Reciprocal Gamma function

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Plot of 1/Γ(x) along the real axis

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.

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

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

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.