1. There are many situations in science where you have a periodic function and you
would like to express it as a sum of sines of different frequencies and amplitudes using a
method called fourier transform. For example, if you’re analyzing the feedback from an
MRI, you may want to break up the electromagnetic wave into ones that are easier to
analyze or if you want to make an electric keyboard that sounds like a violin, you may
want to break up the sound wave of a violin into a sum of simple waves you can replicate
with a computer.
If f (x) is the periodic function you’d like to decompose into a fourier transform, assume
f (x) can be written as
f (x) =
Show that the
∞
X
an sin(nx)
n=1
mth coefficient is
am
= a1 sin(x) + a2 sin(2x) + a3 sin(3x) + . . .
given by
Z
1 π
f (x) sin(mx)dx
=
π −π
Some hints: Take for granted that you can break up the integral of an infinite sum into
an infinite sum of integrals. i.e. assume
Z
Z
Z
Z
g1 (x) + g2 (x) + g3 (x) + · · · = g1 (x) + g2 (x) + g3 (x) + . . .
You may also want to look at problem 68 from section 7.2 of the book for inspiration,
but make sure you show the result from problem 68 if you’re going to use it.
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