1.9 Fourier Integral 111
f′(x)=∫∞
0[
−λA(λ)sin(λx)+λB(λ)cos(λx)]
dλ.Example 5.
Letf(x)=exp(−|x|)as in Example 3. Then its derivative is
f′(x)={
−e−x, 0 <x,
ex, x<0.Clearly,f(x)is continuous, and bothf(x)andf′(x)have Fourier integral rep-
resentations. The one forf(x)is in Example 3. Thus we have
f′(x)=∫∞
02
π−λ
1 +λ^2 sin(λx)dλ. EXERCISES
- Sketch the even and odd extensions of each of the following functions, and
find the Fourier cosine and sine integrals forf.Eachfunctionisgivenin
the interval 0<x<∞.
a. f(x)=e−x;
b. f(x)=
{ 1 , 0 <x<1,
0 , 1 <x;
c. f(x)={π−x, 0 <x<π,
0 ,π<x.- Find the Fourier integral representation of the following functionft(x).
This is sometimes called a “window” because it is “open” fort−h<x<
t+h.
ft(x)=
{
1 , |x−t|<h,
0 , |x−t|>h.- Find the Fourier integral representation of each of the following functions.
a.f(x)= 1 +^1 x 2 ; b.f(x)=sin(x)
x.
(Hint: To evaluate the Fourier integral coefficient functions, consult the
Fourier integral representations found in the examples.)- In Exercise 3b, the integral
∫∞
−∞|f(x)|dxis not finite. Nevertheless,A(λ)
andB(λ)do exist(B(λ)= 0 ). Find a rationale in the convergence theo-
rem for saying that this function can be represented by its Fourier integral.
(Hint: See Example 1.)- Find the Fourier integral representation of each of these functions: