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Well, let's get started.

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The topic for today is --

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Sorry.
Thank you.

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For today and the next two
lectures, we are going to be

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studying Fourier series.
Today will be an introduction

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explaining what they are.
And, I calculate them,

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but I thought before we do that
I ought to least give a couple

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minutes oversight of why and
where we're going with them,

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and why they're coming into the
course at this place at all.

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So, the situation up to now is
that we've been trying to solve

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equations of the form y double
prime plus a y prime,

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constant coefficient
second-order equations,

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and the f of t was the input.
So, we are considering

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inhomogeneous equations.
This is the input.

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And so far, the response,
then, is the solution equals

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the corresponding solution,
y of t,

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maybe with some given initial
conditions to pick out a special

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one we call the response,
the response to that particular

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input.
And now, over the last few

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days, the inputs have been,
however, extremely special.

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For input, the basic input has
been an exponential,

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or sines and cosines.
And, the trouble is that we

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learn how to solve those.
But the point is that those

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seem extremely special.
Now, the point of Fourier

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series is to show you that they
are not as special as they look.

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The reason is that,
let's put it this way,

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that any reasonable f of t
which is periodic,

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it doesn't have to be even very
reasonable.

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It can be somewhat
discontinuous,

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although not terribly
discontinuous,

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which is periodic with period,
maybe not the minimal period,

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but some period two pi.
Of course, sine t

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and cosine t have the
exact period two pi,

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but if I change the frequency
to an integer frequency like

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sine 2t or sine 26 t,

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two pie would still be a
period, although would not be

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the period.
The period would be shorter.

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The point is,
such a thing can always be

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represented as an infinite sum
of sines and cosines.

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So, it's going to look like
this.

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There's a constant term you
have to put out front.

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And then, the rest,
instead of writing,

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it's rather long to write
unless you use summation

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notation.
So, I will.

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So, it's a sum from n equal one
to infinity integer values of n,

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in other words,
of a sine and a cosine.

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It's customary to put the
cosine first,

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and with the frequency,
the n indicates the frequency

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of the thing.
And, the bn is sine nt.

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Now, why does that solve the

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problem of general inputs for
periodic functions,

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at least if the period is two
pi or some fraction of it?

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Well, you could think of it
this way.

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I'll make a little table.
I'll make a little table.

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Let's look at,
let's put over here the input,

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and here, I'll put the
response.

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Okay, suppose the input is the
function sine nt.

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Well, in other words,
if you just solve the problem,

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you put a sine nt
here, you know how to get the

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answer, find a particular
solution, in other words.

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In fact, you do it by
converting this to a complex

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exponential, and then all the
rigmarole we've been going

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through.
So, let's call the response

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something.
Let's call it y.

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I'd better index it by n
because it, of course,

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is a response to this
particular periodic function.

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So, n of t,
and if the input is cosine nt,

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that also will have
a response, yn.

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Now, I really can't call them
both by the same name.

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So, why don't we put a little s
up here to indicate that that's

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the response to the sine.
And here, I'll put a little c

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to indicate what the answer to
the cosine.

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You're feeding cosine nt,
what you get out is

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this function.
Now what?

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Well, by the way,
notice that if n is zero,

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it's going to take care of a
constant term,

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too.
In other words,

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the reason there is a constant
term out front is because that

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corresponds to cosine of zero t,
which is one.

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Now, suppose I input instead an
cosine nt.

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All you do is multiply the
answer by an.

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Same here.
Multiply the input by bn.

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You multiply the response.
That's because the equation is

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a linear equation.
And now, what am I going to do?

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I'm going to add them up.
If I add them up from the

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different ends and take a count
also, the n equals zero

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corresponding to this first
constant term,

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the sum of all these according
to my Fourier formula is going

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to be f of t.
What's the sum of this,

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the corresponding responses?
Well, that's going to be

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summation a n y n c t
plus b n y n,

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the response to the
sine.

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That will be the sum from one
to infinity, and there will be

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some sort of constant term here.
Let's just call it c1.

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So, in other words,
if this input produces that

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response, and these are things
which we can calculate,

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we're led by this formula,
Fourier's formula,

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to the response to things which
otherwise we would have not been

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able to calculate,
namely, any periodic function

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of period two pi will have,
the procedure will be,

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you've got a periodic function
of period two pi.

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Find its Fourier series,
and I'll show you how to do

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that today.
Find its Fourier series,

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and then the response to that
general f of t will be this

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infinite series of functions,
where these things are things

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you already know how to
calculate.

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They are the responses to sines
and cosines.

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And, you just formed the sum
with those coefficients.

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Now, why does that work?
It works by the superposition

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principle.
So, this is true.

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The reason I can do the adding
and multiplying by constant,

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I'm using the superposition
principle.

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If this input produces that
response, then the sum of a

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bunch of inputs produces the sum
of the corresponding responses.

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And, why is that?
Why can I use the superposition

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principle?
Because the ODE is linear.

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It's okay, since the ODE is
linear.

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That's what makes all this
work.

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Now, so what we're going to do
today is I will show you how to

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calculate those Fourier series.
I will not be able to use it to

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actually solve any differential
equation.

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It will take us pretty much all
the period to show how to

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calculate a Fourier series.
And, okay, so I'm going to

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solve differential equations on
Monday.

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Wrong.
I probably won't even get to it

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then because the calculation of
a Fourier series is a sufficient

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amount of work that you really
want to know all the possible

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tricks and shortcuts there are.
Unfortunately,

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they are not very clever
tricks.

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They are just obvious things.
But, it will take me a period

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to point out those obvious
things, obvious in my sense if

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not in yours.
And, finally,

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the third day,
we'll solve differential

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equations.
I will actually carry out the

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program.
But the main thing we're going

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to get out of it is another
approach to resonance because

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the things that we are going to
be interested in are picking out

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which of these terms may
possibly produce resonance,

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and therefore a very crazy
response.

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Some of the terms in the
response suddenly get a much

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bigger amplitude than this than
you would normally have thought

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they had because it's picking
out resonant terms in the

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Fourier series of the input.
Okay, well, that's a big

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mouthfu.
Let's get started on

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calculating.
So, the program today is

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calculate the Fourier series.
Given f of t periodic,

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having two pi as a period,
find its Fourier series.

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How, in other words,
do I calculate those

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coefficients,
an and bn.

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Now, the answer is not
immediately apparent,

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and it's really quite
remarkable.

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I think it's quite remarkable,
anyway.

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It's one of the basic things of
higher mathematics.

