reciprocal gamma function

Following from the infinite product definitions for the gamma function, due to Euler and Weierstrass respectively, we get the following infinite product expansion for the reciprocal gamma function:

:$\backslash begin\{align\}$

\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)} &= z e^{\gamma z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right) e^{-\frac{z}{n}}

\end{align}

where {{math|γ {{=}} 0.577216...}} is the Euler–Mascheroni constant. These expansions are valid for all complex numbers {{math|z}}.

Taylor series expansion around 0 gives:{{cite web |last=Weisstein |first=Eric W. |title=Gamma function |website=mathworld.wolfram.com |lang=en |url=https://mathworld.wolfram.com/GammaFunction.html |access-date=2021-06-15}}

:$\backslash frac\{1\}\{\backslash \; \backslash Gamma(z)\backslash \; \}\; =\; z\; +\; \backslash gamma\backslash \; z^2\; +\; \backslash left(\backslash frac\{\backslash gamma^2\}\{2\}\; -\; \backslash frac\{\backslash pi^2\}\{12\}\backslash right)\backslash \; z^3\; +\; \backslash left(\backslash frac\{\backslash gamma^3\}\{6\}\; -\; \backslash frac\{\backslash gamma\backslash pi^2\}\{12\}\; +\; \backslash frac\{\backslash zeta(3)\}\{3\}\backslash \; \backslash right)z^4\; +\; \backslash cdots\backslash$

where {{math|γ}} is the Euler–Mascheroni constant. For {{math|n > 2}}, the coefficient {{math|a_n}} for the {{math|zⁿ}} term can be computed recursively as{{cite journal |last=Wrench |first=J.W. |year=1968 |title=Concerning two series for the gamma function |journal=Mathematics of Computation |volume=22 |issue=103 |pages=617–626 |doi=10.1090/S0025-5718-1968-0237078-4 |s2cid=121472614 |doi-access=free }} and
{{cite journal |last=Wrench |first=J.W. |year=1973 |title=Erratum: Concerning two series for the gamma function |journal=Mathematics of Computation |volume=27 |issue=123 |pages=681–682 |doi=10.1090/S0025-5718-1973-0319344-9 |doi-access=free }}

:$a\_n\; =\; \backslash frac\{\backslash \; \{a\_2\backslash \; a\_\{n-1\}\; +\; \backslash sum\_\{j=2\}^\{n-1\}\; (-1)^\{j+1\}\backslash \; \backslash zeta(j)\backslash \; a\_\{n-j\}\}\backslash \; \}\{n-1\}\; =\; \backslash frac\{\backslash \; \backslash gamma\backslash \; a\_\{n-1\}\; -\; \backslash zeta(2)\backslash \; a\_\{n-2\}\; +\; \backslash zeta(3)\backslash \; a\_\{n-3\}-\backslash cdots\backslash \; \}\{n-1\}$

where {{math|ζ}} is the Riemann zeta function. An integral representation for these coefficients was recently found by Fekih-Ahmed (2014):{{cite web |last=Fekih-Ahmed |first=L. |year=2014 |title=On the power series expansion of the reciprocal gamma function |website=HAL archives |url=https://hal.archives-ouvertes.fr/hal-01029331v1}}

:$a\_n\; =\; \backslash frac\{(-1)^n\}\{\backslash pi\; n!\}\backslash int\_0^\backslash infty\; e^\{-t\}\backslash \; \backslash operatorname\{Im\}\backslash Bigl[\backslash \; \backslash bigl(\; \backslash log(t)-i\backslash pi\; \backslash bigr)^n\backslash \; \backslash Bigr]\backslash \; \backslash mathrm\{d\}\; t\; ~.$

For small values, these give the following values:

Fekih-Ahmed (2014) also gives an approximation for $a\_n$:

:$a\_n\; \backslash approx\; \backslash frac\{(-1)^n\}\{\backslash \; (n-1)!\backslash \; \}\; \backslash \; \backslash sqrt\{\; \backslash frac\{2\}\{\backslash \; \backslash pi\; n\; \backslash \; \}\backslash \; \}\backslash$

\operatorname{Im} \left( \frac{\ z_0^{\left( 1/2 - n \right)}\ e^{-n z_0}\ }{\sqrt{ 1 + z_0\ }} \right)\ ,

where $z\_0\; =\; -\backslash frac\{1\}\{n\}\; \backslash exp\backslash !\backslash Bigl(\; W\_\{-1\}(-n)\; \backslash Bigr)\backslash \; ,$ and $W\_\{-1\}$ is the minus-first branch of the Lambert W function.

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

:$\backslash frac\{1\}\{\backslash Gamma(1+z)\}\; =\; \backslash frac\{1\}\{z\backslash Gamma(z)\}\; =\; 1\; +\; \backslash gamma\backslash \; z\; +\; \backslash left(\backslash frac\{\backslash gamma^2\}\{2\}\; -\; \backslash frac\{\backslash pi^2\}\{12\}\backslash right)\backslash \; z^2\; +\; \backslash left(\backslash frac\{\backslash gamma^3\}\{6\}\; -\; \backslash frac\{\backslash gamma\backslash pi^2\}\{12\}\; +\; \backslash frac\{\backslash zeta(3)\}\{3\}\backslash \; \backslash right)z^3\; +\; \backslash cdots\backslash$

(the reciprocal of Gauss' pi-function).

