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The gamma function is related to Euler's beta function by the formula (,) = = () (+). The logarithmic derivative of the gamma function is called the digamma function ; higher derivatives are the polygamma functions .
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.
Euler's product formula for the gamma function, combined with the functional equation and an identity for the Euler–Mascheroni constant, yields the following expression for the digamma function, valid in the complex plane outside the negative integers (Abramowitz and Stegun 6.3.16): [1]
The Euler integral of the second kind is the gamma function [2] = For positive integers m and n , the two integrals can be expressed in terms of factorials and binomial coefficients : B ( n , m ) = ( n − 1 ) !
The notation γ appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time, perhaps because of the constant's connection to the gamma function. [3] For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835, [ 4 ] and Augustus De Morgan used it in a textbook published in parts ...
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 ...
A famous relationship is Euler's reflection formula () = (), for the gamma function (), due to Leonhard Euler. There is also a reflection formula for the general n-th order polygamma function ψ (n) (z),
The cumulative distribution function is the regularized gamma function: (; ... (where γ is the Euler–Mascheroni constant), and that for all > ...