Fourier Series January 26, 2009
ME 501B – Engineering Analysis 1
Fourier Series
Fourier Series
Larry Caretto
Mechanical Engineering 501B
Seminar in Engineering Analysis
January 26, 2009
2
Outline
• Review last class
• Fourier series as expansions in periodic
functions
– Comparison to eigenfunction expansions
• Odd and even functions
• Periodic extensions of non-periodic
functions
• Complex Fourier series
3
Review Last Lecture
• Discussed Sturm-Liouville Problem
• Solutions are a set of orthogonal
eigenfunctions, ym(x) that can be used to
express other functions, f(x)
∑
∞
=
=
0
)()(
mmm xyaxf
()
∫
∫
== b
a
mm
b
a
m
mm
m
m
dxxyxyxp
dxxfxyxp
yy
fy
a
)()()(
)()()(
,
),(
• p(x) is
weight
function
• Have to
compute
am
4
Review Orthogonal Functions
• Defined in terms of inner product
• Norm of two like eigenfunctions ||yi||
()
iji
b
a
jiji ydxxpxyxyyy
δ
2
*)()()(, == ∫
()
ij
b
a
jiji dxxpxfxfff
δ
== ∫)()()(, *
• Convert orthogonal eigenfunctions to
orthonormal eigenfunctions
• Orthonormal eigenfunctions
i
i
iy
y
f=
5
Review Sturm-Liouville
General equation whose solutions provide
orthogonal eigenfunctions
– Defined for a ≤x ≤b with p(x), q(x) and r(x)
continuous and p(x) > 0
– (Homogenous) differential equation and
boundary conditions shown below
[]
0)()(
)(
=+
+
⎟
⎠
⎞
⎜
⎝
⎛
yxpxq
dx
dy
xr
dx
d
λ
0)(
0)(
21
21
=+
=+
=
=
bx
ax
dx
dy
by
dx
dy
kayk
ll
6
Review Sturm-Liouville Results
• Eigenvalues are real
• Eigenfunctions defined over a region a ≤
x ≤b form an orthogonal set over that
region.
• Eigenfunctions form a complete set over
an infinite-dimensional vector space
• We can expand any function over the
region in which the Sturm-Liouville
problem is defined in terms of the
eigenfunctions for that problem
