><ul>
<li class="">Wikidata: <a href="https://www.wikidata.org/wiki/Q360074" target="_blank" rel="noopener noreferrer" class="">Q360074</a></li>
<li class="">NIST: <a href="http://dlmf.nist.gov/19.8" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/19.8</a></li>
</ul></section>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="hypergeometric-functions">Hypergeometric Functions<a href="#hypergeometric-functions" class="hash-link" aria-label="Direct link to Hypergeometric Functions" title="Direct link to Hypergeometric Functions" translate="no"></a></h2>
<section id="Hypergeometric2F1" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Hypergeometric2F1</b>(<em>a</em>, <em>b</em>, <em>c</em>, <em>z</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="punctuation_YpjP">{</span><span class="punctuation_YpjP">}</span><span>_2F_1(a, b; c; z)</span><br></div></div><div class="display_sbg0">$$${}_2F_1(a, b; c; z)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Hypergeometric_function" target="_blank" rel="noopener noreferrer" class="">Gauss hypergeometric function</a>,
defined for <code class="language-math math-inline">|z| < 1</code> by the series:</p><div class="language-math math-display codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-math codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">{}_2F_1(a, b; c; z) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n}</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">\frac{z^n}{n!}</span><br></div></code></pre></div></div><p>where <code class="language-math math-inline">(q)_n</code> is the Pochhammer symbol (rising factorial), and extended
elsewhere by analytic continuation.</p><p>Many elementary and special functions are particular cases. For example
<code class="language-math math-inline">\ln(1+z) = z \cdot {}_2F_1(1, 1; 2; -z)</code> and
<code class="language-math math-inline">K(m) = \frac{\pi}{2} \, {}_2F_1\big(\frac12, \frac12; 1; m\big)</code>.</p><p>If <code class="language-math math-inline">a</code> or <code class="language-math math-inline">b</code> is a non-positive integer the series terminates and the
function is a polynomial in <code class="language-math math-inline">z</code>, evaluable for any <code class="language-math math-inline">z</code>. Otherwise the
function evaluates numerically for <code class="language-math math-inline">z \le 1</code> (real) and for complex <code class="language-math math-inline">z</code>
within the unit disk and the Pfaff-transform region.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Hypergeometric2F1"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0.5</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 1.38629436111989 (= 2 ln 2)</span><br></div></code></pre></div></div><ul>
<li class="">Wikidata: <a href="https://www.wikidata.org/wiki/Q672619" target="_blank" rel="noopener noreferrer" class="">Q672619</a></li>
<li class="">NIST: <a href="http://dlmf.nist.gov/15.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/15.2</a></li>
</ul></section>
<section id="Hypergeometric1F1" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>Hypergeometric1F1</b>(<em>a</em>, <em>b</em>, <em>z</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="punctuation_YpjP">{</span><span class="punctuation_YpjP">}</span><span>_1F_1(a; b; z)</span><br></div></div><div class="display_sbg0">$$${}_1F_1(a; b; z)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Confluent_hypergeometric_function" target="_blank" rel="noopener noreferrer" class="">Kummer confluent hypergeometric function</a>,
also written <code class="language-math math-inline">M(a, b, z)</code>:</p><code class="language-math math-display">{}_1F_1(a; b; z) = \sum_{n=0}^{\infty} \frac{(a)_n}{(b)_n} \frac{z^n}{n!}</code><p>It is an entire function of <code class="language-math math-inline">z</code> and evaluates numerically for any real or
complex <code class="language-math math-inline">z</code>.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Hypergeometric1F1"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 3.19452804946533 (= (e² − 1)/2)</span><br></div></code></pre></div></div><ul>
<li class="">Wikidata: <a href="https://www.wikidata.org/wiki/Q1331447" target="_blank" rel="noopener noreferrer" class="">Q1331447</a></li>
<li class="">NIST: <a href="http://dlmf.nist.gov/13.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/13.2</a></li>
