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Reciprocal gamma function
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<h2 id="infinite-product-expansion">Infinite product expansion</h2> <p>Following from the <a href="/facts/Infinite_product/qaaNh5PM">infinite product</a> definitions for the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>, due to <a href="/facts/Leonhard_Euler/z2fsbjgj">Euler</a> and <a href="/facts/Karl_Weierstrass/4U8vdQzL">Weierstrass</a> respectively, we get the following infinite product expansion for the reciprocal gamma function: </p> 1 Γ ( z ) = z ∏ n = 1 ∞ 1 + z n ( 1 + 1 n ) z 1 Γ ( z ) = z e γ z ∏ n = 1 ∞ ( 1 + z n ) e − z n {\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}}} <p>where <i>γ</i> = 0.577216... is the <a href="/facts/Euler%25E2%2580%2593Mascheroni_constant/2kEgRvjC">Euler–Mascheroni constant</a>. These expansions are valid for all complex numbers <i>z</i>. </p> <h2 id="taylor-series">Taylor series</h2> <p><a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> expansion around 0 gives:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> 1 Γ ( z ) = z + γ z 2 + ( γ 2 2 − π 2 12 ) z 3 + ( γ 3 6 − γ π 2 12 + ζ ( 3 ) 3 ) z 4 + ⋯ {\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 \ } <p>where <i>γ</i> is the <a href="/facts/Euler%25E2%2580%2593Mascheroni_constant/2kEgRvjC">Euler–Mascheroni constant</a>. For <i>n</i> > 2, the coefficient <i>a</i><i>n</i> for the <i>z</i><i>n</i> term can be computed recursively as<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> a n = a 2 a n − 1 + ∑ j = 2 n − 1 ( − 1 ) j + 1 ζ ( j ) a n − j n − 1 = γ a n − 1 − ζ ( 2 ) a n − 2 + ζ ( 3 ) a n − 3 − ⋯ n − 1 {\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}}} <p>where <i>ζ</i> is the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>. An integral representation for these coefficients was recently found by Fekih-Ahmed (2014):<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> a n = ( − 1 ) n π n ! ∫ 0 ∞ e − t Im [ ( log ( t ) − i π ) n ] d t . {\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~.} <p>For small values, these give the following values: </p> <table><tbody><tr><th><i>n</i></th><th><i>a</i><i>n</i></th></tr><tr><td>1</td><td>+1.0000000000000000000000000000000000000000</td></tr><tr><td>2</td><td>+0.5772156649015328606065120900824024310422</td></tr><tr><td>3</td><td>−0.6558780715202538810770195151453904812798</td></tr><tr><td>4</td><td>−0.0420026350340952355290039348754298187114</td></tr><tr><td>5</td><td>+0.1665386113822914895017007951021052357178</td></tr><tr><td>6</td><td>−0.0421977345555443367482083012891873913017</td></tr><tr><td>7</td><td>−0.0096219715278769735621149216723481989754</td></tr><tr><td>8</td><td>+0.0072189432466630995423950103404465727099</td></tr><tr><td>9</td><td>−0.0011651675918590651121139710840183886668</td></tr><tr><td>10</td><td>−0.0002152416741149509728157299630536478065</td></tr><tr><td>11</td><td>+0.0001280502823881161861531986263281643234</td></tr><tr><td>12</td><td>−0.0000201348547807882386556893914210218184</td></tr><tr><td>13</td><td>−0.0000012504934821426706573453594738330922</td></tr><tr><td>14</td><td>+0.0000011330272319816958823741296203307449</td></tr><tr><td>15</td><td>−0.0000002056338416977607103450154130020573</td></tr><tr><td>16</td><td>+0.0000000061160951044814158178624986828553</td></tr><tr><td>17</td><td>+0.0000000050020076444692229300556650480600</td></tr><tr><td>18</td><td>−0.0000000011812745704870201445881265654365</td></tr><tr><td>19</td><td>+0.0000000001043426711691100510491540332312</td></tr><tr><td>20</td><td>+0.0000000000077822634399050712540499373114</td></tr><tr><td>21</td><td>−0.0000000000036968056186422057081878158781</td></tr><tr><td>22</td><td>+0.0000000000005100370287454475979015481323</td></tr><tr><td>23</td><td>−0.0000000000000205832605356650678322242954</td></tr><tr><td>24</td><td>−0.0000000000000053481225394230179823700173</td></tr><tr><td>25</td><td>+0.0000000000000012267786282382607901588938</td></tr><tr><td>26</td><td>−0.0000000000000001181259301697458769513765</td></tr><tr><td>27</td><td>+0.0000000000000000011866922547516003325798</td></tr><tr><td>28</td><td>+0.0000000000000000014123806553180317815558</td></tr><tr><td>29</td><td>−0.0000000000000000002298745684435370206592</td></tr><tr><td>30</td><td>+0.0000000000000000000171440632192733743338</td></tr></tbody></table> <p>Fekih-Ahmed (2014)<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> also gives an approximation for a n {\displaystyle a_{n}} : </p> a n ≈ ( − 1 ) n ( n − 1 ) ! 