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And, what it depends upon are
certain things called the

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orthogonality relations.
So, this is the place where

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you've got to learn what such
things are.

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Well, I think it would be a
good idea to have a general

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definition, rather than
immediately get into the

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specifics.
So, I'm going to call u of x,

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u of t, I think I will use,

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since Fourier analysis is most
often applied when the variable

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is time, I think I will stick to
independent variable t all

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period long, if I remember to,
at any rate.

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So, these are two continuous,
or not very discontinuous

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functions on minus pi.
Let's make them periodic.

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Let's say two pi is a period.
So, functions,

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for example like those guys,
sine t, sine nt,

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sine 22t,

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and so on, say two pi is a
period.

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Well, I want them really on the
whole real axis,

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not there.
Define for all real numbers.

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Then, I say that they are
orthogonal, perpendicular.

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But nobody says perpendicular.
Orthogonal is the word,

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orthogonal on the interval
minus pi to pi

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if the integral,
so, two are orthogonal.

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Well, these two functions,
if the integral from minus pi

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to pi of u of t v of t,
the product is zero,

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that's called the orthogonality
condition on minus pi to pi.

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Now, well, it's just the

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definition.
I would love to go into a

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little song and dance now on
what the definition really

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means, and what its application,
why the word orthogonal is

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used, because it really does
have something to do with two

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vectors being orthogonal in the
sense in which you live it in

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18.02.
I'll have to put that on the

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ice for the moment,
and whether I get to it or not

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depends on how fast I talk.
But, you probably prefer I talk

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slowly.
So, let's compromise.

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Anyway, that's the condition.
And now, what I say is that

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that Fourier,
that blue Fourier series,

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--
-- what finding the

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coefficients an and bn depends
upon is this theorem that the

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collection of functions,
as I look at this collection of

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functions, sine nt
for any value of the integer,

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n, of course I can assume n is
a positive integer because sine

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of minus nt is the same as sine
of nt.

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And, cosine mt,
let's give it a different,

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so I don't want you to think
they are exactly the same

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integers.
So, this is a big collection of

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functions, as n runs from one to
infinity-- Here,

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I could let m be run from zero
to infinity because cosine of

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zero t means something.

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It's a constant,
one-- that any two distinct

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ones, two distinct,
you know, how can two things be

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not different?
Well, you know,

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you talk about two coincident
roots.

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I'm just killing,
doing a little overkill.

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Any two distinct ones of these,
two distinct members of the set

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of this collection of,
I don't know,

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there's no way to say that,
any two distinct ones are

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orthogonal on this interval.
Of course, they all have two pi

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as a period for all of them.
So, they form into this general

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category that I'm talking about,
but any two distinct ones are

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orthogonal on the interval for
minus pi to pi.

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00:16:51,000 --> 00:16:57,000
So, if I integrate from minus

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pi to pi sine of three t times
cosine of four t dt,

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00:16:57,000 --> 00:17:03,000
answer is zero.

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00:17:00,000 --> 00:17:06,000
If I integrate sine of 3t times

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00:17:05,000 --> 00:17:11,000
the sine of 60t,
answer is zero.

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00:17:09,000 --> 00:17:15,000
The same thing with two
cosines, or a sine and a cosine.

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00:17:13,000 --> 00:17:19,000
The only time you don't get
zero is if you integrate,

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00:17:17,000 --> 00:17:23,000
if you make the two functions
the same.

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Now, how do you know that you
could not possibly get the

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answer is zero if the two
functions are the same?

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If the two functions are the
same, then I'm integrating a

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square.
A square is always positive.

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00:17:38,000 --> 00:17:44,000
I'm integrating a square.
A square is always positive,

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and therefore I cannot get the
answer, zero.

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00:17:47,000 --> 00:17:53,000
But, in the other cases,
I might get the answer zero.

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And the theorem is you always
do.

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Okay, so, why is this?
Well, there are three ways to

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prove this.
It's like many fundamental

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00:18:03,000 --> 00:18:09,000
facts in mathematics.
There are different ways of

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00:18:07,000 --> 00:18:13,000
going about it.
By the way, along with the

235
00:18:10,000 --> 00:18:16,000
theorem, I probably should have
included, so,

236
00:18:14,000 --> 00:18:20,000
I'm far away.
But you might as well include,

237
00:18:18,000 --> 00:18:24,000
because we're going to need it.
What happens if you use the

238
00:18:23,000 --> 00:18:29,000
same function?
If I take U equal to V,

239
00:18:26,000 --> 00:18:32,000
and in that case,
as I've indicated,

240
00:18:29,000 --> 00:18:35,000
you're not going to get the
answer, zero.

241
00:18:34,000 --> 00:18:40,000
But, what you will get is,
so, in other words,

242
00:18:37,000 --> 00:18:43,000
I'm just asking,
what is the sine of

243
00:18:41,000 --> 00:18:47,000
n t squared.
That's a case where two of them

244
00:18:44,000 --> 00:18:50,000
are the same.
I use the same function.

245
00:18:47,000 --> 00:18:53,000
What's that?
Well, the answer is,

246
00:18:50,000 --> 00:18:56,000
it's the same as what you will
get if you integrate the cosine,

247
00:18:54,000 --> 00:19:00,000
cosine squared n t dt.

248
00:18:58,000 --> 00:19:04,000
And, the answer to either one

249
00:19:02,000 --> 00:19:08,000
of these is pi.
That's something you know how

250
00:19:06,000 --> 00:19:12,000
to do from 18.01 or the
equivalent thereof.

251
00:19:09,000 --> 00:19:15,000
You can integrate sine squared.
It's one of the things you had

252
00:19:14,000 --> 00:19:20,000
to learn for whatever exam you
took on methods of integration.

253
00:19:19,000 --> 00:19:25,000
Anyway, so I'm not going to
calculate this out.

254
00:19:23,000 --> 00:19:29,000
The answer turns out to be pi.
All right, now,

255
00:19:27,000 --> 00:19:33,000
the ways to prove it are you
can use trig identities.

256
00:19:33,000 --> 00:19:39,000
And, I'm asking you in one of
the early problems in the

257
00:19:37,000 --> 00:19:43,000
problem set, identities,
identities for the product of

258
00:19:41,000 --> 00:19:47,000
sine and cosine,
expressing it in a form in

259
00:19:44,000 --> 00:19:50,000
which it's easy to integrate,
and you can prove it that way.

