WEBVTT

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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

00:16:22.000 --> 00:16:28.000
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

00:16:36.000 --> 00:16:42.000
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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So, if I integrate from minus

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

00:16:57.000 --> 00:17:03.000
answer is zero.

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If I integrate sine of 3t times

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the sine of 60t,
answer is zero.

00:17:09.000 --> 00:17:15.000
The same thing with two
cosines, or a sine and a cosine.

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The only time you don't get
zero is if you integrate,

00:17:17.000 --> 00:17:23.000
if you make the two functions
the same.

00:17:20.000 --> 00:17:26.000
Now, how do you know that you
could not possibly get the

00:17:25.000 --> 00:17:31.000
answer is zero if the two
functions are the same?

00:17:30.000 --> 00:17:36.000
If the two functions are the
same, then I'm integrating a

00:17:35.000 --> 00:17:41.000
square.
A square is always positive.

00:17:38.000 --> 00:17:44.000
I'm integrating a square.
A square is always positive,

00:17:43.000 --> 00:17:49.000
and therefore I cannot get the
answer, zero.

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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facts in mathematics.
There are different ways of

00:18:07.000 --> 00:18:13.000
going about it.
By the way, along with the

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theorem, I probably should have
included, so,

00:18:14.000 --> 00:18:20.000
I'm far away.
But you might as well include,

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because we're going to need it.
What happens if you use the

00:18:23.000 --> 00:18:29.000
same function?
If I take U equal to V,

00:18:26.000 --> 00:18:32.000
and in that case,
as I've indicated,

00:18:29.000 --> 00:18:35.000
you're not going to get the
answer, zero.

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But, what you will get is,
so, in other words,

00:18:37.000 --> 00:18:43.000
I'm just asking,
what is the sine of

00:18:41.000 --> 00:18:47.000
n t squared.
That's a case where two of them

00:18:44.000 --> 00:18:50.000
are the same.
I use the same function.

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What's that?
Well, the answer is,

00:18:50.000 --> 00:18:56.000
it's the same as what you will
get if you integrate the cosine,

00:18:54.000 --> 00:19:00.000
cosine squared n t dt.

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And, the answer to either one

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of these is pi.
That's something you know how

00:19:06.000 --> 00:19:12.000
to do from 18.01 or the
equivalent thereof.

00:19:09.000 --> 00:19:15.000
You can integrate sine squared.
It's one of the things you had

00:19:14.000 --> 00:19:20.000
to learn for whatever exam you
took on methods of integration.

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Anyway, so I'm not going to
calculate this out.

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The answer turns out to be pi.
All right, now,

00:19:27.000 --> 00:19:33.000
the ways to prove it are you
can use trig identities.

00:19:33.000 --> 00:19:39.000
And, I'm asking you in one of
the early problems in the

00:19:37.000 --> 00:19:43.000
problem set, identities,
identities for the product of

00:19:41.000 --> 00:19:47.000
sine and cosine,
expressing it in a form in

00:19:44.000 --> 00:19:50.000
which it's easy to integrate,
and you can prove it that way.

00:19:48.000 --> 00:19:54.000
Or, you can use,
if you have forgotten the

00:19:51.000 --> 00:19:57.000
trigonometric identities and
want to get some more exercise

00:19:56.000 --> 00:20:02.000
with complex-- you can use
complex exponentials.

00:20:01.000 --> 00:20:07.000
So, I'm asking you how to,
in another part of the same

00:20:05.000 --> 00:20:11.000
problem I'm asking you how to do
it, do one of these,

00:20:09.000 --> 00:20:15.000
at any rate,
using complex exponentials.

00:20:13.000 --> 00:20:19.000
And now, I'm going to use a
mysterious third method another

00:20:18.000 --> 00:20:24.000
way.
I'm going to use the ODE.

00:20:20.000 --> 00:20:26.000
I'm going to do that because
this is the method.

00:20:24.000 --> 00:20:30.000
It's not just sines and cosines
which are orthogonal.

00:20:30.000 --> 00:20:36.000
There are masses of orthogonal
functions out there.

00:20:33.000 --> 00:20:39.000
And, the way they are
discovered, and the way you

00:20:36.000 --> 00:20:42.000
prove they're orthogonal is not
with trig identities and complex

00:20:40.000 --> 00:20:46.000
exponentials because those only
work with sines and cosines.

