Fourier Series and Partial Differential Equations Lecture Notes

Chapter 2. Fourier series 20
Formally, we define g(X) = f(x) = f(LX/π) so that
L(X +2π) LX
g(X +2π) = f = f
π π +2L
= f
LX
= g(X), (2.60)
π
and g is 2π-periodic. Hence the previous theory holds for g, i.e. if we can write
then
and
So if we can write
then (2.61) holds, so
g(X) = 1
2 a0 +
∞
[ancos(nX)+bnsin(nX)], (2.61)
n=1
an = 1
π
π
= 1
π
−π
g(X)cos(nX)dX,
L
πx
nπx
π
g cos
−L L L L dx,
= 1
L
nπx
f(x)cos dx, (2.62)
L L
−L
bn = 1
π
g(X)sin(nX)dX,
π −π
= 1
L
πx
nπx
π
g sin
π −L L L L dx,
= 1
L
nπx
f(x)sin dx. (2.63)
L L
f(x) = 1
2 a0 +
an = 1
L
f(x)cos
L −L
∞
n=1
−L
ancos
nπx
+bnsin
L
nπx
dx, bn =
L
1
L
f(x)sin
L −L
nπx
, (2.64)
L
nπx
dx. (2.65)
L
The series in equation (2.64) is called the Fourier series for f and an and bn are the
Fourier coefficients of f. Again, we use ∼ if we do not know whether or not it converges.
By Theorem 2.2, under suitable conditions the series in equation (2.61) converges to
so we obtain
g(X+)+g(X−)
, (2.66)
2
Theorem 2.4 Let f beaperiodicfunction of period2L which is sufficiently well-behaved.
Then the Fourier series of f at x converges to
so equation (2.64) holds at any point where f is continuous.
f(x+)+f(x−)
, (2.67)
2