260
00:19:48,000 --> 00:19:54,000
Or, you can use,
if you have forgotten the

261
00:19:51,000 --> 00:19:57,000
trigonometric identities and
want to get some more exercise

262
00:19:56,000 --> 00:20:02,000
with complex-- you can use
complex exponentials.

263
00:20:01,000 --> 00:20:07,000
So, I'm asking you how to,
in another part of the same

264
00:20:05,000 --> 00:20:11,000
problem I'm asking you how to do
it, do one of these,

265
00:20:09,000 --> 00:20:15,000
at any rate,
using complex exponentials.

266
00:20:13,000 --> 00:20:19,000
And now, I'm going to use a
mysterious third method another

267
00:20:18,000 --> 00:20:24,000
way.
I'm going to use the ODE.

268
00:20:20,000 --> 00:20:26,000
I'm going to do that because
this is the method.

269
00:20:24,000 --> 00:20:30,000
It's not just sines and cosines
which are orthogonal.

270
00:20:30,000 --> 00:20:36,000
There are masses of orthogonal
functions out there.

271
00:20:33,000 --> 00:20:39,000
And, the way they are
discovered, and the way you

272
00:20:36,000 --> 00:20:42,000
prove they're orthogonal is not
with trig identities and complex

273
00:20:40,000 --> 00:20:46,000
exponentials because those only
work with sines and cosines.

274
00:20:44,000 --> 00:20:50,000
It is, instead,
by going back to the

275
00:20:46,000 --> 00:20:52,000
differential equation that they
solve.

276
00:20:48,000 --> 00:20:54,000
And that's, therefore,
the method here that I'm going

277
00:20:52,000 --> 00:20:58,000
to use here because this is the
method which generalizes to many

278
00:20:56,000 --> 00:21:02,000
other differential equations
other than the simple ones

279
00:20:59,000 --> 00:21:05,000
satisfied by sines and cosines.
But anyway, that is the source.

280
00:21:05,000 --> 00:21:11,000
So, the way the proof of these
orthogonality conditions goes,

281
00:21:09,000 --> 00:21:15,000
so I'm not going to do that.
And, I'm going to assume that m

282
00:21:14,000 --> 00:21:20,000
is different from n so that I'm
not in either of these two

283
00:21:18,000 --> 00:21:24,000
cases.
What it depends on is,

284
00:21:20,000 --> 00:21:26,000
what's the differential
equation that all these

285
00:21:23,000 --> 00:21:29,000
functions satisfy?
Well, it's a different

286
00:21:26,000 --> 00:21:32,000
differential equation depending
upon the value of n,

287
00:21:30,000 --> 00:21:36,000
--
-- but they look at essentially

288
00:21:35,000 --> 00:21:41,000
the same.
These satisfy the differential

289
00:21:38,000 --> 00:21:44,000
equation, in other words,
what they have in common.

290
00:21:43,000 --> 00:21:49,000
The differential equation is,
let's call it u.

291
00:21:48,000 --> 00:21:54,000
It looks better.
It's going to look better if

292
00:21:52,000 --> 00:21:58,000
you let me call it u.
u double prime plus,

293
00:21:56,000 --> 00:22:02,000
well, n squared,
so for the function sine n t

294
00:22:00,000 --> 00:22:06,000
cosine n t, satisfy u double

295
00:22:05,000 --> 00:22:11,000
prime plus n squared times u.

296
00:22:11,000 --> 00:22:17,000
In other words,
the frequency is n,

297
00:22:13,000 --> 00:22:19,000
and therefore,
this is a square of the

298
00:22:16,000 --> 00:22:22,000
frequency is what you put here,
equals zero.

299
00:22:19,000 --> 00:22:25,000
In other words,
what these functions have in

300
00:22:22,000 --> 00:22:28,000
common is that they satisfy
differential equations that look

301
00:22:26,000 --> 00:22:32,000
like that.
And the only thing that's

302
00:22:28,000 --> 00:22:34,000
allowed to vary is the
frequency, which is allowed to

303
00:22:32,000 --> 00:22:38,000
change.
The frequency is in this

304
00:22:36,000 --> 00:22:42,000
coefficient of u.
Now, the remarkable thing is

305
00:22:42,000 --> 00:22:48,000
that's all you need to know.
The fact that they satisfy the

306
00:22:49,000 --> 00:22:55,000
differential equation,
that's all you need to know to

307
00:22:55,000 --> 00:23:01,000
prove the orthogonality
relationship.

308
00:22:59,000 --> 00:23:05,000
Okay, let's try to do it.
Well, I need some notation.

309
00:23:06,000 --> 00:23:12,000
So, I'm going to let un and vm
be any two of the functions.

310
00:23:11,000 --> 00:23:17,000
In other words,
I'll assume m is different from

311
00:23:16,000 --> 00:23:22,000
n.
For example,

312
00:23:17,000 --> 00:23:23,000
this one could be sine nt,
and that could be

313
00:23:22,000 --> 00:23:28,000
sine of mt,
or this could be sine nt

314
00:23:26,000 --> 00:23:32,000
and that could be
cosine of mt.

315
00:23:33,000 --> 00:23:39,000
You get the idea.
Any two of those in the

316
00:23:35,000 --> 00:23:41,000
subscript indicates whether what
the n or the m is that are in

317
00:23:40,000 --> 00:23:46,000
that.
Any two, and I mean really two,

318
00:23:42,000 --> 00:23:48,000
distinct, well,
if I say that m is not n,

319
00:23:45,000 --> 00:23:51,000
then they positively have to be
different.

320
00:23:48,000 --> 00:23:54,000
So, again, it's overkill with
my two's-ness.

321
00:23:51,000 --> 00:23:57,000
And, what I'm going to
calculate, well,

322
00:23:53,000 --> 00:23:59,000
first of all,
from the equation,

323
00:23:56,000 --> 00:24:02,000
I'm going to write the equation
this way.

324
00:24:00,000 --> 00:24:06,000
It says that u double prime is
equal to minus n squared u.

325
00:24:07,000 --> 00:24:13,000
That's true for any of these

326
00:24:11,000 --> 00:24:17,000
guys.
Of course, here,

327
00:24:13,000 --> 00:24:19,000
it would be v double prime is
equal to minus m squared

328
00:24:20,000 --> 00:24:26,000
times v.
You have to make those simple

329
00:24:26,000 --> 00:24:32,000
adjustments.
And now, what we're going to

330
00:24:30,000 --> 00:24:36,000
calculate is the integral from
minus pi to pi of un double

331
00:24:37,000 --> 00:24:43,000
prime times vm dt.