</ul></section>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="theta-and-modular-functions">Theta and Modular Functions<a href="#theta-and-modular-functions" class="hash-link" aria-label="Direct link to Theta and Modular Functions" title="Direct link to Theta and Modular Functions" translate="no"></a></h2>
<section id="JacobiTheta" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>JacobiTheta</b>(<em>j</em>, <em>z</em>, <em>τ</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\theta</span><span>_j(z, </span><span class="command_yVKk">\tau</span><span>)</span><br></div></div><div class="display_sbg0">$$$\theta_j(z, \tau)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Theta_function" target="_blank" rel="noopener noreferrer" class="">Jacobi theta functions</a>
<code class="language-math math-inline">\theta_j(z, \tau)</code> for <code class="language-math math-inline">j \in \{1, 2, 3, 4\}</code>, with nome <code class="language-math math-inline">q = e^{i\pi\tau}</code>
and <code class="language-math math-inline">\operatorname{Im}(\tau) > 0</code>:</p><code class="language-math math-display">\theta_3(z, \tau) = 1 + 2\sum_{n=1}^{\infty} q^{n^2} \cos(2n\pi z)</code><p>and similarly for <code class="language-math math-inline">\theta_1</code>, <code class="language-math math-inline">\theta_2</code>, <code class="language-math math-inline">\theta_4</code>.</p><p><strong>Convention:</strong> the trigonometric argument is a multiple of <code class="language-math math-inline">\pi z</code> (the
functions have period 1 in <code class="language-math math-inline">z</code>), matching <a href="https://fungrim.org" target="_blank" rel="noopener noreferrer" class="">Fungrim</a>
and mpmath — not the classical convention with period <code class="language-math math-inline">\pi</code>.</p><p>An optional fourth argument indicates the order of differentiation with
respect to <code class="language-math math-inline">z</code>; only order 0 currently evaluates numerically.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"JacobiTheta"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"ImaginaryUnit"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 1.08643481121331 (= π^(1/4)/Γ(3/4))</span><br></div></code></pre></div></div><ul>
<li class="">Wikidata: <a href="https://www.wikidata.org/wiki/Q1154532" target="_blank" rel="noopener noreferrer" class="">Q1154532</a></li>
<li class="">NIST: <a href="http://dlmf.nist.gov/20.2" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/20.2</a></li>
</ul></section>
<section id="DedekindEta" class="functionDefinition_O4AF"><p class="signature_CWyf"><b>DedekindEta</b>(<em>τ</em>)</p><div class="latexWrapper_OyfQ"><div class="source_Stpb language-latex"><div class="line_GvIm"><span class="command_yVKk">\eta</span><span>(</span><span class="command_yVKk">\tau</span><span>)</span><br></div></div><div class="display_sbg0">$$$\eta(\tau)$$</div></div><p>The <a href="https://en.wikipedia.org/wiki/Dedekind_eta_function" target="_blank" rel="noopener noreferrer" class="">Dedekind eta function</a>,
defined on the upper half-plane (<code class="language-math math-inline">\operatorname{Im}(\tau) > 0</code>) by:</p><div class="language-math math-display codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-math codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">\eta(\tau) = e^{i\pi\tau/12} \prod_{k=1}^{\infty} \left(1 - e^{2\pi i k</span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">\tau}\right)</span><br></div></code></pre></div></div><p>It is a modular form of weight <code class="language-math math-inline">\frac12</code>, central to the theory of modular
functions and integer partitions.</p><div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"DedekindEta"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"ImaginaryUnit"</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ 0.768225422326057 (= Γ(1/4)/(2π^(3/4)))</span><br></div></code></pre></div></div><ul>
<li class="">Wikidata: <a href="https://www.wikidata.org/wiki/Q1187208" target="_blank" rel="noopener noreferrer" class="">Q1187208</a></li>
<li class="">NIST: <a href="http://dlmf.nist.gov/23.15" target="_blank" rel="noopener noreferrer" class="">http://dlmf.nist.gov/23.15</a></li>
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