2 π n Im ( z 0 ( 1 / 2 − n ) e − n z 0 1 + z 0 ) , {\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)\ ,} <p>where z 0 = − 1 n exp ( W − 1 ( − n ) ) , {\displaystyle z_{0}=-{\frac {1}{n}}\exp \!{\Bigl (}W_{-1}(-n){\Bigr )}\ ,} and W − 1 {\displaystyle W_{-1}} is the minus-first branch of the <a href="/facts/Lambert_W_function/NG1wd71f">Lambert W function</a>. </p><p>The Taylor expansion around 1 has the same (but shifted) coefficients, i.e.: </p> 1 Γ ( 1 + z ) = 1 z Γ ( z ) = 1 + γ z + ( γ 2 2 − π 2 12 ) z 2 + ( γ 3 6 − γ π 2 12 + ζ ( 3 ) 3 ) z 3 + ⋯ {\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 \ } <p>(the reciprocal of <a href="/facts/Gamma_function/SjGQcnyw">Gauss' pi-function</a>). </p> <h2 id="asymptotic-expansion">Asymptotic expansion</h2> <p>As |<i>z</i>| goes to infinity at a constant arg(<i>z</i>) we have: </p> ln ( 1 / Γ ( z ) ) ∼ − z ln ( z ) + z + 1 2 ln ( z 2 π ) − 1 12 z + 1 360 z 3 − 1 1260 z 5 for | arg ( z ) | < π {\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 } <h2 id="contour-integral-representation">Contour integral representation</h2> <p>An integral representation due to <a href="/facts/Hermann_Hankel/rWmnbQC8">Hermann Hankel</a> is </p> 1 Γ ( z ) = i 2 π ∮ H ( − t ) − z e − t d t , {\displaystyle {\frac {1}{\Gamma (z)}}={\frac {i}{2\pi }}\oint _{H}(-t)^{-z}e^{-t}\,dt,} <p>where <i>H</i> is the <a href="/facts/Hankel_contour/cGD27Hus">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="/facts/Branch_cut/lJ7Y6Iht">branch cut</a> along the positive real axis. According to Schmelzer & Trefethen,<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> numerical evaluation of Hankel's integral is the basis of some of the best methods for computing the gamma function. </p> <h2 id="integral-representations-at-the-positive-integers">Integral representations at the positive integers</h2> <p>For positive integers n ≥ 1 {\displaystyle n\geq 1} , there is an integral for the reciprocal <a href="/facts/Factorial/kkfy1Bwe">factorial</a> function given by<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> 1 n ! = 1 2 π ∫ − π π e − n i t e e i t d t . {\displaystyle {\frac {1}{n!}}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }e^{-nit}e^{e^{it}}\ dt.} <p>Similarly, for any real c > 0 {\displaystyle c>0} and z ∈ C {\displaystyle z\in \mathbb {C} } such that R e ( z ) > 0 {\displaystyle Re(z)>0} we have the next integral for the reciprocal gamma function along the real axis in the form of:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> 1 Γ ( z ) = 1 2 π ∫ − ∞ ∞ ( c + i t ) − z e c + i t d t , {\displaystyle {\frac {1}{\Gamma (z)}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }(c+it)^{-z}e^{c+it}dt,} <p>where the particular case when z = n + 1 / 2 {\displaystyle z=n+1/2} provides a corresponding relation for the reciprocal <a href="/facts/Double_factorial/VSU8AgzB">double factorial</a> function, 1 ( 2 n − 1 ) ! ! = π 2 n ⋅ Γ ( n + 1 2 ) . {\displaystyle {\frac {1}{(2n-1)!!}}={\frac {\sqrt {\pi }}{2^{n}\cdot \Gamma \left(n+{\frac {1}{2}}\right)}}.} </p> <h2 id="integral-along-the-real-axis">Integral along the real axis</h2> <p>Integration of the reciprocal gamma function along the positive real axis gives the value </p> ∫ 0 ∞ 1 Γ ( x ) d x ≈ 2.80777024 , {\displaystyle \int _{0}^{\infty }{\frac {1}{\Gamma (x)}}\,dx\approx 2.80777024,} <p>which is known as the <a href="/facts/Frans%25C3%25A9n%25E2%2580%2593Robinson_constant/GBlrQlAq">Fransén–Robinson constant</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p>We have the following formula (<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> chapter 9, exercise 100) </p> ∫ 0 ∞ a x Γ ( x ) d x = a e a + a ∫ 0 ∞ e − a x log 2 ( x ) + π 2 d x {\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} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Bessel%25E2%2580%2593Clifford_function/BWX2owAe">Bessel–Clifford function</a></li> <li><a href="/facts/Inverse-gamma_distribution/zpyktNyN">Inverse-gamma distribution</a></li></ul> <ul><li>Mette Lund, <a href="http://www.nbi.dk/~polesen/borel/node14.html">An integral for the reciprocal Gamma function</a></li> <li>Milton Abramowitz & Irene A. Stegun, <i><a href="/facts/Handbook_of_Mathematical_Functions_with_Formulas%2c_Graphs%2c_and_Mathematical_Tables/CNGSV3Ed">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a></i></li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Eric W. Weisstein</a>, <i><a href="http://mathworld.wolfram.com/GammaFunction.html">Gamma Function</a></i>, MathWorld</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Weisstein, Eric W. "Gamma function". mathworld.wolfram.com. Retrieved 2021-06-15. <a href="https://mathworld.wolfram.com/GammaFunction.html" target="_blank">https://mathworld.wolfram.com/GammaFunction.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Wrench, J.W. (1968). "Concerning two series for the gamma function". Mathematics of Computation. 22 (103): 617–626. doi:10.1090/S0025-5718-1968-0237078-4. S2CID 121472614. and Wrench, J.W. (1973). "Erratum: Concerning two series for the gamma function". Mathematics of Computation. 27 (123): 681–682. doi:10.1090/S0025-5718-1973-0319344-9. <a href="https://doi.org/10.1090%2FS0025-5718-1968-0237078-4" target="_blank">https://doi.org/10.1090%2FS0025-5718-1968-0237078-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Fekih-Ahmed, L. (2014). "On the power series expansion of the reciprocal gamma function". HAL archives. <a href="https://hal.archives-ouvertes.fr/hal-01029331v1" target="_blank">https://hal.archives-ouvertes.fr/hal-01029331v1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Fekih-Ahmed, L. (2014). "On the power series expansion of the reciprocal gamma function". HAL archives. <a href="https://hal.archives-ouvertes.fr/hal-01029331v1" target="_blank">https://hal.archives-ouvertes.fr/hal-01029331v1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Fekih-Ahmed, L. (2014). "On the power series expansion of the reciprocal gamma function". HAL archives. <a href="https://hal.archives-ouvertes.fr/hal-01029331v1" target="_blank">https://hal.archives-ouvertes.fr/hal-01029331v1</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Schmelzer, Thomas; Trefethen, Lloyd N. (2007). "Computing the Gamma function using contour integrals and rational approximations". SIAM Journal on Numerical Analysis. 45 (2). Society for Industrial and Applied Mathematics: 558–571. doi:10.1137/050646342.; "Copy on Trefethen's academic website" (PDF). Mathematics, Oxford, UK. Retrieved 2020-08-03.; "Link to two other copies". CiteSeerX 10.1.1.210.299. <a href="/wiki/Lloyd_N._Trefethen" target="_blank">/wiki/Lloyd_N._Trefethen</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Graham, Knuth, and Patashnik (1994). Concrete Mathematics. Addison-Wesley. p. 566.{{cite book}}: CS1 maint: multiple names: authors list (link) <a href="/wiki/Template:Cite_book" target="_blank">/wiki/Template:Cite_book</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Schmidt, Maxie D. (2019-05-19). "A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions". Axioms. 8 (2): 62. arXiv:1809.03933. doi:10.3390/axioms8020062. <a href="https://doi.org/10.3390%2Faxioms8020062" target="_blank">https://doi.org/10.3390%2Faxioms8020062</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Sloane, N. J. A. (ed.). "Sequence A058655 (Decimal expansion of area under the curve 1/Gamma(x) from zero to infinity)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. <a href="/wiki/Neil_Sloane" target="_blank">/wiki/Neil_Sloane</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Henri Cohen (2007). Number Theory Volume II: Analytic and Modern Tools. Graduate Texts in Mathematics. Vol. 240. doi:10.1007/978-0-387-49894-2. ISBN 978-0-387-49893-5. ISSN 0072-5285. <a href="978-0-387-49893-5" target="_blank">978-0-387-49893-5</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> </ol>