332
00:24:43,000 --> 00:24:49,000
Now, just bear with me.

333
00:24:48,000 --> 00:24:54,000
Why am I going to do that?
I can't explain what I'm going

334
00:24:53,000 --> 00:24:59,000
to do that.
But you won't ask me the

335
00:24:56,000 --> 00:25:02,000
question in five minutes.
But the point is,

336
00:24:59,000 --> 00:25:05,000
this is highly un-symmetric.
The u is differentiated twice.

337
00:25:05,000 --> 00:25:11,000
The v isn't.
So, those two functions-- but

338
00:25:08,000 --> 00:25:14,000
there is a way of turning them
into an expression which looks

339
00:25:12,000 --> 00:25:18,000
extremely symmetric,
where they are the same.

340
00:25:16,000 --> 00:25:22,000
And the way to do that is I
want to get rid of one of these

341
00:25:20,000 --> 00:25:26,000
primes here and put one on here.
The way to do that is if you

342
00:25:25,000 --> 00:25:31,000
want to integrate one of these
guys, and differentiate this one

343
00:25:29,000 --> 00:25:35,000
to make them look the same,
that's called integration by

344
00:25:33,000 --> 00:25:39,000
parts, the most important
theoretical method you learned

345
00:25:38,000 --> 00:25:44,000
in 18.01 even though you didn't
know that it was the most

346
00:25:42,000 --> 00:25:48,000
important theoretical method.
Okay, we're going to use it now

347
00:25:47,000 --> 00:25:53,000
as a basis for Fourier series.
Okay, so I'm going to integrate

348
00:25:51,000 --> 00:25:57,000
by parts.
Now, the first thing you do,

349
00:25:53,000 --> 00:25:59,000
of course, when you integrate
by parts is you just do the

350
00:25:56,000 --> 00:26:02,000
integration.
You don't do differentiation.

351
00:25:59,000 --> 00:26:05,000
So, the first thing looks like
this.

352
00:26:02,000 --> 00:26:08,000
And, that's to be evaluated
between negative pi and pi.

353
00:26:08,000 --> 00:26:14,000
In doing integration by parts
between limits,

354
00:26:12,000 --> 00:26:18,000
minus what you get by doing
both.

355
00:26:16,000 --> 00:26:22,000
You do both,
the integration and the

356
00:26:20,000 --> 00:26:26,000
differentiation.
And, again, evaluate that

357
00:26:24,000 --> 00:26:30,000
between limits.
Now, I'm just going to BS my

358
00:26:29,000 --> 00:26:35,000
way through this.
This is zero.

359
00:26:34,000 --> 00:26:40,000
I don't care what the un's,
which un you picked and which

360
00:26:39,000 --> 00:26:45,000
vm you picked.
The answer here is always going

361
00:26:43,000 --> 00:26:49,000
to be zero.
Instead of wasting six boards

362
00:26:46,000 --> 00:26:52,000
trying to write out the
argument, let me wave my hands.

363
00:26:51,000 --> 00:26:57,000
Okay, it's clear,
for example,

364
00:26:54,000 --> 00:27:00,000
that a v is a sine, sine mt.

365
00:26:57,000 --> 00:27:03,000
Of course it's zero because the
sine vanishes at both pi and

366
00:27:02,000 --> 00:27:08,000
minus pi.
If the un were a cosine,

367
00:27:06,000 --> 00:27:12,000
after I differentiate it,
it became a sine.

368
00:27:09,000 --> 00:27:15,000
And so, now it's this side guy
that's zero at both ends.

369
00:27:14,000 --> 00:27:20,000
So, the only case in which we
might have a little doubt is if

370
00:27:18,000 --> 00:27:24,000
this is a cosine,
and after differentiation,

371
00:27:21,000 --> 00:27:27,000
this is also a cosine.
In other words,

372
00:27:24,000 --> 00:27:30,000
it might look like cosine,
after, this cosine nt times

373
00:27:28,000 --> 00:27:34,000
cosine mt.
But, I claim that that's zero,

374
00:27:34,000 --> 00:27:40,000
too.
Why?

375
00:27:35,000 --> 00:27:41,000
Because the cosines are even
functions, and therefore,

376
00:27:39,000 --> 00:27:45,000
they have the same value at
both ends.

377
00:27:42,000 --> 00:27:48,000
So, if I subtract the value
evaluated at pi,

378
00:27:46,000 --> 00:27:52,000
and subtract the value of minus
pi, again zero because I have

379
00:27:51,000 --> 00:27:57,000
the same value at both ends.
So, by this entirely convincing

380
00:27:56,000 --> 00:28:02,000
argument, no matter what
combination of sines and cosines

381
00:28:00,000 --> 00:28:06,000
I have here, the answer to that
part will always be zero.

382
00:28:07,000 --> 00:28:13,000
So, by calculation,
but thought calculation;

383
00:28:11,000 --> 00:28:17,000
it's just a waste of time to
write anything out.

384
00:28:16,000 --> 00:28:22,000
You stare at it until you agree
that it's so.

385
00:28:20,000 --> 00:28:26,000
And now, I've taken,
by this integration by parts,

386
00:28:25,000 --> 00:28:31,000
I've taken this highly
un-symmetric expression and

387
00:28:30,000 --> 00:28:36,000
turned it into something in
which the u and the v are

388
00:28:35,000 --> 00:28:41,000
treated exactly alike.
Well, good, that's nice,

389
00:28:40,000 --> 00:28:46,000
but why?
Why did I go to this trouble?

390
00:28:43,000 --> 00:28:49,000
Okay, now we're going to use
the fact that this satisfies the

391
00:28:47,000 --> 00:28:53,000
differential equation,
in other words,

392
00:28:50,000 --> 00:28:56,000
that u double prime is equal to
minus n,

393
00:28:53,000 --> 00:28:59,000
I'm sorry, I should have
subscripted this.

394
00:28:56,000 --> 00:29:02,000
If that's the solution,
then this is equal to,

395
00:29:00,000 --> 00:29:06,000
times.
You have to put in a subscript

396
00:29:02,000 --> 00:29:08,000
otherwise.
The n wouldn't matter.

397
00:29:06,000 --> 00:29:12,000
All right, I'm now going to
take that expression,

398
00:29:10,000 --> 00:29:16,000
and evaluate it differently.
un double prime vm dt

399
00:29:15,000 --> 00:29:21,000
is equal to,
well, un double prime,

400
00:29:18,000 --> 00:29:24,000
because it satisfies the
differential equation is equal

401
00:29:22,000 --> 00:29:28,000
to that.
So, what is this?

