1.10 Complex Methods 115
Fourier integral
The Fourier integral of a functionf(x)defined in the entire interval−∞<
x<∞can also be cast in complex form:
f(x)=∫∞
−∞C(λ)eiλxdλ. (5)The complex Fourier integral coefficient function is given by
C(λ)= 21 π∫∞
−∞f(x)e−iλxdx. (6)It is simple to show that
C(λ)=^1
2(
A(λ)−iB(λ))
, (7)
whereAandBare the usual Fourier integral coefficients. The complex Fourier
integral coefficient is often called theFourier transformof the functionf(x).
Example.
Find the complex Fourier integral representation of
f(x)={
1 , −a<x<a,
0 , x<|a|.The coefficient function (or transform) offis
C(λ)= 21 π∫a−ae−iλxdx= 21 πe−iλx
−iλ∣∣
∣∣
a
−a= 21 πeiλa−e−iλa
iλ =sin(λa)
πλ.The representation offis
f(x)=∫∞
−∞sin(λa)
πλeiλxdλ, −∞<x<∞.Of course, atx=±a,theintegralconvergesto1/2.
This example brings out a fact about symmetry: Iff(x)is even,C(λ)is real;
iff(x)is odd,C(λ)is imaginary.
The Fourier integral or transform may be used to solve differential equations
on the interval−∞<x<∞, in much the same way that Laplace transform
is used.