Gamma function

Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral:


$${\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}{\text{ d}}t,\ \qquad \Re (z)>0\,.}$$


The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.


The gamma function has no zeros, so the reciprocal gamma function 1/Γ(z) is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function:


$${\displaystyle \Gamma (z)={\mathcal {M}}\{e^{-x}\}(z)\,.}$$


Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics.

Practical implementations[edit]

Unlike many other functions, such as a Normal Distribution, no obvious fast, accurate implementation that is easy to implement for the Gamma Function ${\displaystyle \Gamma (z)}$ is easily found. Therefore, it is worth investigating potential solutions. For the case that speed is more important than accuracy, published tables for ${\displaystyle \Gamma (z)}$ are easily found in an Internet search, such as the Online Wiley Library. Such tables may be used with linear interpolation. Greater accuracy is obtainable with the use of cubic interpolation at the cost of more computational overhead. Since ${\displaystyle \Gamma (z)}$ tables are usually published for argument values between 1 and 2, the property ${\displaystyle \Gamma (z+1)=z\ \Gamma (z)}$ may be used to quickly and easily translate all real values ${\displaystyle z<1}$ and ${\displaystyle z>2}$ into the range ${\displaystyle 1\leq z\leq 2}$, such that only tabulated values of ${\displaystyle z}$ between 1 and 2 need be used.^[55]


If interpolation tables are not desirable, then the Lanczos approximation mentioned above works well for 1 to 2 digits of accuracy for small, commonly used values of z. If the Lanczos approximation is not sufficiently accurate, the Sterling's formula for the Gamma Function may be used. A more complete and accurate solution to Sterling's Approximation may be found at Math2.org, and is reproduced below in terms of ${\displaystyle \Gamma (z)}$^[56] for the first 8 (${\displaystyle z^{0}{\text{ to }}z^{-7}}$) terms. Based on experimental findings against known values of ${\displaystyle \Gamma (z)}$ for 1, 1.5, and 2 (1, ${\displaystyle {\sqrt {\pi }}/2}$, and 1 respectively), this solution is accurate to 9 digits for values of z above 5 and 16 digits for z above 20. Smaller values of z are less accurate, but the simple translation of ${\displaystyle \Gamma (z)=\Gamma (z+1)/z}$ may be used to easily translate higher, more accurate values to lower values when needed. Less accurate needs may also be accommodated by simply using fewer terms of the infinite series.


Stirling's Asymptotic Series for ${\displaystyle \Gamma (z)}$ using the first 8 terms:


${\displaystyle \Gamma (z)={\sqrt {2\pi /z}}{\Bigl (}z/e{\Big )}^{z}{\Bigl (}1+{\frac {1}{12z}}+{\frac {1}{288z^{2}}}-{\frac {139}{51840z^{3}}}-{\frac {571}{2488320z^{4}}}+{\frac {163879}{209018880z^{5}}}+{\frac {5246819}{75246796800z^{6}}}-{\frac {534703531}{902961561600z^{7}}}-...{\Bigl )}}$