402
00:29:25,000 --> 00:29:31,000
This is minus n squared
times the integral from

403
00:29:29,000 --> 00:29:35,000
negative pi to pi,
and I'm replacing un double

404
00:29:33,000 --> 00:29:39,000
prime by minus n
squared un.

405
00:29:39,000 --> 00:29:45,000
I pulled the minus n squared
out.

406
00:29:43,000 --> 00:29:49,000
So, it's un here,
and the other factor is vm dt.

407
00:29:47,000 --> 00:29:53,000
Now, that's the proof.
Huh?

408
00:29:50,000 --> 00:29:56,000
What do you mean that's the
proof?

409
00:29:54,000 --> 00:30:00,000
Okay, well, I'll first state
it, why intuitively that's the

410
00:29:59,000 --> 00:30:05,000
end of the argument.
And then, I'll spell it out a

411
00:30:06,000 --> 00:30:12,000
little more detail,
but the more detail you make

412
00:30:11,000 --> 00:30:17,000
for this, the more obscure it
gets instead of,

413
00:30:16,000 --> 00:30:22,000
look, I just showed you that
this is symmetric in u and v,

414
00:30:22,000 --> 00:30:28,000
after you massage it a little
bit.

415
00:30:26,000 --> 00:30:32,000
Here, I'm calculating it a
different way.

416
00:30:30,000 --> 00:30:36,000
Is this symmetric in u and v?
Well, the answer is yes or no.

417
00:30:37,000 --> 00:30:43,000
Is this symmetric at u and v?
No.

418
00:30:40,000 --> 00:30:46,000
Why?
Because of the n.

419
00:30:42,000 --> 00:30:48,000
The n favors u.
We have what is called a

420
00:30:46,000 --> 00:30:52,000
paradox.
This thing is symmetric in u

421
00:30:50,000 --> 00:30:56,000
and v because I can show it is.
And, it's not symmetric in u

422
00:30:55,000 --> 00:31:01,000
and v because I can show it is.
I can show it's not symmetric

423
00:31:01,000 --> 00:31:07,000
because it favors the n.
Now, there's only one possible

424
00:31:09,000 --> 00:31:15,000
resolution of that paradox.
Both would be symmetric if what

425
00:31:19,000 --> 00:31:25,000
were true?
Pardon?

426
00:31:22,000 --> 00:31:28,000
Negative pi.
All right, let me write it this

427
00:31:29,000 --> 00:31:35,000
way.
Okay, never mind.

428
00:31:32,000 --> 00:31:38,000
You see, the only way this can
happen is if this expression is

429
00:31:37,000 --> 00:31:43,000
zero.
In other words,

430
00:31:39,000 --> 00:31:45,000
the only way something can be
both symmetric and not symmetric

431
00:31:44,000 --> 00:31:50,000
is if it's zero all the time.
And, that's what we're trying

432
00:31:48,000 --> 00:31:54,000
to prove, that this is zero.
But, instead of doing it that

433
00:31:53,000 --> 00:31:59,000
way, let me show you.
This is equal to that,

434
00:31:57,000 --> 00:32:03,000
and therefore,
two things according to Euclid,

435
00:32:00,000 --> 00:32:06,000
two things equal to the same
thing are equal to each other.

436
00:32:07,000 --> 00:32:13,000
So, this equals that,
which, in turn,

437
00:32:09,000 --> 00:32:15,000
therefore, equals what I would
have gotten.

438
00:32:12,000 --> 00:32:18,000
I'm just saying the symmetry of
different way,

439
00:32:15,000 --> 00:32:21,000
what I would have gotten if I
had done this calculation.

440
00:32:19,000 --> 00:32:25,000
And, that turns out to be minus
m squared times the integral

441
00:32:23,000 --> 00:32:29,000
from minus pi to pi
of un vm dt.

442
00:32:28,000 --> 00:32:34,000
So, these two are equal because

443
00:32:33,000 --> 00:32:39,000
they are both equal to this.
This is equal to that.

444
00:32:38,000 --> 00:32:44,000
This equals that.
Therefore, how can this equal

445
00:32:42,000 --> 00:32:48,000
that unless the integral is
zero?

446
00:32:46,000 --> 00:32:52,000
How's that?
Remember, m is different from

447
00:32:50,000 --> 00:32:56,000
n.
So, what this proves is,

448
00:32:52,000 --> 00:32:58,000
therefore, the integral from
negative pi to pi of un vm dt is

449
00:32:59,000 --> 00:33:05,000
equal to zero,

450
00:33:05,000 --> 00:33:11,000
at least if m is different from
n.

451
00:33:10,000 --> 00:33:16,000
Now, there is one case I didn't
include.

452
00:33:12,000 --> 00:33:18,000
Which case didn't I include?
un times un is not supposed to

453
00:33:16,000 --> 00:33:22,000
be zero.
So, in that case,

454
00:33:18,000 --> 00:33:24,000
I don't have to worry about,
but there is a case that I

455
00:33:22,000 --> 00:33:28,000
didn't.
For example,

456
00:33:24,000 --> 00:33:30,000
something like the cosine of nt
times the sine of nt.

457
00:33:28,000 --> 00:33:34,000
Here, the m is the same as the

458
00:33:32,000 --> 00:33:38,000
n.
Nonetheless,

459
00:33:34,000 --> 00:33:40,000
I am claiming that this is zero
because these aren't the same

460
00:33:39,000 --> 00:33:45,000
function.
One is a cosine.

461
00:33:42,000 --> 00:33:48,000
Why is that zero?
Can you see mentally that

462
00:33:46,000 --> 00:33:52,000
that's zero?
Mentally?

463
00:33:48,000 --> 00:33:54,000
Well, this is trying to be in
another life,

464
00:33:52,000 --> 00:33:58,000
it's trying to be one half the
sine of two nt, right?

465
00:33:57,000 --> 00:34:03,000
And obviously the integral of

466
00:34:02,000 --> 00:34:08,000
sine of two nt is zero between
minus pi and pi

467
00:34:06,000 --> 00:34:12,000
because you integrate it,

468
00:34:09,000 --> 00:34:15,000
and it turns out to be zero.
You integrate it to a cosine,

469
00:34:13,000 --> 00:34:19,000
which has the same value of
both ends.

470
00:34:16,000 --> 00:34:22,000
Well, that was a lot of
talking.

