GAMMA FUNCTION

generalization of the factorial function to nonintegral values. (The factorial is written as !, with n! defined as the product 1 2 3 . . . n). If a graph is drawn (see graph) of the function y = x! when x = 0, 1, 2, 3, . . . etc., the points can be joined by a curve, and the gamma function gives a precise way of defining and calculating these intermediate points on the curve. The gamma function G(z) can be defined as the value that is approached by the quotient n!nz/z(z + 1)(z + 2) . . . (z + n) as n gets larger and larger, and is equivalent to its definition as a type of infinite sum given by an integral. For z = 1, this integral equals 1, and an operation with integrals known as integration by parts gives the recursion relation G(z + 1) = zG(z). Using these two facts together, G(2) = 1G(1) = 1, G(3) = 2G(2) = 2, etc., can be calculated, giving the result that G(n) = (n - 1)! when n is an integer. The gamma function is useful more for its relationship to other functions than as a solution by itself of some problem. It arises in simplifying the evaluation of some infinite or improper integrals and in the solution of differential and difference equations arising in probability theory, statistics, mathematical physics, and engineering mathematics.