Reciprocal Gamma function

In mathematics, the reciprocal Gamma function is the function

:$f(z)\; =\; frac\{1\}\{Gamma(z)\},$

where $Gamma(z)$ denotes the Gamma function. Since the Gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. The reciprocal is sometimes used as a starting point for numerical computation of the Gamma function, and a few software libraries provide it separately from the regular Gamma function.

Karl Weierstrass called the reciprocal Gamma function the "factorielle" and used it in his development of the Weierstrass factorization theorem.

Taylor series

Taylor series expansion around 0 gives

:$frac\{1\}\{Gamma(z)\}\; =\; z\; +\; gamma\; z^2\; +\; left(frac\{gamma^2\}\{2\}\; -\; frac\{pi^2\}\{12\}\; ight)z^3\; +\; ldots$

where $gamma$ is the Euler-Mascheroni constant. For "k" > 2, the coefficient "a_k" for the "z^k" term can be computed recursively as

:$a\_k\; =\; k\; a\_1\; a\_k\; -\; a\_2\; a\_\{k-1\}\; +\; sum\_\{j=2\}^\{k-1\}\; (-1)^j\; ,\; zeta(j)\; ,\; a\_\{k-j\}$

where ζ("s") is the Riemann zeta function.

Contour integral representation

An integral representation due to Hermann Hankel is

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

where "C" is a 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, numerical evaluation of Hankel's integral is the basis of some of the best methods for computing the Gamma function.

Integral along the real axis

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

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

which is known as the Fransén-Robinson constant.

ee also

* Inverse-gamma distribution

References

* Thomas Schmelzer & Lloyd N. Trefethen, [http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/gamma.pdf Computing the Gamma function using contour integrals and rational approximations]
* 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