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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 ...
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.
For example, the gamma function is a function that satisfies the functional equation (+) = and the initial value () = There are many functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically convex for x real and positive ( Bohr–Mollerup theorem ).
The Gamma function can be defined for any complex value in the plane if we evaluate the integral along the Hankel contour. The Hankel contour is especially useful for expressing the Gamma function for any complex value because the end points of the contour vanish, and thus allows the fundamental property of the Gamma function to be satisfied ...
The Kissing Number Problem. A broad category of problems in math are called the Sphere Packing Problems. They range from pure math to practical applications, generally putting math terminology to ...
The cumulative distribution function is the regularized gamma function ... the problem of determining an ... the gamma function, based on solving the ...
Gautschi's inequality is specific to a quotient of gamma functions evaluated at two real numbers having a small difference. However, there are extensions to other situations. If x {\displaystyle x} and y {\displaystyle y} are positive real numbers , then the convexity of ψ {\displaystyle \psi } leads to the inequality: [ 6 ]