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<title>Reciprocal gamma function</title>

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Reciprocal gamma function

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the reciprocal gamma function is the <a href="Special_function" class="mw-redirect" title="Special function">function</a>

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<annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {1}{\Gamma (z)}},}</annotation>

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</math><img src="./dff2eb8df2bf924238048da4bc055a3b4e5ddf4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.108ex; height:6.009ex;" alt="{\displaystyle f(z)={\frac {1}{\Gamma (z)}},}" loading="lazy"></dd></dl>

where Γ(z) denotes the <a href="Gamma_function" title="Gamma function">gamma function</a>. Since the gamma function is <a href="Meromorphic_function" title="Meromorphic function">meromorphic</a> and nonzero everywhere in the <a href="Complex_plane" title="Complex plane">complex plane</a>, its reciprocal is an <a href="Entire_function" title="Entire function">entire function</a>. As an entire function, it is of order 1 (meaning that log log |⁠1/Γ(z)⁠| grows no faster than log |z|), but of infinite type (meaning that log |⁠1/Γ(z)⁠| grows faster than any multiple of |z|, since its growth is approximately proportional to |z| log |z| in the left-half plane).

The reciprocal is sometimes used as a starting point for <a href="Numerical_analysis" title="Numerical analysis">numerical computation</a> of the gamma function, and a few software libraries provide it separately from the regular gamma function.

<a href="Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> called the reciprocal gamma function the "factorielle" and used it in his development of the <a href="Weierstrass_factorization_theorem" title="Weierstrass factorization theorem">Weierstrass factorization theorem</a>.

<div class="mw-heading mw-heading2"><h2 id="Infinite_product_expansion">Infinite product expansion</h2></div>

Following from the <a href="Infinite_product" title="Infinite product">infinite product</a> definitions for the <a href="Gamma_function" title="Gamma function">gamma function</a>, due to <a href="Leonhard_Euler" title="Leonhard Euler">Euler</a> and <a href="Karl_Weierstrass" title="Karl Weierstrass">Weierstrass</a> respectively, we get the following infinite product expansion for the reciprocal gamma function:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {1}{\Gamma (z)}}&=z\prod _{n=1}^{\infty }{\frac {1+{\frac {z}{n}}}{\left(1+{\frac {1}{n}}\right)^{z}}}\\{\frac {1}{\Gamma (z)}}&=ze^{\gamma z}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)e^{-{\frac {z}{n}}}\end{aligned}}}</annotation>

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</math><img src="./da75e0f641bec367b64f49ddd9b6156095e4bc32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:30.59ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {1}{\Gamma (z)}}&=z\prod _{n=1}^{\infty }{\frac {1+{\frac {z}{n}}}{\left(1+{\frac {1}{n}}\right)^{z}}}\\{\frac {1}{\Gamma (z)}}&=ze^{\gamma z}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)e^{-{\frac {z}{n}}}\end{aligned}}}" loading="lazy"></dd></dl>

where γ = 0.577216... is the <a href="Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>. These expansions are valid for all complex numbers z.

<div class="mw-heading mw-heading2"><h2 id="Taylor_series">Taylor series</h2></div>

<a href="Taylor_series" title="Taylor series">Taylor series</a> expansion around 0 gives:<a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\ \Gamma (z)\ }}=z+\gamma \ z^{2}+\left({\frac {\gamma ^{2}}{2}}-{\frac {\pi ^{2}}{12}}\right)\ z^{3}+\left({\frac {\gamma ^{3}}{6}}-{\frac {\gamma \pi ^{2}}{12}}+{\frac {\zeta (3)}{3}}\ \right)z^{4}+\cdots \ }</annotation>

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</math><img src="./33c4be19764d1d482f5743d4e18094f893903459.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:70.117ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\ \Gamma (z)\ }}=z+\gamma \ z^{2}+\left({\frac {\gamma ^{2}}{2}}-{\frac {\pi ^{2}}{12}}\right)\ z^{3}+\left({\frac {\gamma ^{3}}{6}}-{\frac {\gamma \pi ^{2}}{12}}+{\frac {\zeta (3)}{3}}\ \right)z^{4}+\cdots \ }" loading="lazy"></dd></dl>

where γ is the <a href="Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>. For n > 2, the coefficient an for the zn term can be computed recursively as<a href="#cite_note-2">[2]</a><a href="#cite_note-Fekih-Ahmed2014-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle a_{n}={\frac {\ {a_{2}\ a_{n-1}+\sum _{j=2}^{n-1}(-1)^{j+1}\ \zeta (j)\ a_{n-j}}\ }{n-1}}={\frac {\ \gamma \ a_{n-1}-\zeta (2)\ a_{n-2}+\zeta (3)\ a_{n-3}-\cdots \ }{n-1}}}</annotation>

