Reciprocal Gamma function

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Plot of 1/Γ(x) along the real axis

Reciprocal Gamma function 1/Γ(z) in the complex plane. The color of a point z encodes the value of 1/Γ(z). Strong colors denote values close to zero and hue encodes the value's argument.

In mathematics, the reciprocal Gamma function is the function

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

where $Γ(z)$ 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.

1 Taylor series
2 Contour integral representation
3 Integral along the real axis
4 See also
5 References

[edit] Taylor series

Taylor series expansion around 0 gives

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

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

$a_k = k a_1 a_k - a_2 a_{k-1} + \sum_{j=2}^k (-1)^j \, \zeta(j) \, a_{k-j}$

where ζ(s) is the Riemann zeta function.

[edit] Contour integral representation

An integral representation due to Hermann Hankel is

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

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.