1.11 Applications of Fourier Series and Integrals 117
- Find the functionf(x)whose complex Fourier coefficient function is
given.
a. C(λ)={ 1 , − 1 <λ<1,
0 , otherwise;
b. C(λ)=e−|λ|.- Show that the complex Fourier coefficient off(x)=e−x^2 is
C(λ)=e−λ^2 / 4
2√
π
Use a change of variable in the exponent. You need to know that
∫∞−∞e−z^2 dz=√
π.1.11 Applications of Fourier Series and Integrals
Fourier series and integrals are among the most basic tools of applied mathe-
matics. In what follows, we give just a few applications that do not fall within
the scope of the rest of this book.
A. Nonhomogeneous Differential Equation
Many mechanical and electrical systems may be described by the differential
equation
y ̈+α ̇y+βy=f(t).The functionf(t)is called the “forcing function,”βythe “restoring term,” and
αy ̇the “damping term.” It is known (see Section 0.2) that: (1) a sine or cosine
inf(t)will cause functions of the same period iny(t);(2)iff(t)is broken
down as a sum of simpler functions,y(t)canbebrokendowninthesameway.
Suppose thatf(t)is periodic with period 2π, and let its Fourier series be
f(t)=a 0 +∑∞
n= 1ancos(nt)+bnsin(nt).Then a particular solutiony(t)will be periodic with period 2π; it and its deriv-
atives have Fourier series
y(t)=A 0 +∑∞
n= 1Ancos(nt)+Bnsin(nt),