Mathematical Methods of Physics – Winter 2010-11
Week 7 – Tuesday 15 Feb. 2011
Dr. E.J. Zita
Egyptians’ liberation day: 11 Feb. 2011. Next: Iran? Yemen? Bahrain?
Ch.7: Fourier Series and transforms
Review: Two wave speeds for y(x,t) = sin(kx – wt)
v = ω/k
v = dy/dt
1.0
0.5
1
2
3
4
5
6
-0.5
-1.0
Fourier Series: useful for PERIODIC functions – expand in sin(x) and cox(x) instead of xn:
f(x) = ½ a0 + a1 cos(x) + a2 cos(2x) + a3 cos(3x) + a4 cos(4x) + … + an cos(nx)
+ b1 sin(x) + b2 sin(2x) + b3 sin(3x) + b4 sin(4x) + … + bn sin(nx)
a0 =
1
π
π
∫
−π
f ( x) dx an =
1
π
π
∫
−π
f ( x) cos nx dx bn =
1
π
π
∫
f ( x)sin nx dx
−π
⎧0, − π < x < 0
Ex: square wave – a classic in electronics: ON – off – ON – off: f ( x) = ⎨
⎩ 1, 0 < x < π
1 2 ⎛ sin x sin 3 x sin 5 x
sin mx ⎞
f ( x) = + ⎜
+
+
+ ⋅⋅⋅
⎟ , m = odd
m ⎠
2 π⎝ 1
3
5
•
Problems Sec.5, p.354: choose (1,2,3), DO (8, 11, 12, 13) (Draw together)
MMP2011/lectures/Boas7Fourier
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Complex Form of Fourier Series – very useful generalization
f ( x) = c0 + c1eix + c−1e− ix + c2e2ix + c−2e −2ix + ⋅ ⋅ ⋅ =
1
c0 =
2π
∞
∑ce
n =−∞
inx
n
π
1
cn =
2π
∫π f ( x) dx
−
π
∫π f ( x) e
− inx
dx
−
CONTINUING…
7.11 Parseval’s Theorem or Completeness relation between average of f(x)2 and coefficients.
Average of [ f ( x) ] (over a period) =
2
( 12 a0 )
2
∞
∞
∞
1
−∞
+ 12 ∑ an2 + 12 ∑ bn2 = ∑ cn2
1
The set of all cos(nx) and sin(nx), or equivalently all e
period of f(x).
± inx
, is a complete basis set over the
You can sometimes use Parseval’s theorem to find sums of infinite series.
If you can find the average of a function2 over its period, you have the sum of its FS coefficients.
Ex: You can show that f(x) = x on the interval (-1,1) is represented by the Fourier series
f ( x) =
=
−i ⎛ iπ x
1
1
1
1
⎞
− iπ x
+ ei 2π x − e −2iπ x + e3iπ x − e −3iπ x + ... ⎟
⎜e − e
2
2
3
3
π⎝
⎠
−i
(c e
π
1
iπ x
+ c−1e − iπ x + c2ei 2π x + c−2e −2iπ x + c3e3iπ x + c−3e−3iπ x + ...)
1
1
1 2
1 ⎡ x3 ⎤
1
The average of [ f ( x) ] on (-1,1) is ∫ x dx = ⎢ ⎥ =
2 −1
2 ⎣ 3 ⎦ −1 3
2
∞
The average of [ f ( x) ] = ∑ cn2 =
2
−∞
1 ⎛
1 1 1 1
⎞ 2 ∞ 1 1
+
+
+
+
+
+
1
1
...
⎟= 2∑ 2 =
π 2 ⎜⎝
4 4 9 9
3
⎠ π 1 n
∞
Therefore the sum of the series is
1
∑n
1
2
=
1π2 π2
=
3 2
6
This is a pretty involved way to get a series sum, but sometimes the best way.
Application of Fourier Series
Electromagnetism Ch.3 (EM3.ppt), Worksheet for Ex. 3.3 and Prob.12 (EM3_12.pdf)
Solve Laplace’s equation in 2D Cartesian using separation of variables,
then solve for V in waveguide using Fourier Series
MMP2011/lectures/Boas7Fourier
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Boas 7.12 Fourier Transforms – the piece de resistance
Fourier series are useful for writing periodic, analytic functions in terms of sines and cosines.
What if you have a function (or signal) that is not analytic (e.g. data), and not periodic?