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</math><img src="./cd7cdbaaca0c6d8f1e44c1dc715a3ede844c0e15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:82.301ex; height:6.676ex;" alt="{\displaystyle a_{n}={\frac {\ {a_{2}\ a_{n-1}+\sum _{j=2}^{n-1}(-1)^{j+1}\ \zeta (j)\ a_{n-j}}\ }{n-1}}={\frac {\ \gamma \ a_{n-1}-\zeta (2)\ a_{n-2}+\zeta (3)\ a_{n-3}-\cdots \ }{n-1}}}" loading="lazy"></dd></dl>

where ζ is the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>. An integral representation for these coefficients was recently found by Fekih-Ahmed (2014):<a href="#cite_note-Fekih-Ahmed2014-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle a_{n}={\frac {(-1)^{n}}{\pi n!}}\int _{0}^{\infty }e^{-t}\ \operatorname {Im} {\Bigl [}\ {\bigl (}\log(t)-i\pi {\bigr )}^{n}\ {\Bigr ]}\ \mathrm {d} t~.}</annotation>

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</math><img src="./b9a0fc18163e2f8015de9562b7e290afa4b3ac29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.441ex; height:6.176ex;" alt="{\displaystyle a_{n}={\frac {(-1)^{n}}{\pi n!}}\int _{0}^{\infty }e^{-t}\ \operatorname {Im} {\Bigl [}\ {\bigl (}\log(t)-i\pi {\bigr )}^{n}\ {\Bigr ]}\ \mathrm {d} t~.}" loading="lazy"></dd></dl>

For small values, these give the following values:

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<td>+1.0000000000000000000000000000000000000000

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<td>+0.5772156649015328606065120900824024310422

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<td>−0.6558780715202538810770195151453904812798

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<td>−0.0420026350340952355290039348754298187114

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<td>+0.1665386113822914895017007951021052357178

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<td>−0.0421977345555443367482083012891873913017

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<td>−0.0096219715278769735621149216723481989754

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<td>+0.0072189432466630995423950103404465727099

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<td>−0.0011651675918590651121139710840183886668

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<td>−0.0002152416741149509728157299630536478065

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<td>+0.0001280502823881161861531986263281643234

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<td>−0.0000201348547807882386556893914210218184

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<td>−0.0000012504934821426706573453594738330922

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<td>+0.0000011330272319816958823741296203307449

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<td>−0.0000002056338416977607103450154130020573

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<td>+0.0000000061160951044814158178624986828553

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<td>+0.0000000050020076444692229300556650480600

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<td>−0.0000000011812745704870201445881265654365

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<td>+0.0000000001043426711691100510491540332312

</td></tr>

<tr>

<td>+0.0000000000077822634399050712540499373114

</td></tr>

<tr>

<td>−0.0000000000036968056186422057081878158781

</td></tr>

<tr>

<td>+0.0000000000005100370287454475979015481323

</td></tr>

<tr>

<td>−0.0000000000000205832605356650678322242954

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<td>−0.0000000000000053481225394230179823700173

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<td>+0.0000000000000012267786282382607901588938

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<td>−0.0000000000000001181259301697458769513765

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<td>+0.0000000000000000011866922547516003325798

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<td>+0.0000000000000000014123806553180317815558

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<td>−0.0000000000000000002298745684435370206592

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<td>+0.0000000000000000000171440632192733743338

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Fekih-Ahmed (2014)<a href="#cite_note-Fekih-Ahmed2014-3">[3]</a> also gives an approximation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle a_{n}\approx {\frac {(-1)^{n}}{\ (n-1)!\ }}\ {\sqrt {{\frac {2}{\ \pi n\ }}\ }}\ \operatorname {Im} \left({\frac {\ z_{0}^{\left(1/2-n\right)}\ e^{-nz_{0}}\ }{\sqrt {1+z_{0}\ }}}\right)\ ,}</annotation>

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</math><img src="./5eba12bf13a02c6e207629f81fc6de86a0f84eb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.346ex; height:7.843ex;" alt="{\displaystyle a_{n}\approx {\frac {(-1)^{n}}{\ (n-1)!\ }}\ {\sqrt {{\frac {2}{\ \pi n\ }}\ }}\ \operatorname {Im} \left({\frac {\ z_{0}^{\left(1/2-n\right)}\ e^{-nz_{0}}\ }{\sqrt {1+z_{0}\ }}}\right)\ ,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}=-{\frac {1}{n}}\exp \!{\Bigl (}W_{-1}(-n){\Bigr )}\ ,}">

