Gammafüggvényre

The Gamma function, often denoted by the Greek letter Gamma followed by a variable like x, is a fundamental mathematical function with broad applications in various fields of mathematics, physics, and engineering. It is essentially an extension of the factorial function to complex and real numbers. For positive integers n, the Gamma function is defined as Gamma(n) = (n-1)!. This means Gamma(1) = 0! = 1, Gamma(2) = 1! = 1, Gamma(3) = 2! = 2, and so on.

However, the true power of the Gamma function lies in its ability to be defined for non-integer values. The most common definition for complex numbers z with a positive real part is given by an integral: Gamma(z) = integral from 0 to infinity of t^(z-1) * e^(-t) dt. This integral converges for Re(z) > 0. The Gamma function possesses an important property known as the recurrence relation: Gamma(z+1) = z * Gamma(z). This relation is crucial for extending the definition of the Gamma function to the entire complex plane, except for non-positive integers where it has simple poles.

The Gamma function is closely related to other important mathematical concepts, including the Beta function and various probability distributions like the Gamma distribution, which is named after it. Its analytical properties, such as its behavior at its poles and its relation to the sine function through Euler's reflection formula, make it a cornerstone in the study of special functions and advanced calculus.