471
00:34:18,000 --> 00:34:24,000
If this proof is too abstract
for you, I won't ask you to

472
00:34:22,000 --> 00:34:28,000
reproduce it on an exam.
You can go with the proofs

473
00:34:25,000 --> 00:34:31,000
using trigonometric identities,
and/or complex exponentials.

474
00:34:31,000 --> 00:34:37,000
But, you ought to know at least
one of those,

475
00:34:34,000 --> 00:34:40,000
and for the problem set I'm
asking you to fool around a

476
00:34:39,000 --> 00:34:45,000
little with at least two of
them.

477
00:34:41,000 --> 00:34:47,000
Okay, now, what has this got to
do with the problem we started

478
00:34:47,000 --> 00:34:53,000
with originally?
The problem is to explain this

479
00:34:50,000 --> 00:34:56,000
blue series.
So, our problem is,

480
00:34:53,000 --> 00:34:59,000
how, from this,
am I going to get the terms of

481
00:34:57,000 --> 00:35:03,000
this blue series?
So, given f of t,

482
00:35:02,000 --> 00:35:08,000
two pi s a period.
Find the an and the bn.

483
00:35:06,000 --> 00:35:12,000
Okay, let's focus on the an.
The bn is the same.

484
00:35:11,000 --> 00:35:17,000
Once you know how to do one,
you know how to do the other.

485
00:35:16,000 --> 00:35:22,000
So, here's the idea.
Again, it goes back to the

486
00:35:21,000 --> 00:35:27,000
something you learned at the
very beginning of 18.02,

487
00:35:26,000 --> 00:35:32,000
but I don't think it took.
But maybe some of you will

488
00:35:32,000 --> 00:35:38,000
recognize it.
So, what I'm going to do is

489
00:35:36,000 --> 00:35:42,000
write it.
Here's the term we're looking

490
00:35:40,000 --> 00:35:46,000
for here, this one.
Okay, and there are others.

491
00:35:45,000 --> 00:35:51,000
It's an infinite series that
goes on forever.

492
00:35:50,000 --> 00:35:56,000
And now, to make the argument,
I've got to put it one more

493
00:35:56,000 --> 00:36:02,000
term here.
So, I'm going to put in ak

494
00:36:00,000 --> 00:36:06,000
cosine kt.
I don't mean to imply that that

495
00:36:07,000 --> 00:36:13,000
k could be more than n,
in which case I should have

496
00:36:11,000 --> 00:36:17,000
written it here.
I could have also used equally

497
00:36:16,000 --> 00:36:22,000
well bk sine kt
here, and I could have put it

498
00:36:22,000 --> 00:36:28,000
there.
This is just some other term.

499
00:36:25,000 --> 00:36:31,000
This is the an,
and this is the one we want.

500
00:36:30,000 --> 00:36:36,000
And, this is some other term.
Okay, all right,

501
00:36:35,000 --> 00:36:41,000
now, what you do is,
to get the an,

502
00:36:38,000 --> 00:36:44,000
what you do is you multiply
everything through by,

503
00:36:42,000 --> 00:36:48,000
you focus on the one you want,
so it's dot,

504
00:36:46,000 --> 00:36:52,000
dot, dot, dot,
dot, and you multiply by cosine

505
00:36:50,000 --> 00:36:56,000
nt.
So, it's ak cosine kt times

506
00:36:54,000 --> 00:37:00,000
cosine nt.

507
00:36:57,000 --> 00:37:03,000
Of course, that gets
multiplied, too.

508
00:37:02,000 --> 00:37:08,000
But, the one we want also gets
multiplied, an.

509
00:37:06,000 --> 00:37:12,000
And, it becomes,
when I multiply by cosine nt,

510
00:37:11,000 --> 00:37:17,000
cosine squared nt,

511
00:37:16,000 --> 00:37:22,000
and now, I hope you can see
what's going to happen.

512
00:37:21,000 --> 00:37:27,000
Now, oops, I didn't multiply
the f of t,

513
00:37:26,000 --> 00:37:32,000
sorry.
It's the oldest trick in the

514
00:37:30,000 --> 00:37:36,000
book.
I now integrate everything from

515
00:37:35,000 --> 00:37:41,000
minus, so I don't endlessly
recopy.

516
00:37:38,000 --> 00:37:44,000
I'll integrate by putting it up
in yellow chalk,

517
00:37:42,000 --> 00:37:48,000
and you are left to your own
devices.

518
00:37:46,000 --> 00:37:52,000
This is definitely a colored
pen type of course.

519
00:37:50,000 --> 00:37:56,000
Okay, so, you want to integrate
from minus pi to pi?

520
00:37:55,000 --> 00:38:01,000
Good.
Just integrate everything on

521
00:37:59,000 --> 00:38:05,000
the right hand side,
also, from minus pi to pi.

522
00:38:05,000 --> 00:38:11,000
Plus, these are the guys just
to indicate that I haven't,

523
00:38:10,000 --> 00:38:16,000
they are out there,
too.

524
00:38:13,000 --> 00:38:19,000
And now, what happens?
What's this?

525
00:38:16,000 --> 00:38:22,000
Zero.
Every term is zero because of

526
00:38:20,000 --> 00:38:26,000
the orthogonality relations.
They are all of the form,

527
00:38:25,000 --> 00:38:31,000
a constant times cosine nt
times something different from

528
00:38:31,000 --> 00:38:37,000
cosine nt, sine kt,

529
00:38:35,000 --> 00:38:41,000
cosine kt,
or even that constant term.

530
00:38:42,000 --> 00:38:48,000
All of the other terms are
zero, and the only one which

531
00:38:46,000 --> 00:38:52,000
survives is this one.
And, what's its value?

532
00:38:50,000 --> 00:38:56,000
The integral from minus pi to
pi of cosine squared,

533
00:38:54,000 --> 00:39:00,000
I put that up somewhere.
It's right here,

534
00:38:57,000 --> 00:39:03,000
down there?
It is pi.

535
00:39:00,000 --> 00:39:06,000
So, this term turns into an pi,
an, dragged along,

536
00:39:04,000 --> 00:39:10,000
but this, the integral of the
square of the cosine turns out

537
00:39:10,000 --> 00:39:16,000
to be pi.
And so, the end result is that

538
00:39:14,000 --> 00:39:20,000
we get a formula for an.
What is an?

