Complex Fourier Series Worksheet

Complex Fourier Series Page 1
Recall, Euler’s Formula:
e iθ = cos θ + i sin θ e −iθ = cos θ − i sin θ
1. Use Euler’s Formulas above to verify the following definitions:
cos θ = eiθ + e −iθ
2
2. The Fourier Series for a 2π−periodic function is given by
and sin θ = eiθ − e −iθ
f(x) = a0 +
∞
an cos nx + bn sin nx
Use the definitions from #1 to rewrite the Fourier Series using e inx and e −inx .
Distribute and collect all the e inx and e −inx terms.
n=1
What is the coefficient of the e inx term? [Rationalize it if necessary.]
What is the coefficient of the e −inx term? [Rationalize it if necessary.]
How are these two coefficients related? i.e. What do we call them?
2i

Complex Fourier Series Page 2
Let
c0 = a0
cn = coefficient of e inx
cn = coefficient of e −inx cn denotes complex conjugate
Then if you rewrite the Fourier Series in terms using complex exponentials and these coefficients, you should get
f(x) = c0 +
Clearly define c0, cn, and cn in terms of an and bn.
∞
n=1
cne inx + cn e −inx
3. Let cn be denoted by c−n since it corresponds to the coefficients preceding e−inx (i.e. term involving −n).
Now
∞
f(x) = c0 + cne inx + c−n e −inx
n=1
which can be condensed into the single term series below
[Fill in the bounds – it may be helpful to write out a few of the terms in the series.]
This form is called the Complex Fourier Series for f(x).
f(x) =
cne inx
n=

Complex Fourier Series Page 3
4. Homework: Use the relationship cn = an − ibn
2
for the complex Fourier coefficients:
cn = 1
2π
π
−π
and the integrals for an and bn to derive the following formula
f(x)e −inx dx which is valid for all integers n (e.g. negative, positive, and zero).
5. Find the Complex Fourier Coefficients for f(x) =
1 0 < x < π
−1 −π < x < 0

Complex Fourier Series Page 4
6. Given f(x) = e αx on −π < x < π.
(a). Find the Complex Fourier Coefficients for
(b). Find an and bn from part (a) using the relationship cn = an − ibn
2
(found in #2)