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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.
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 ...
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.
where is the gamma function. This form can be useful when combining a linear denominator A {\displaystyle A} with a quadratic denominator B {\displaystyle B} , such as in heavy quark effective theory (HQET).
Interpolated approximations and bounds are all of the form ~ () + (~ ()) where ~ is an interpolating function running monotonially from 0 at low α to 1 at high α, approximating an ideal, or exact, interpolator (): = () () For the simplest interpolating function considered, a first-order rational function ~ = + the tightest lower bound has ...
The reciprocal is sometimes used as a starting point for numerical computation of the gamma function, and a few software libraries provide it separately from the regular gamma function. Karl Weierstrass called the reciprocal gamma function the "factorielle" and used it in his development of the Weierstrass factorization theorem.
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) .
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.