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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.
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 ...
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. In mathematics, the multiple gamma function is a generalization of the Euler gamma function and the Barnes G-function.
Graphs of the polygamma functions ψ, ψ (1), ψ (2) and ψ (3) of real arguments Plot of the digamma function, the first polygamma function, in the complex plane from −2−2i to 2+2i with colors created by Mathematica's function ComplexPlot3D showing one cycle of phase shift around each pole and the zero
In mathematics, the elliptic gamma function is a generalization of the q-gamma function, which is itself the q-analog of the ordinary gamma function. It is closely related to a function studied by Jackson (1905) , and can be expressed in terms of the triple gamma 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.
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.
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 ...