∫x0 e−x2 dx
called the “error-function integral,” and denoted by “Erf x.” All these functions have been tabulated (see [Tables, Mathematical]).
51. Eulerian integrals.New functions can be introduced also by means of the definite integrals of functions of two or more variables with respect to one of the variables, the limits of integration being fixed. Prominent among such functions are the Beta and Gamma functions expressed by the equations
B(l, m) = ∫10 xl−1 (1 − x)m−1 dx,
Γ(n) = ∫∞0 e−t tn−1 dt.
When n is a positive integer Γ(n + 1) = n!. The Beta function (or “Eulerian integral of the first kind”) is expressible in terms of Gamma functions (or “Eulerian integrals of the second kind”) by the formula
B(l, m) · Γ(l + m) = Γ(l) · Γ(m).
The Gamma function satisfies the difference equation
Γ(x + 1) = x Γ(x),
and also the equation