18.03 Lecture #22 Oct. 28, 2009: draft notes
I mentioned explicitly that there was an error in the version of PS 6 posted from noon to
evening on Sunday (and then corrected): that the exponential in the formula for a complex Fourier
coefficient
c
n
should be
e
−
int
(not
int
). I also explicitly corrected the formula for the Fourier series
of the square wave function (which appears correctly in the notes for Lecture 22, but had been
messed up a bit at the end of the lecture).
I’m out of time to write detailed notes, so here is a more telegraphic summary of the lecture.
Possibly I’ll fill in more details later.
Topic for today was meant to be (first) behavior of Fourier series when the function changes,
and (second) a little more about resonance in differential equations. In fact I spoke only about how
to compute the Fourier series of the derivative of
f
. The final theorem was
Theorem.
Suppose that the function
f
has the following properties:
(1)
f
is
2
π
periodic;
(2)
f
is continuous, and has a derivative at all but a finite number of points between

π
and
π
;
(3) the derivative has only “jump discontinuities” (as described in section 8.1 of EP).
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 Fall '09
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 Fourier Series

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