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Physics 401  Homework #2
1)
Two forms of Fourier Series (three points).
If f(x) is periodic, with period 2L, and square
integrable between (L,L), then we can represent it as a linear combination of the
functions
L
x
in
/
exp(
(a Fourier Series):
L
x
in
n
n
n
e
c
x
f
/
)
(
for some set of coefficients {c
n
}. If f(x) is real, then we can also represent the function this way:
1
0
sin
cos
2
)
(
n
n
n
L
x
n
b
L
x
n
a
a
x
f
where the relationship between the two forms is:
0
0
)
(
)
(
2
)
(
c
a
c
c
i
b
c
c
a
n
n
n
n
n
n
Show that these two forms are equivalent by substituting for a
n
, b
n
, and a
0
in the second form and
recovering the first form. Hint: use Euler's formula to convert sines and cosines into exponentials.
2)
Fourier Series of a square wave.
We can write a square wave so it's an odd function:
L
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 Fall '11
 HALL
 Work, Quantum Physics

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