As {{math|{{abs|z}}}} goes to infinity at a constant {{math|arg(z)}} we have:

:

An integral representation due to Hermann Hankel is

:$\backslash frac\{1\}\{\backslash Gamma(z)\}\; =\; \backslash frac\{i\}\{2\backslash pi\}\; \backslash oint\_H\; (-t)^\{-z\}\; e^\{-t\}\; \backslash ,\; dt,$

where {{math|H}} is the Hankel contour, that is, the path encircling 0 in the positive direction, beginning at and returning to positive infinity with respect for the branch cut along the positive real axis. According to Schmelzer & Trefethen,{{cite journal |first1=Thomas |last1=Schmelzer |author2-link=Lloyd N. Trefethen |first2=Lloyd N. |last2=Trefethen |year=2007 |title=Computing the Gamma function using contour integrals and rational approximations |journal=SIAM Journal on Numerical Analysis |volume=45 |issue=2 |pages=558–571 |doi=10.1137/050646342 |publisher=Society for Industrial and Applied Mathematics |url=https://epubs.siam.org/doi/abs/10.1137/050646342}};

{{cite web |title=Copy on Trefethen's academic website |website=Mathematics, Oxford, UK |url=https://people.maths.ox.ac.uk/trefethen/publication/PDF/2007_122.pdf |access-date=2020-08-03}};

{{cite CiteSeerX |title=Link to two other copies |citeseerx=10.1.1.210.299 }} numerical evaluation of Hankel's integral is the basis of some of the best methods for computing the gamma function.

For positive integers $n\; \backslash geq\; 1$, there is an integral for the reciprocal factorial function given by{{cite book|last1=Graham, Knuth, and Patashnik|title=Concrete Mathematics|date=1994|publisher=Addison-Wesley|page=566}}

:$\backslash frac\{1\}\{n!\}\; =\; \backslash frac\{1\}\{2\backslash pi\}\; \backslash int\_\{-\backslash pi\}^\{\backslash pi\}\; e^\{-nit\}\; e^\{e^\{it\}\}\backslash \; dt.$

Similarly, for any real $c\; >\; 0$ and $z\; \backslash in\; \backslash mathbb\{C\}$ such that $Re(z)\; >\; 0$ we have the next integral for the reciprocal gamma function along the real axis in the form of:

{{cite journal |last1= Schmidt|first1= Maxie D.|date= 2019-05-19|title=A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions |journal=Axioms |volume= 8|issue=2 |pages=62 |doi=10.3390/axioms8020062 |doi-access= free|arxiv=1809.03933}}

:$\backslash frac\{1\}\{\backslash Gamma(z)\}\; =\; \backslash frac\{1\}\{2\backslash pi\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; (c+\; it)^\{-z\}\; e^\{c+it\}\; dt,$

where the particular case when $z\; =\; n\; +\; 1/2$ provides a corresponding relation for the reciprocal double factorial function, $\backslash frac\{1\}\{(2n-1)!!\}\; =\; \backslash frac\{\backslash sqrt\{\backslash pi\}\}\{2^n\; \backslash cdot\; \backslash Gamma\backslash left(n+\backslash frac\{1\}\{2\}\backslash right)\}.$

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

:$\backslash int\_\{0\}^\backslash infty\; \backslash frac\{1\}\{\backslash Gamma(x)\}\backslash ,\; dx\; \backslash approx\; 2.80777024,$

which is known as the Fransén–Robinson constant.{{cite OEIS|A058655|Decimal expansion of area under the curve 1/Gamma(x) from zero to infinity}}

We have the following formula ({{Cite book |author= Henri Cohen|date=2007 |title=Number Theory Volume II: Analytic and Modern Tools |url=https://doi.org/10.1007/978-0-387-49894-2 |series=Graduate Texts in Mathematics |volume=240 |language=en |doi=10.1007/978-0-387-49894-2 |isbn=978-0-387-49893-5 |issn=0072-5285}} chapter 9, exercise 100)

:$\backslash int\_0^\backslash infty\; \backslash dfrac\{a^x\}\{\backslash Gamma(x)\}\backslash ,dx=ae^a+a\backslash int\_0^\backslash infty\backslash dfrac\{e^\{-ax\}\}\{\backslash log^2(x)+\backslash pi^2\}\backslash ,dx$

• Bessel–Clifford function
• Inverse-gamma distribution

{{reflist|25em}}

• Mette Lund, [http://www.nbi.dk/~polesen/borel/node14.html An integral for the reciprocal Gamma function]
• Milton Abramowitz & Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
• Eric W. Weisstein, [http://mathworld.wolfram.com/GammaFunction.html Gamma Function], MathWorld

Category:Gamma and related functions

Category:Analytic functions