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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.
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.
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When k = 1 the standard Pochhammer symbol and gamma function are obtained. Díaz and Pariguan use these definitions to demonstrate a number of properties of the hypergeometric function . Although Díaz and Pariguan restrict these symbols to k > 0, the Pochhammer k -symbol as they define it is well-defined for all real k, and for negative k ...
Ackermann function: in the theory of computation, a computable function that is not primitive recursive. Dirac delta function : everywhere zero except for x = 0; total integral is 1. Not a function but a distribution , but sometimes informally referred to as a function, particularly by physicists and engineers.
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 ...
In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. [1] The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. [2]
In q-analog theory, the -gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by Jackson (1905) .