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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.
one of the Gegenbauer functions in analytic number theory (may be replaced by the capital form of the Latin letter P). represents: one of the Gegenbauer functions in analytic number theory. the Dickman–de Bruijn function; the radius in a polar, cylindrical, or spherical coordinate system; the correlation coefficient in statistics
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.
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 ...
Plot of the Barnes G aka double gamma function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D The Barnes G function along part of the real axis. In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers.
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.
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) .
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 ...