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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.
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
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 ...
Plot of the hypergeometric function 2F1(a,b; c; z) with a=2 and b=3 and c=4 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D. In mathematics, the Gaussian or ordinary hypergeometric function 2 F 1 (a,b;c;z) is a special function represented by the hypergeometric series, that includes many ...
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
the gamma function, a generalization of the factorial [14] the upper incomplete gamma function; the modular group, the group of fractional linear transformations; the gamma distribution, a continuous probability distribution defined using the gamma function [15] second-order sensitivity to price in mathematical finance
Plot of the Jacobi polynomial function (,) with = and = and = in the complex plane from to + with colors created with Mathematica 13.1 function ComplexPlot3D In mathematics , Jacobi polynomials (occasionally called hypergeometric polynomials ) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are a class of classical orthogonal ...
It is a multivariate generalization of the beta distribution, [1] hence its alternative name of multivariate beta distribution (MBD). [2] Dirichlet distributions are commonly used as prior distributions in Bayesian statistics , and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial ...