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PROFESSOR: Hi everyone.

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Welcome back.

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So today I'd like
to look at a problem

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on manipulating Fourier series.

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Specifically, we're
asked to find the Fourier

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series of cosine 2t minus pi/4.

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And then the second problem
is given a square wave

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function, which takes on
the value of minus 1 and 1,

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and it's 2pi periodic.

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We're also told that the square
wave function has the Fourier

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series 4/pi, 1/n sine n*t.

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And the question is to find the
Fourier series of three related

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functions.

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So the first function takes
on the value of 0 and 4

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on the intervals minus
pi to 0 and 0 to pi.

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Note that this function
is also 2pi periodic.

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The second problem it is
to find the Fourier series

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of a function which is
minus 1 from minus 1 to 0,

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and 1 from 0 to 1.

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So this is the
square wave function,

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but now has period two.

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And then lastly
we're asked to find

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the Fourier series of a function
f of t, which is absolute t.

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And it's defined as
absolute t on the interval

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from minus pi to pi.

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However this function
f is going to be

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extended on the entire
domain with period 2pi.

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So I'll let you think
about this problem

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and I'll be back in a moment.

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Hi everyone.

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Welcome back.

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So let's take a look
at the first problem,

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finding the Fourier series
of cosine 2t minus pi/4.

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So in this class when we say
find the Fourier series, what

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we're really looking for are
coefficients a_0, a_n and b_n,

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such that, for example, we
can write our function out

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as a sum of sine and cosines
with these values a_n and b_n.

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Now, what we could do is we
could take this function,

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plug it into the
integral formulas

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and compute, using integrals,
the value a_0, a_n, and b_n.

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However, there's
a bit of a trick.

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And that's to realize that this
function, cosine 2t minus pi/4,

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is really just a
sinusoidal function

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with a single frequency.

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And it's currently in
an amplitude-phase form.

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And we can always convert
an amplitude-phase form

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of a sinusoidal function into
some amplitude times a sine

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and some amplitude times a
cosine and add the two up.

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So specifically,
this can be done

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by recalling the
cosine identity,

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cosine A minus B is
cosine A times cosine

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B plus sine of A sine of B.

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And if we identify A
with 2t and B with pi/4,

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we obtain that f of
t is cosine 2t cosine

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pi/4 plus sine 2t sine of pi/4.

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And now sine pi/4 and cosine
pi/4 are both 1 over root 2.

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So f of t becomes 1
over root 2 cosine 2t

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plus 1 over root 2 sine 2t.

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And that's it.

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So when we take a look at
this expression for f of t,

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we see that this is actually
in exactly the same form

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that we were seeking in the
beginning, the Fourier series

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representation for f.

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So this is the end
of the problem.

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We don't have to actually
compute any integrals.

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So now let's take
a look at part two.

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So for part two, we're asked
to find the Fourier series

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of several different functions.

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And I'll just take
a look at part one.

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So in this case, f of t
takes on the value of 0

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from minus pi to zero, and
the value of 4 from 0 to pi.

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And if I were to quickly
sketch this function,

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it would look like a square
wave, 0, pi, minus pi.

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This is f of t.

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And we see that it's actually
related-- I can draw in dashes

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just to be clear-- what
the original square wave

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function might look like.

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So this is square wave of t.

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We see that if we were to
take the square wave function

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and shift it up, the square
wave takes on values of plus 1

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and minus 1.

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But if we take the
square wave function

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and shift it up 1 unit,
and then rescale it,

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we'll then get the function
we're looking for, f of t.

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So specifically, we can
write f of t in terms

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of the square wave as well.

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We have to shift the
square wave function up 1,

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and then we have to amplify
it by a factor of 2.

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So then when we seek the Fourier
series for f, all we have to do

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is substitute the Fourier series
for the square wave function

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in.

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So let's do this.

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So we have 1 plus 4/pi.

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So I'm writing here this Fourier
series for the square wave

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function, n odd, 1/n sine n*t.

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So by manipulating the Fourier
series for the square wave

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function, we can arrive at
a Fourier series for f of t,

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n odd, 1/n sine n*t.

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So this is the first function.

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The second function f
of t-- I'll just rewrite

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it again-- f of t takes
on the value of negative 1

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from negative 1 to 0, and
takes on the value of plus 1

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from 0 to 1.

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So it's somewhat useful to
draw, just to quickly sketch,

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this function.

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And I'll draw in my axes here.

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So this is f of t.

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This is t.

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And it transitions.

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It makes its jumps
at the integers.

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So for example, it jumps
from plus 1 to minus 1 at 1,

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and from plus 1 to minus
1 at minus 1 as well.

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And if I were to just quickly
sketch the square wave function

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underneath of it, the
square wave function

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makes its transitions
at minus pi and pi.

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So this is the
square wave function.

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This is t.

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So we see that we can take
the square wave function

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and squeeze it to obtain
this function f of t.

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And the question
now is, what factor

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do we have to squeeze the square
wave function by to get f of t?

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Well, usually what
I like to do is

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take one characteristic
point in the sketch

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for in this case the
square wave function.

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So a nice
characteristic point is

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when the square wave
function has a 0,

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so when it goes
through the origin,

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and compare it to the same
point on this function f of t.

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So we see that the
point pi has to get

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stretched to the point 1.

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So we basically need to take
the time axis in the square wave

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function and squeeze
it by a factor of pi.

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So what this means
is that f of t

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is going to be the square wave.

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And when we do what
mathematicians sometimes

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called a dilatation, a
stretch or a squeeze.

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If we're squeezing
by some factor,

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that factor always appears in
the argument of the function

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we're squeezing.

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If it's a stretch, then it would
be t over the stretch factor.

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And we can quickly check
here that if t equals 1,

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so as t goes from 0
to 1, so when t is 1,

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the square wave function
is being evaluated at pi.

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And I'll now write out
the Fourier series.

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So this is going to be 4/pi, sum
of n odd, 1/n sine of n*pi*t.

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And there's another
quick check we can do.

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And that's this
function f we were told

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is periodic with period 2.

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And if we just take a look
at this function here,

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the Fourier series
representation of it,

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we see that each sine n*pi*t is
in fact periodic with period 2.

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So that's just another quick,
back of the envelope check

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that we've done our
calculation correctly.

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So this concludes part two.

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And then lastly than
for part three, f of t

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is the absolute value of
t from minus pi to pi.

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And it's extended
beyond minus pi to pi

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to make it 2pi periodic.

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And at first it might look
like this function f of t

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is not related to the
square wave function.

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But if we look at it a
little more carefully,

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and analyze the
derivative of f of t,

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we see that this function
takes on the value plus 1

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on the interval of 0 to pi and
takes on the value of minus 1

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on the interval
of minus pi to 0.

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And this is exactly
the square wave.

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So what we can do
is we can write

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f of t-- it's going to be an
integral of the square wave

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function.

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So specifically, we have
the integral of square wave,

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which is going to be the
integral of 4/pi 1/n sine n*t,

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summation n odd.

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And I'm going to leave this
as an indefinite integral.

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So when we integrate, we have
a constant of integration

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c plus 4/pi sum of 1
over now n squared.

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And we have negative cosine n*t.

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And this is n odd again.

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And now we have one
last question to answer.

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What is the constant
of integration?

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Well, notice how the
constant of integration

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can be identified with the
a_0 term of a Fourier series.

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So if we want the
constant of integration,

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we can just
calculate it directly

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by the integral formula for
the a_0 of this function.

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So what this means is c
is going to be 1/(2pi)--

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essentially it's just the
average of the function--

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but it's pi to pi, the absolute
value of t, which is 1/pi,

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pi squared over 2.

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So we get pi/2.

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So just to quickly
recap, in this problem

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we were given several
functions which were related

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to the square wave function.

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And we started off
knowing what the Fourier

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series for the square
wave function was.

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And by manipulating or
rewriting these functions

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in terms of the
square wave function,

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we were able to compute
their Fourier series.

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And we were able to
do this essentially

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without having to
evaluate any integrals.

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OK.

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So I hope you
enjoyed this problem,

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and I'll see you next time.