539
00:39:18,000 --> 00:39:24,000
an is, well,
an times pi,

540
00:39:20,000 --> 00:39:26,000
all these terms of zero,
and nothing is left but this

541
00:39:25,000 --> 00:39:31,000
left-hand side.
And therefore,

542
00:39:28,000 --> 00:39:34,000
an times pi is the integral
from negative pi to pi of f of t

543
00:39:34,000 --> 00:39:40,000
times cosine nt dt.

544
00:39:40,000 --> 00:39:46,000
But, that's an times pi.

545
00:39:45,000 --> 00:39:51,000
Therefore, if I want just an,
I have to divide it by pi.

546
00:39:50,000 --> 00:39:56,000
And, that's the formula for the
coefficient an.

547
00:39:54,000 --> 00:40:00,000
The argument is exactly the
same if you want bn,

548
00:39:57,000 --> 00:40:03,000
but I will write it down for
the sake of completeness,

549
00:40:02,000 --> 00:40:08,000
as they say,
and to give you a chance to

550
00:40:05,000 --> 00:40:11,000
digest what I've done,
you know, 30 seconds to digest

551
00:40:09,000 --> 00:40:15,000
it. Sine nt dt.

552
00:40:12,000 --> 00:40:18,000
And, that's because the
argument is the same.

553
00:40:16,000 --> 00:40:22,000
And, the integral of sine
squared nt is also

554
00:40:20,000 --> 00:40:26,000
pi. So, there's no difference

555
00:40:22,000 --> 00:40:28,000
there.
Now, there's only one little

556
00:40:24,000 --> 00:40:30,000
caution.
It have to be a little careful.

557
00:40:27,000 --> 00:40:33,000
This is n one,
two, and so on,

558
00:40:29,000 --> 00:40:35,000
and this is also n one,
two, and unfortunately,

559
00:40:33,000 --> 00:40:39,000
the constant term is a slight
exception.

560
00:40:35,000 --> 00:40:41,000
We better look at that
specifically because if you

561
00:40:39,000 --> 00:40:45,000
forget it, you can get them to
gross, gross,

562
00:40:42,000 --> 00:40:48,000
gross errors.
How about the constant term?

563
00:40:48,000 --> 00:40:54,000
Suppose I repeat the argument
for that in miniature.

564
00:40:54,000 --> 00:41:00,000
There is a constant term plus
other stuff, a typical other

565
00:41:01,000 --> 00:41:07,000
stuff, an cosine,
let's say.

566
00:41:06,000 --> 00:41:12,000
How am I going to get that
constant term?

567
00:41:10,000 --> 00:41:16,000
Well, if you think of this as
sort of like a constant times,

568
00:41:16,000 --> 00:41:22,000
the reason is the constant is
because it's being multiplied by

569
00:41:22,000 --> 00:41:28,000
cosine zero t.
So, that suggests I should

570
00:41:27,000 --> 00:41:33,000
multiply by one.
In other words,

571
00:41:31,000 --> 00:41:37,000
what I should do is simply
integrate this from negative pi

572
00:41:36,000 --> 00:41:42,000
to pi, f of t dt.

573
00:41:40,000 --> 00:41:46,000
What's the answer?
Well, this integrated from

574
00:41:44,000 --> 00:41:50,000
minus pi to pi is how much?
It's c zero times two pi,

575
00:41:49,000 --> 00:41:55,000
right?
And, the other terms all give

576
00:41:52,000 --> 00:41:58,000
me zero.
Every other term is zero

577
00:41:55,000 --> 00:42:01,000
because if you integrate cosine
nt or sine nt

578
00:42:00,000 --> 00:42:06,000
over a complete
period, you always get zero.

579
00:42:06,000 --> 00:42:12,000
There is as much area above the
axis or below.

580
00:42:10,000 --> 00:42:16,000
Or, you can look at two special
cases.

581
00:42:13,000 --> 00:42:19,000
Anyway, you always get zero.
It's the same thing with sine

582
00:42:18,000 --> 00:42:24,000
here.
So, the answer is that c zero

583
00:42:21,000 --> 00:42:27,000
is equal to,
is a little special.

584
00:42:24,000 --> 00:42:30,000
You don't just put n equals
zero here because then

585
00:42:30,000 --> 00:42:36,000
you would lose a factor of two.
So, c zero should be one

586
00:42:36,000 --> 00:42:42,000
over two pi times
this integral.

587
00:42:40,000 --> 00:42:46,000
Now, there are two kinds of
people in the world,

588
00:42:44,000 --> 00:42:50,000
the ones who learn two separate
formulas, and the ones who just

589
00:42:50,000 --> 00:42:56,000
learn two separate notations.
So, what most people do is they

590
00:42:55,000 --> 00:43:01,000
say, look, I want this to be
always the formula for a zero.

591
00:43:02,000 --> 00:43:08,000
That means, even when n
is zero, I want this to be the

592
00:43:07,000 --> 00:43:13,000
formula.
Well, then you are not going to

593
00:43:10,000 --> 00:43:16,000
get the right leading term.
Instead of getting c zero,

594
00:43:14,000 --> 00:43:20,000
you're going to get
twice it, and therefore,

595
00:43:18,000 --> 00:43:24,000
the formula is,
the Fourier series,

596
00:43:21,000 --> 00:43:27,000
therefore, isn't written this
way.

597
00:43:24,000 --> 00:43:30,000
It's written-- If you want an a
zero there,

598
00:43:28,000 --> 00:43:34,000
calculate it by this formula.
Then, you've got to write not c

599
00:43:34,000 --> 00:43:40,000
zero, but a zero over two.

600
00:43:37,000 --> 00:43:43,000
I think you will be happiest if
I have to give you advice.

601
00:43:41,000 --> 00:43:47,000
I think you'll be happiest
remembering a single formula for

602
00:43:45,000 --> 00:43:51,000
the an's and bn's,
in which case you have to

603
00:43:48,000 --> 00:43:54,000
remember that the constant
leading term is a zero over two

604
00:43:52,000 --> 00:43:58,000
if you insist on
using that formula.

605
00:43:55,000 --> 00:44:01,000
Otherwise, you have to learn a
special formula for the leading

606
00:43:59,000 --> 00:44:05,000
coefficient, namely one over two
pi instead of one

607
00:44:03,000 --> 00:44:09,000
over pi.
Well, am I really going to

608
00:44:08,000 --> 00:44:14,000
calculate a Fourier series in
four minutes?