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<annotation encoding="application/x-tex">{\displaystyle z_{0}=-{\frac {1}{n}}\exp \!{\Bigl (}W_{-1}(-n){\Bigr )}\ ,}</annotation>

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</math><img src="./2e12b32f3fd2a1c40e5b31d7111073449d30ccaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.754ex; height:5.176ex;" alt="{\displaystyle z_{0}=-{\frac {1}{n}}\exp \!{\Bigl (}W_{-1}(-n){\Bigr )}\ ,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{-1}}</annotation>

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</math><img src="./bdf91805ba18586ff58b3035ff4b8fe48d39dd67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.526ex; height:2.509ex;" alt="{\displaystyle W_{-1}}" loading="lazy"> is the minus-first branch of the <a href="Lambert_W_function" title="Lambert W function">Lambert W function</a>.

The Taylor expansion around 1 has the same (but shifted) coefficients, i.e.:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\Gamma (1+z)}}={\frac {1}{z\Gamma (z)}}=1+\gamma \ z+\left({\frac {\gamma ^{2}}{2}}-{\frac {\pi ^{2}}{12}}\right)\ z^{2}+\left({\frac {\gamma ^{3}}{6}}-{\frac {\gamma \pi ^{2}}{12}}+{\frac {\zeta (3)}{3}}\ \right)z^{3}+\cdots \ }</annotation>

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</math><img src="./9b0591c6a56334c376a1a6e208501a900437d9f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:81.35ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\Gamma (1+z)}}={\frac {1}{z\Gamma (z)}}=1+\gamma \ z+\left({\frac {\gamma ^{2}}{2}}-{\frac {\pi ^{2}}{12}}\right)\ z^{2}+\left({\frac {\gamma ^{3}}{6}}-{\frac {\gamma \pi ^{2}}{12}}+{\frac {\zeta (3)}{3}}\ \right)z^{3}+\cdots \ }" loading="lazy"></dd></dl>

(the reciprocal of <a href="Gamma_function#Pi_function" title="Gamma function">Gauss' pi-function</a>).

<div class="mw-heading mw-heading2"><h2 id="Asymptotic_expansion">Asymptotic expansion</h2></div>

As |z| goes to infinity at a constant arg(z) we have:

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<annotation encoding="application/x-tex">{\displaystyle \ln(1/\Gamma (z))\sim -z\ln(z)+z+{\tfrac {1}{2}}\ln \left({\frac {z}{2\pi }}\right)-{\frac {1}{12z}}+{\frac {1}{360z^{3}}}-{\frac {1}{1260z^{5}}}\qquad {\text{for}}~\left|\arg(z)\right|<\pi }</annotation>

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</math><img src="./7dfa9e66211c0928e0a3e54755a67082ded3a2c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:85.628ex; height:5.509ex;" alt="{\displaystyle \ln(1/\Gamma (z))\sim -z\ln(z)+z+{\tfrac {1}{2}}\ln \left({\frac {z}{2\pi }}\right)-{\frac {1}{12z}}+{\frac {1}{360z^{3}}}-{\frac {1}{1260z^{5}}}\qquad {\text{for}}~\left|\arg(z)\right|<\pi }" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Contour_integral_representation">Contour integral representation</h2></div>

An integral representation due to <a href="Hermann_Hankel" title="Hermann Hankel">Hermann Hankel</a> is

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\Gamma (z)}}={\frac {i}{2\pi }}\oint _{H}(-t)^{-z}e^{-t}\,dt,}</annotation>

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</math><img src="./de194bade98289ede89e61d0738819148af914a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.001ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{\Gamma (z)}}={\frac {i}{2\pi }}\oint _{H}(-t)^{-z}e^{-t}\,dt,}" loading="lazy"></dd></dl>

where H is the <a href="Hankel_contour" title="Hankel contour">Hankel contour</a>, that is, the path encircling 0 in the positive direction, beginning at and returning to positive infinity with respect for the <a href="Branch_cut" class="mw-redirect" title="Branch cut">branch cut</a> along the positive real axis. According to Schmelzer & Trefethen,<a href="#cite_note-4">[4]</a> numerical evaluation of Hankel's integral is the basis of some of the best methods for computing the gamma function.

