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The gamma function is an important special function in mathematics.Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general.
Print/export Download as PDF ... Special functions and miscellaneous topics related to the Gamma function ... Particular values of the gamma function;
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles. The gamma function has no zeros, so the reciprocal gamma function 1 / Γ(z) is an entire function.
Repeated application of the recurrence relation for the lower incomplete gamma function leads to the power series expansion: [2] (,) = = (+) (+) = = (+ +). Given the rapid growth in absolute value of Γ(z + k) when k → ∞, and the fact that the reciprocal of Γ(z) is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all ...
This has led to much research and generalization. In particular there is an analog of the Chowla–Selberg formula for p-adic numbers, involving a p-adic gamma function, called the Gross–Koblitz formula. The Chowla–Selberg formula gives a formula for a finite product of values of the eta functions.
Thus the -gamma function can be considered as an extension of the -factorial function to the real numbers. The relation to the ordinary gamma function is made explicit in the limit = (). There is a simple proof of this limit by Gosper.
Download as PDF; Printable version; In other projects ... is the standard normal probability density function, ... Owen, D. (1980). "A table of normal integrals".
Thus computing the gamma function becomes a matter of evaluating only a small number of elementary functions and multiplying by stored constants. The Lanczos approximation was popularized by Numerical Recipes , according to which computing the gamma function becomes "not much more difficult than other built-in functions that we take for granted ...