609
00:44:11,000 --> 00:44:17,000
Not very likely,
but I'll give it a brave

610
00:44:14,000 --> 00:44:20,000
college try.
Anyway, you will be doing a

611
00:44:17,000 --> 00:44:23,000
great deal of it,
and your book has lots and lots

612
00:44:21,000 --> 00:44:27,000
of examples, too many,
in fact.

613
00:44:23,000 --> 00:44:29,000
It ruined all the good examples
by calculating them for you.

614
00:44:28,000 --> 00:44:34,000
But, I will at least outline.
Do you want me to spend three

615
00:44:34,000 --> 00:44:40,000
minutes outlining a calculation
just so you have something to

616
00:44:38,000 --> 00:44:44,000
work on in the next boring class
you are in?

617
00:44:42,000 --> 00:44:48,000
Let's see, so I'll just put a
few key things on the board.

618
00:44:46,000 --> 00:44:52,000
I would advise you to sit still
for this.

619
00:44:49,000 --> 00:44:55,000
Otherwise you're going to hack
it, and take twice as long as

620
00:44:54,000 --> 00:45:00,000
you should, even though I knew
you've been up to 3:00 in the

621
00:44:58,000 --> 00:45:04,000
morning doing your problem set.
Cheer up.

622
00:45:03,000 --> 00:45:09,000
I got up at 6:00 to make up the
new one.

623
00:45:08,000 --> 00:45:14,000
So, we're even.
This should be zero here.

624
00:45:13,000 --> 00:45:19,000
So, here's minus pi.
Here's pi.

625
00:45:17,000 --> 00:45:23,000
Here's one, negative one.
The function starts out like

626
00:45:24,000 --> 00:45:30,000
that, and now to be periodic,
it then has to continue on in

627
00:45:31,000 --> 00:45:37,000
the same way.
So, I think that's enough of

628
00:45:37,000 --> 00:45:43,000
its path through life to
indicate how it runs.

629
00:45:42,000 --> 00:45:48,000
This is a typical square-away
function, sometimes it's called.

630
00:45:48,000 --> 00:45:54,000
It's an odd function.
It goes equally above and below

631
00:45:53,000 --> 00:45:59,000
the axis.
Now, the integrals,

632
00:45:56,000 --> 00:46:02,000
when you calculate them,
the an is going to be,

633
00:46:00,000 --> 00:46:06,000
okay, look, the an is going to
turn out to be zero.

634
00:46:08,000 --> 00:46:14,000
Let me, instead,
and you will get that with a

635
00:46:11,000 --> 00:46:17,000
little hacking.
I'm much more worried about

636
00:46:14,000 --> 00:46:20,000
what you'll do with the bn's.
Also, next Monday you'll see

637
00:46:17,000 --> 00:46:23,000
intuitively that the an is zero,
in which case you won't even

638
00:46:22,000 --> 00:46:28,000
bother trying to calculate it.
How about the bn,

639
00:46:25,000 --> 00:46:31,000
though?
Well, you see,

640
00:46:26,000 --> 00:46:32,000
because the function is
discontinuous,

641
00:46:29,000 --> 00:46:35,000
so, this is my input.
My f of t is that

642
00:46:32,000 --> 00:46:38,000
orange discontinuous function.
The bn is going to be,

643
00:46:37,000 --> 00:46:43,000
I have to break it into two
parts.

644
00:46:40,000 --> 00:46:46,000
In the first part,
the function is negative one.

645
00:46:43,000 --> 00:46:49,000
And there, I will be
integrating from minus pi to pi

646
00:46:47,000 --> 00:46:53,000
of the function,
which is minus one times the

647
00:46:50,000 --> 00:46:56,000
sine of nt dt.

648
00:46:54,000 --> 00:47:00,000
And then, there's another part,

649
00:46:57,000 --> 00:47:03,000
sorry, minus pi to zero.
The other part I integrate from

650
00:47:02,000 --> 00:47:08,000
zero to pi of what?
Well, f of t is now plus one.

651
00:47:06,000 --> 00:47:12,000
And so, I simply integrate sine

652
00:47:10,000 --> 00:47:16,000
nt dt.
Now, each of these is a

653
00:47:14,000 --> 00:47:20,000
perfectly simple integral.
The only question is how you

654
00:47:19,000 --> 00:47:25,000
combine them.
So, this is,

655
00:47:21,000 --> 00:47:27,000
after you calculate it,
it will be (one minus cosine n

656
00:47:26,000 --> 00:47:32,000
pi) all over n.

657
00:47:29,000 --> 00:47:35,000
And, this part will turn out to
be (one minus cosine n pi) over

658
00:47:34,000 --> 00:47:40,000
n also. And therefore,

659
00:47:40,000 --> 00:47:46,000
the answer will be two minus
two cosine, two over n times,

660
00:47:48,000 --> 00:47:54,000
right, two minus,
two times (one minus cosine n

661
00:47:55,000 --> 00:48:01,000
pi) over n.

662
00:48:01,000 --> 00:48:07,000
No, okay, now,
what's this?

663
00:48:03,000 --> 00:48:09,000
This is minus one if n is odd.
It's plus one if n is even.

664
00:48:09,000 --> 00:48:15,000
Now, either you can work with
it this way, or you can combine

665
00:48:15,000 --> 00:48:21,000
the two of them into a single
expression.

666
00:48:19,000 --> 00:48:25,000
Its minus one to the nth power
takes care of both of

667
00:48:26,000 --> 00:48:32,000
them.
But, the way the answer is

668
00:48:29,000 --> 00:48:35,000
normally expressed,
it would be minus two over n,

669
00:48:34,000 --> 00:48:40,000
two over n times,
if n is even,

670
00:48:37,000 --> 00:48:43,000
I get zero.
If n is odd,

671
00:48:41,000 --> 00:48:47,000
I get two.
So, times two,

672
00:48:43,000 --> 00:48:49,000
if n is odd,
and zero if n is even.

673
00:48:46,000 --> 00:48:52,000
So, it's four over n,
or it's zero,

674
00:48:50,000 --> 00:48:56,000
and the final series is a sum
of those coefficients times the

675
00:48:55,000 --> 00:49:01,000
appropriate-- cosine or sine?
Sine terms because the cosine

676
00:49:01,000 --> 00:49:07,000
terms were all coefficients,
all turned out to be zero.

677
00:49:08,000 --> 00:49:14,000
I'm sorry I didn't have the
chance to do that calculation in

678
00:49:13,000 --> 00:49:19,000
detail.
But, I think that's enough

679
00:49:16,000 --> 00:49:22,000
sketch for you to be able to do
the rest of it yourself.