Fourier Series and Partial Differential Equations Lecture Notes

Chapter 2. Fourier series 12
Notes. If f, f1 are odd functions and g, g1 are even functions then:
1. f(0) = 0 because f(0) = −f(−0) = −f(0);
y
2. for any α ∈ R, α
f(x)dx = 0; (2.17)
−α
3. for any α ∈ R, α α
g(x)dx = 2 g(x)dx; (2.18)
−α 0
4. the functions h(x) = f(x)g(x), h1(x) = f(x)f1(x) and h2(x) = g(x)g1(x) are odd,
even and even, respectively.
2.2 Fourier series for functions of period 2π
Let f be a function of period 2π. We would like to get an expansion for f of the form
f(x) = 1
2 a0 +
∞
[ancos(nx)+bnsin(nx)], (2.19)
n=1
where the an and bn are constants. Remember that sin(nx) and cos(nx) are periodic with
period 2π. We have two questions to answer.
Question 1: if equation (2.19) is true, can we find the an and bn in terms of f?
Question 2: with these an, bn, when, if ever, is equation (2.19) true?
2.2.1 Question 1
Suppose equation (2.19) is true and that we can integrate it term by term, i.e.
π
−π
f(x)dx = 1
2 a0
π
−π
Since π
−π
dx = 2π,
dx+
∞
n=1
an
π π
cos(nx)dx+bn
−π
π
cos(nx)dx = 0,
−π
−π
x
sin(nx)dx . (2.20)
π
sin(nx)dx = 0, (2.21)
−π