00:20:44.000 --> 00:20:50.000
It is, instead,
by going back to the

00:20:46.000 --> 00:20:52.000
differential equation that they
solve.

00:20:48.000 --> 00:20:54.000
And that's, therefore,
the method here that I'm going

00:20:52.000 --> 00:20:58.000
to use here because this is the
method which generalizes to many

00:20:56.000 --> 00:21:02.000
other differential equations
other than the simple ones

00:20:59.000 --> 00:21:05.000
satisfied by sines and cosines.
But anyway, that is the source.

00:21:05.000 --> 00:21:11.000
So, the way the proof of these
orthogonality conditions goes,

00:21:09.000 --> 00:21:15.000
so I'm not going to do that.
And, I'm going to assume that m

00:21:14.000 --> 00:21:20.000
is different from n so that I'm
not in either of these two

00:21:18.000 --> 00:21:24.000
cases.
What it depends on is,

00:21:20.000 --> 00:21:26.000
what's the differential
equation that all these

00:21:23.000 --> 00:21:29.000
functions satisfy?
Well, it's a different

00:21:26.000 --> 00:21:32.000
differential equation depending
upon the value of n,

00:21:30.000 --> 00:21:36.000
--
-- but they look at essentially

00:21:35.000 --> 00:21:41.000
the same.
These satisfy the differential

00:21:38.000 --> 00:21:44.000
equation, in other words,
what they have in common.

00:21:43.000 --> 00:21:49.000
The differential equation is,
let's call it u.

00:21:48.000 --> 00:21:54.000
It looks better.
It's going to look better if

00:21:52.000 --> 00:21:58.000
you let me call it u.
u double prime plus,

00:21:56.000 --> 00:22:02.000
well, n squared,
so for the function sine n t

00:22:00.000 --> 00:22:06.000
cosine n t, satisfy u double

00:22:05.000 --> 00:22:11.000
prime plus n squared times u.

00:22:11.000 --> 00:22:17.000
In other words,
the frequency is n,

00:22:13.000 --> 00:22:19.000
and therefore,
this is a square of the

00:22:16.000 --> 00:22:22.000
frequency is what you put here,
equals zero.

00:22:19.000 --> 00:22:25.000
In other words,
what these functions have in

00:22:22.000 --> 00:22:28.000
common is that they satisfy
differential equations that look

00:22:26.000 --> 00:22:32.000
like that.
And the only thing that's

00:22:28.000 --> 00:22:34.000
allowed to vary is the
frequency, which is allowed to

00:22:32.000 --> 00:22:38.000
change.
The frequency is in this

00:22:36.000 --> 00:22:42.000
coefficient of u.
Now, the remarkable thing is

00:22:42.000 --> 00:22:48.000
that's all you need to know.
The fact that they satisfy the

00:22:49.000 --> 00:22:55.000
differential equation,
that's all you need to know to

00:22:55.000 --> 00:23:01.000
prove the orthogonality
relationship.

00:22:59.000 --> 00:23:05.000
Okay, let's try to do it.
Well, I need some notation.

00:23:06.000 --> 00:23:12.000
So, I'm going to let un and vm
be any two of the functions.

00:23:11.000 --> 00:23:17.000
In other words,
I'll assume m is different from

00:23:16.000 --> 00:23:22.000
n.
For example,

00:23:17.000 --> 00:23:23.000
this one could be sine nt,
and that could be

00:23:22.000 --> 00:23:28.000
sine of mt,
or this could be sine nt

00:23:26.000 --> 00:23:32.000
and that could be
cosine of mt.

00:23:33.000 --> 00:23:39.000
You get the idea.
Any two of those in the

00:23:35.000 --> 00:23:41.000
subscript indicates whether what
the n or the m is that are in

00:23:40.000 --> 00:23:46.000
that.
Any two, and I mean really two,

00:23:42.000 --> 00:23:48.000
distinct, well,
if I say that m is not n,

00:23:45.000 --> 00:23:51.000
then they positively have to be
different.

00:23:48.000 --> 00:23:54.000
So, again, it's overkill with
my two's-ness.

