Reciprocal gamma function

Infinite product expansion

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: \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 is the Euler–Mascheroni constant. These expansions are valid for all complex numbers z.

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 + \left(\frac{\gamma^3}{6} - \frac{\gamma\pi^2}{12} + \frac{\zeta(3)}{3}\ \right)z^4 + \cdots\ where γ is the Euler–Mascheroni constant. For n 2, the coefficient a**n for the z**n term can be computed recursively as 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} where ζ is the Riemann zeta function. An integral representation for these coefficients was recently found by Fekih-Ahmed (2014): 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 ~.

For small values, these give the following values:

Fekih-Ahmed (2014) also gives an approximation for a**n:

a_n \approx (-1)^n \sqrt{\frac{ 2 }{ \pi }\ } \frac{\sqrt{n\ }}{n!} \ \operatorname\mathcal{I_m} \left( \frac{\ z_0^{(1/2-n)}\ e^{-n z_0}\ }{\sqrt{ 1 + z_0\ }} \right)\ , -- 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^{-n z_0}\ }{\sqrt{ 1 + z_0\ }} \right)\ , where , and W−1 is the negative-first branch of the Lambert W function.

The Taylor expansion around 1 has the same (but shifted) coefficients, i.e.: \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\ (the reciprocal of Gauss's pi function).

Asymptotic expansion

As goes to infinity at a constant arg(z) we have: \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}{1260 z^5}\qquad \text{for}~ \left|\arg(z)\right|

Contour integral representation

An integral representation due to Hermann Hankel is \frac{1}{\Gamma(z)} = \frac{i}{2\pi} \oint_H (-t)^{-z} e^{-t} , dt, where 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,; ; numerical evaluation of Hankel's integral is the basis of some of the best methods for computing the gamma function.

Integral representations at the positive integers

For positive integers n ≥ 1, there is an integral for the reciprocal factorial function given by \frac{1}{n!} = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{-nit} e^{e^{it}}\ dt.

Similarly, for any real c 0 and z ∈ C such that Re(z) 0 we have the next integral for the reciprocal gamma function along the real axis in the form of: \frac{1}{\Gamma(z)} = \frac{1}{2\pi} \int_{-\infty}^{\infty} (c+ it)^{-z} e^{c+it} dt, where the particular case when provides a corresponding relation for the reciprocal double factorial function, \frac{1}{(2n-1)!!} = \frac{\sqrt{\pi}}{2^n \cdot \Gamma\left(n+\frac{1}{2}\right)}.

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.

We have the following formula \int_0^\infty \dfrac{a^x}{\Gamma(x)},dx=ae^a+a\int_0^\infty\dfrac{e^{-ax}}{\log^2(x)+\pi^2},dx

References

• Mette Lund, 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, Gamma Function, MathWorld

References

1. Weisstein, Eric W.. "Gamma function".
2. Wrench, J.W.. (1968). "Concerning two series for the gamma function". Mathematics of Computation.
3. Fekih-Ahmed, L.. (2014). "On the power series expansion of the reciprocal gamma function".
4. (1994). "Concrete Mathematics". Addison-Wesley.
5. {{cite OEIS. A058655. Decimal expansion of area under the curve 1/Gamma(x) from zero to infinity
6. Henri Cohen. (2007). "Number Theory Volume II: Analytic and Modern Tools".