E. Kreyszig, Advanced Engineering Mathematics, 9th ed. Section 11.4, pgs. 496-499
Lecture: Complex Fourier Series Module: 10
Suggested Problem Set: {2, 9, 11 }March 4, 2009
Quote of Lecture 10
Juliet: What’s in a name? That which we call a rose by any other name would smell as
sweet.
Shakespeare : Romeo and Juliet ( 1591)
1. Review
So, at this point we have the following,
f(x) = a0+
∞
X
n=1
ancos “nπ
Lx”+bnsin “nπ
Lx”,(1)
a0=1
2LZL
−L
f(x)dx,(2)
an=1
LZL
−L
f(x) cos “nπ
Lx”dx,(3)
bn=1
LZL
−L
f(x) sin “nπ
Lx”dx,(4)
which defines the Fourier series and it’s associated coefficients for a 2L-periodic function, where Lis a scaling
parameter introduced to control the length of the period. We also have the following important results:
•Any function for, which the integrals (2)-(4) are defined has a Fourier series representation. Notice
that this does not require the function to be periodic, but the Fourier series will induce this function
to be periodic with principle domain (-L,L).
•The Fourier series may actually differ from the function f(x) at a countably infinite amount of
points. We can know where this might occur by knowing the jump-discontinuities of fand we have
that the Fourier series will average the right and left hand limits at these points.
•The Fourier series represents the function fin terms of it’s oscillatory features for which the data
fsupplies the amplitudes for each oscillatory mode. 1
•The Fourier series represents the function fin terms of it’s even components and odd components.
2
2. Lecture Overview
Now we are going to make use of the well celebrated Euler’s formula,
eiθ = cos(θ) + isin(θ), i =√−1,(5)
so that we can rewrite (1)-(4) in its complex form,
f(x) =
∞
X
n=−∞
cne−inπ
Lx,(6)
cn=1
2LZL
−L
f(x)einπ
Lxdx,(7)
1Each term in the series is called a Fourier mode and the lowest order term is often called the Fundamental mode.
2If the function fhas symmetry then the equations (1)-(4) simplify according to the intal properties of symmetric
functions.
1