00:23:51.000 --> 00:23:57.000
And, what I'm going to
calculate, well,

00:23:53.000 --> 00:23:59.000
first of all,
from the equation,

00:23:56.000 --> 00:24:02.000
I'm going to write the equation
this way.

00:24:00.000 --> 00:24:06.000
It says that u double prime is
equal to minus n squared u.

00:24:07.000 --> 00:24:13.000
That's true for any of these

00:24:11.000 --> 00:24:17.000
guys.
Of course, here,

00:24:13.000 --> 00:24:19.000
it would be v double prime is
equal to minus m squared

00:24:20.000 --> 00:24:26.000
times v.
You have to make those simple

00:24:26.000 --> 00:24:32.000
adjustments.
And now, what we're going to

00:24:30.000 --> 00:24:36.000
calculate is the integral from
minus pi to pi of un double

00:24:37.000 --> 00:24:43.000
prime times vm dt.

00:24:43.000 --> 00:24:49.000
Now, just bear with me.

00:24:48.000 --> 00:24:54.000
Why am I going to do that?
I can't explain what I'm going

00:24:53.000 --> 00:24:59.000
to do that.
But you won't ask me the

00:24:56.000 --> 00:25:02.000
question in five minutes.
But the point is,

00:24:59.000 --> 00:25:05.000
this is highly un-symmetric.
The u is differentiated twice.

00:25:05.000 --> 00:25:11.000
The v isn't.
So, those two functions-- but

00:25:08.000 --> 00:25:14.000
there is a way of turning them
into an expression which looks

00:25:12.000 --> 00:25:18.000
extremely symmetric,
where they are the same.

00:25:16.000 --> 00:25:22.000
And the way to do that is I
want to get rid of one of these

00:25:20.000 --> 00:25:26.000
primes here and put one on here.
The way to do that is if you

00:25:25.000 --> 00:25:31.000
want to integrate one of these
guys, and differentiate this one

00:25:29.000 --> 00:25:35.000
to make them look the same,
that's called integration by

00:25:33.000 --> 00:25:39.000
parts, the most important
theoretical method you learned

00:25:38.000 --> 00:25:44.000
in 18.01 even though you didn't
know that it was the most

00:25:42.000 --> 00:25:48.000
important theoretical method.
Okay, we're going to use it now

00:25:47.000 --> 00:25:53.000
as a basis for Fourier series.
Okay, so I'm going to integrate

00:25:51.000 --> 00:25:57.000
by parts.
Now, the first thing you do,

00:25:53.000 --> 00:25:59.000
of course, when you integrate
by parts is you just do the

00:25:56.000 --> 00:26:02.000
integration.
You don't do differentiation.

00:25:59.000 --> 00:26:05.000
So, the first thing looks like
this.

00:26:02.000 --> 00:26:08.000
And, that's to be evaluated
between negative pi and pi.

00:26:08.000 --> 00:26:14.000
In doing integration by parts
between limits,

00:26:12.000 --> 00:26:18.000
minus what you get by doing
both.

00:26:16.000 --> 00:26:22.000
You do both,
the integration and the

00:26:20.000 --> 00:26:26.000
differentiation.
And, again, evaluate that

00:26:24.000 --> 00:26:30.000
between limits.
Now, I'm just going to BS my

00:26:29.000 --> 00:26:35.000
way through this.
This is zero.

00:26:34.000 --> 00:26:40.000
I don't care what the un's,
which un you picked and which

00:26:39.000 --> 00:26:45.000
vm you picked.
The answer here is always going

00:26:43.000 --> 00:26:49.000
to be zero.
Instead of wasting six boards

00:26:46.000 --> 00:26:52.000
trying to write out the
argument, let me wave my hands.

00:26:51.000 --> 00:26:57.000
Okay, it's clear,
for example,

00:26:54.000 --> 00:27:00.000
that a v is a sine, sine mt.

00:26:57.000 --> 00:27:03.000
Of course it's zero because the
sine vanishes at both pi and

00:27:02.000 --> 00:27:08.000
minus pi.
If the un were a cosine,

00:27:06.000 --> 00:27:12.000
after I differentiate it,
it became a sine.

00:27:09.000 --> 00:27:15.000
And so, now it's this side guy
that's zero at both ends.

