Fourier Integral

a formula for the decomposition of a nonperiodic function into harmonic components whose frequencies range over a continuous set of values.

If a function f(x) satisfies the Dirichlet condition on every finite interval and if the integral

converges, then

The formula was first introduced in 1811 by J. Fourier in connection with the solution of certain heat conduction problems but was proved later by other mathematicians.

Formula (1) can also be given in the form

where

In particular, for even functions

where

Formula (2) may be viewed as the limiting form of the Fourier series for functions with period 2T as T → ∞. Then, a(u) and b(u) are analogues of the Fourier coefficients of f(x).

Using complex numbers, we can replace formula (1) with

Formula (1) can also be written as

(simple Fourier integral).

If the integrals in formulas (2) and (3) diverge (seeIMPROPER INTEGRALS), then, in many cases, they nevertheless converge to f(x) if we use appropriate summability methods. The solution of many problems involves the use of Fourier integrals of functions of two and more variables.