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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.
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.
the Pi function, i.e. the Gamma function when offset to coincide with the factorial; the complete elliptic integral of the third kind; the fundamental groupoid; osmotic pressure; represents: Archimedes' constant (more commonly just called Pi), the ratio of a circle's circumference to its diameter; the prime-counting function
as the only positive function f , with domain on the interval x > 0, that simultaneously has the following three properties: f (1) = 1, and f (x + 1) = x f (x) for x > 0 and f is logarithmically convex. A treatment of this theorem is in Artin's book The Gamma Function, [4] which has been reprinted by the AMS in a collection of Artin's writings.
In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers for which > by = +,as the only function defined on the half-plane := {: >} such that:
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 ]
He was awarded the Chauvenet Prize for mathematical writing in 1963 for an article on the gamma function, [4] and won numerous other prizes, including being chosen to deliver the 1991 Hendrick Lectures of the MAA (which became the basis for his book Spirals: From Theodorus to Chaos). He was a frequent invited lecturer and authored several books.
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 ...