00:27:14.000 --> 00:27:20.000
So, the only case in which we
might have a little doubt is if

00:27:18.000 --> 00:27:24.000
this is a cosine,
and after differentiation,

00:27:21.000 --> 00:27:27.000
this is also a cosine.
In other words,

00:27:24.000 --> 00:27:30.000
it might look like cosine,
after, this cosine nt times

00:27:28.000 --> 00:27:34.000
cosine mt.
But, I claim that that's zero,

00:27:34.000 --> 00:27:40.000
too.
Why?

00:27:35.000 --> 00:27:41.000
Because the cosines are even
functions, and therefore,

00:27:39.000 --> 00:27:45.000
they have the same value at
both ends.

00:27:42.000 --> 00:27:48.000
So, if I subtract the value
evaluated at pi,

00:27:46.000 --> 00:27:52.000
and subtract the value of minus
pi, again zero because I have

00:27:51.000 --> 00:27:57.000
the same value at both ends.
So, by this entirely convincing

00:27:56.000 --> 00:28:02.000
argument, no matter what
combination of sines and cosines

00:28:00.000 --> 00:28:06.000
I have here, the answer to that
part will always be zero.

00:28:07.000 --> 00:28:13.000
So, by calculation,
but thought calculation;

00:28:11.000 --> 00:28:17.000
it's just a waste of time to
write anything out.

00:28:16.000 --> 00:28:22.000
You stare at it until you agree
that it's so.

00:28:20.000 --> 00:28:26.000
And now, I've taken,
by this integration by parts,

00:28:25.000 --> 00:28:31.000
I've taken this highly
un-symmetric expression and

00:28:30.000 --> 00:28:36.000
turned it into something in
which the u and the v are

00:28:35.000 --> 00:28:41.000
treated exactly alike.
Well, good, that's nice,

00:28:40.000 --> 00:28:46.000
but why?
Why did I go to this trouble?

00:28:43.000 --> 00:28:49.000
Okay, now we're going to use
the fact that this satisfies the

00:28:47.000 --> 00:28:53.000
differential equation,
in other words,

00:28:50.000 --> 00:28:56.000
that u double prime is equal to
minus n,

00:28:53.000 --> 00:28:59.000
I'm sorry, I should have
subscripted this.

00:28:56.000 --> 00:29:02.000
If that's the solution,
then this is equal to,

00:29:00.000 --> 00:29:06.000
times.
You have to put in a subscript

00:29:02.000 --> 00:29:08.000
otherwise.
The n wouldn't matter.

00:29:06.000 --> 00:29:12.000
All right, I'm now going to
take that expression,

00:29:10.000 --> 00:29:16.000
and evaluate it differently.
un double prime vm dt

00:29:15.000 --> 00:29:21.000
is equal to,
well, un double prime,

00:29:18.000 --> 00:29:24.000
because it satisfies the
differential equation is equal

00:29:22.000 --> 00:29:28.000
to that.
So, what is this?

00:29:25.000 --> 00:29:31.000
This is minus n squared
times the integral from

00:29:29.000 --> 00:29:35.000
negative pi to pi,
and I'm replacing un double

00:29:33.000 --> 00:29:39.000
prime by minus n
squared un.

00:29:39.000 --> 00:29:45.000
I pulled the minus n squared
out.

00:29:43.000 --> 00:29:49.000
So, it's un here,
and the other factor is vm dt.

00:29:47.000 --> 00:29:53.000
Now, that's the proof.
Huh?

00:29:50.000 --> 00:29:56.000
What do you mean that's the
proof?

00:29:54.000 --> 00:30:00.000
Okay, well, I'll first state
it, why intuitively that's the

00:29:59.000 --> 00:30:05.000
end of the argument.
And then, I'll spell it out a

00:30:06.000 --> 00:30:12.000
little more detail,
but the more detail you make

00:30:11.000 --> 00:30:17.000
for this, the more obscure it
gets instead of,

00:30:16.000 --> 00:30:22.000
look, I just showed you that
this is symmetric in u and v,

00:30:22.000 --> 00:30:28.000
after you massage it a little
bit.

00:30:26.000 --> 00:30:32.000
Here, I'm calculating it a
different way.

00:30:30.000 --> 00:30:36.000
Is this symmetric in u and v?
Well, the answer is yes or no.

00:30:37.000 --> 00:30:43.000
Is this symmetric at u and v?
No.