You can still find out what frequencies make up the signal, using Fourier Transforms, and
express the signal in terms of an integral.
Start with the Fourier series in complex form, for a function f(x) with period 2 A :
∞
∞
inπ x
− inπ x
1
A
f ( x) e
dx .
Recall that f ( x) = ∑ cn e A , where the coefficients are cn =
∫
2π −∞
n =−∞
∞
Similarly, f ( x) =
∫
g (α ) eiα x dx has Fourier Transform g (α ) =
−∞
1
2π
∞
∫
f ( x) e − iα x dx ,
−∞
and f(x) is the Fourier Transform of g(α) as well.
∞
Parseval’s Thm. for FT:
∫
2
g (α ) dα =
−∞
1
2π
∞
∫
2
f ( x) dx
−∞
You can use Fourier transforms to evaluate some integrals. See Ex. 1 & 2:
⎧1, −1 < x < 1
Ex.1: If f(x) is the even square pulse: f ( x) = ⎨
⎩ 0, | x |> 1
then g (α ) =
sin(α )
πα
∞
and the inverse transform is f ( x) =
∫
−∞
sin(α )
πα
iα x
e dx =
∞
2 sin α cos α x
π
∫
0
α
dα .
Ex.2: Using the original definition of f(x) as the square pulse, we can evaluate the integral:
∞
∫
0
sin α cos α x
α
⎧ 1 for x < 1
∞
π ⎪⎪ 1
sin α
π
dα =
dα = f ( x ) = * ⎨
for x = 1 , and for x=0, cos=1 → ∫
2
2
2 ⎪
α
2
0
π
⎪⎩ 0 for x > 1
Try some examples from HW: 3-12
MMP2011/lectures/Boas7Fourier
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Laplace transforms Boas Ch.8.8 preview (p.437)
Like Fourier transforms, these are another example of integral transforms of functions.
What was the purpose of the Fourier Transform?
The purpose of the Laplace Transform is to convert a function(time) → Function(period).
In that sense, it’s also quite similar to the Fourier series/transforms, which tells us what waves
(either frequencies or wavenumbers) make up some periodic or nonperiodic function.
∞
L( f ) = ∫ f (t ) e − pt dt = F ( p )
0
Laplace transforms can also be useful in solving differential equations, which is the context in
which they are discussed in Blanchard, Devaney, and Hall. We’ll return to that after we have
learned the basic technique.
Ex.1 and Ex.2 start out generating a series of solutions for Ln
1
1
For f1(t) = 1, L1 ( f ) = F1 ( p) = ; for f 2 (t ) = e − at , L2 ( f ) = F2 ( p) =
p
p+a
Instead of solving n integrals to find Ln, note that:
Laplace transform of a sum = sum of Laplace transforms
and
transform of c f(t) = c* L(f) (when c = constant)
that is, Laplace transforms are linear.
First calculate L3 directly: for f3 (t ) = sin α t =
∞
eiat − e− iat
(from Table, p.469)
2i
∞
eiat − e− iat − pt
e dt =
i
2
0
L3 ( f ) = F3 ( p) = ∫ f (t ) e− pt dt = ∫
0
MMP2011/lectures/Boas7Fourier
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Two ways to find L3 (and L4): by direct calculation, and by Ex. 3.
Work through Ex. 3 together (below).
Then one team calculate L3, another calculate L4, using method of Ex.3
Replace (a) with (-ia) in f 2 (t ) = e − at , L2 ( f ) = F2 ( p) =
1
to get
p+a
f3 (t ) = eiat = cos α t + i sin α t , L3 ( f ) = F3 ( p) = ____
(8.6) Expand L(cos α t + i sin α t ) = L(cos α t ) + L(i sin α t ) = ___________ + ____________
Similarly, replace (a) with (+ia) in f 2 (t ) = e − at , L2 ( f ) = F2 ( p ) =
1
to get
p+a
f 4 (t ) = e −iat = cos α t − i sin α t , L4 ( f ) = F4 ( p ) = ____
(8.7) Expand L(cos α t − i sin α t ) = L(cos α t ) − L(i sin α t ) = ___________ + ____________
Add (8.6) and (8.7) to get L4, add to get L3.
Now we’ll be ahead of the game when we start solving DiffEqs with Laplace transforms,
next week.
MMP2011/lectures/Boas7Fourier
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