<div class="mw-heading mw-heading2"><h2 id="Integral_representations_at_the_positive_integers">Integral representations at the positive integers</h2></div>

For positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}">

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<annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation>

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</math><img src="./d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}" loading="lazy">, there is an integral for the reciprocal <a href="Factorial" title="Factorial">factorial</a> function given by<a href="#cite_note-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n!}}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }e^{-nit}e^{e^{it}}\ dt.}</annotation>

</semantics>

</math><img src="./6a1ceb79d501895b44ac5c3664baf629966ddd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.04ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{n!}}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }e^{-nit}e^{e^{it}}\ dt.}" loading="lazy"></dd></dl>

Similarly, for any real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c>0}</annotation>

</semantics>

</math><img src="./2ba126f626d61752f62eaacaf11761a54de4dc84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {C} }</annotation>

</semantics>

</math><img src="./169fae60c23a2027ece2aa7fd4b5047492887e91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {C} }" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Re(z)>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Re(z)>0}</annotation>

</semantics>

</math><img src="./0ddebb2f5a486b91450ccc69278d92ff5125e46b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.006ex; height:2.843ex;" alt="{\displaystyle Re(z)>0}" loading="lazy"> we have the next integral for the reciprocal gamma function along the real axis in the form of:<a href="#cite_note-6">[6]</a>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\Gamma (z)}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }(c+it)^{-z}e^{c+it}dt,}</annotation>

</semantics>

</math><img src="./12955973efad2d46b85945753b0363e35963f30f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.197ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\Gamma (z)}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }(c+it)^{-z}e^{c+it}dt,}" loading="lazy"></dd></dl>

where the particular case when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=n+1/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=n+1/2}</annotation>

</semantics>

</math><img src="./1a156a8b0681031d711ef5ed311baad6c096923f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.909ex; height:2.843ex;" alt="{\displaystyle z=n+1/2}" loading="lazy"> provides a corresponding relation for the reciprocal <a href="Double_factorial" title="Double factorial">double factorial</a> function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{(2n-1)!!}}={\frac {\sqrt {\pi }}{2^{n}\cdot \Gamma \left(n+{\frac {1}{2}}\right)}}.}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msqrt>

</msqrt>

<mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(2n-1)!!}}={\frac {\sqrt {\pi }}{2^{n}\cdot \Gamma \left(n+{\frac {1}{2}}\right)}}.}</annotation>

</semantics>

</math><img src="./b2b0781df62a47fa30accc69381efd829a74f646.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:29.004ex; height:7.343ex;" alt="{\displaystyle {\frac {1}{(2n-1)!!}}={\frac {\sqrt {\pi }}{2^{n}\cdot \Gamma \left(n+{\frac {1}{2}}\right)}}.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Integral_along_the_real_axis">Integral along the real axis</h2></div>

Integration of the reciprocal gamma function along the positive real axis gives the value

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }{\frac {1}{\Gamma (x)}}\,dx\approx 2.80777024,}</annotation>

</semantics>

</math><img src="./94877063dba841477dc14e4314a26c6e70e00f23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.941ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{\infty }{\frac {1}{\Gamma (x)}}\,dx\approx 2.80777024,}" loading="lazy"></dd></dl>

which is known as the <a href="Frans%C3%A9n%E2%80%93Robinson_constant" title="Fransén–Robinson constant">Fransén–Robinson constant</a>.<a href="#cite_note-7">[7]</a>

We have the following formula (<a href="#cite_note-8">[8]</a> chapter 9, exercise 100)

</mrow>

</mrow>

</msubsup>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }{\dfrac {a^{x}}{\Gamma (x)}}\,dx=ae^{a}+a\int _{0}^{\infty }{\dfrac {e^{-ax}}{\log ^{2}(x)+\pi ^{2}}}\,dx}</annotation>

</semantics>

</math><img src="./4df45a2889e1f9edb99accf3cf92e5560bd8a25c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.947ex; height:6.676ex;" alt="{\displaystyle \int _{0}^{\infty }{\dfrac {a^{x}}{\Gamma (x)}}\,dx=ae^{a}+a\int _{0}^{\infty }{\dfrac {e^{-ax}}{\log ^{2}(x)+\pi ^{2}}}\,dx}" loading="lazy"></dd></dl>

<ul><li><a href="Bessel%E2%80%93Clifford_function" title="Bessel–Clifford function">Bessel–Clifford function</a></li>

<li><a href="Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse-gamma distribution</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<ul><li>Mette Lund, <a rel="nofollow" class="external text" href="http://www.nbi.dk/~polesen/borel/node14.html">An integral for the reciprocal Gamma function</a></li>

<li>Milton Abramowitz & Irene A. Stegun, <a href="Handbook_of_Mathematical_Functions_with_Formulas%2C_Graphs%2C_and_Mathematical_Tables" class="mw-redirect" title="Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a></li>

<li><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/GammaFunction.html">Gamma Function</a>, MathWorld</li></ul></div><div><div class="zim-footer">

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