00:30:40.000 --> 00:30:46.000
Why?
Because of the n.

00:30:42.000 --> 00:30:48.000
The n favors u.
We have what is called a

00:30:46.000 --> 00:30:52.000
paradox.
This thing is symmetric in u

00:30:50.000 --> 00:30:56.000
and v because I can show it is.
And, it's not symmetric in u

00:30:55.000 --> 00:31:01.000
and v because I can show it is.
I can show it's not symmetric

00:31:01.000 --> 00:31:07.000
because it favors the n.
Now, there's only one possible

00:31:09.000 --> 00:31:15.000
resolution of that paradox.
Both would be symmetric if what

00:31:19.000 --> 00:31:25.000
were true?
Pardon?

00:31:22.000 --> 00:31:28.000
Negative pi.
All right, let me write it this

00:31:29.000 --> 00:31:35.000
way.
Okay, never mind.

00:31:32.000 --> 00:31:38.000
You see, the only way this can
happen is if this expression is

00:31:37.000 --> 00:31:43.000
zero.
In other words,

00:31:39.000 --> 00:31:45.000
the only way something can be
both symmetric and not symmetric

00:31:44.000 --> 00:31:50.000
is if it's zero all the time.
And, that's what we're trying

00:31:48.000 --> 00:31:54.000
to prove, that this is zero.
But, instead of doing it that

00:31:53.000 --> 00:31:59.000
way, let me show you.
This is equal to that,

00:31:57.000 --> 00:32:03.000
and therefore,
two things according to Euclid,

00:32:00.000 --> 00:32:06.000
two things equal to the same
thing are equal to each other.

00:32:07.000 --> 00:32:13.000
So, this equals that,
which, in turn,

00:32:09.000 --> 00:32:15.000
therefore, equals what I would
have gotten.

00:32:12.000 --> 00:32:18.000
I'm just saying the symmetry of
different way,

00:32:15.000 --> 00:32:21.000
what I would have gotten if I
had done this calculation.

00:32:19.000 --> 00:32:25.000
And, that turns out to be minus
m squared times the integral

00:32:23.000 --> 00:32:29.000
from minus pi to pi
of un vm dt.

00:32:28.000 --> 00:32:34.000
So, these two are equal because

00:32:33.000 --> 00:32:39.000
they are both equal to this.
This is equal to that.

00:32:38.000 --> 00:32:44.000
This equals that.
Therefore, how can this equal

00:32:42.000 --> 00:32:48.000
that unless the integral is
zero?

00:32:46.000 --> 00:32:52.000
How's that?
Remember, m is different from

00:32:50.000 --> 00:32:56.000
n.
So, what this proves is,

00:32:52.000 --> 00:32:58.000
therefore, the integral from
negative pi to pi of un vm dt is

00:32:59.000 --> 00:33:05.000
equal to zero,

00:33:05.000 --> 00:33:11.000
at least if m is different from
n.

00:33:10.000 --> 00:33:16.000
Now, there is one case I didn't
include.

00:33:12.000 --> 00:33:18.000
Which case didn't I include?
un times un is not supposed to

00:33:16.000 --> 00:33:22.000
be zero.
So, in that case,

00:33:18.000 --> 00:33:24.000
I don't have to worry about,
but there is a case that I

00:33:22.000 --> 00:33:28.000
didn't.
For example,

00:33:24.000 --> 00:33:30.000
something like the cosine of nt
times the sine of nt.

00:33:28.000 --> 00:33:34.000
Here, the m is the same as the

00:33:32.000 --> 00:33:38.000
n.
Nonetheless,

00:33:34.000 --> 00:33:40.000
I am claiming that this is zero
because these aren't the same

00:33:39.000 --> 00:33:45.000
function.
One is a cosine.

00:33:42.000 --> 00:33:48.000
Why is that zero?
Can you see mentally that

00:33:46.000 --> 00:33:52.000
that's zero?
Mentally?

00:33:48.000 --> 00:33:54.000
Well, this is trying to be in
another life,

00:33:52.000 --> 00:33:58.000
it's trying to be one half the
sine of two nt, right?

00:33:57.000 --> 00:34:03.000
And obviously the integral of

00:34:02.000 --> 00:34:08.000
sine of two nt is zero between
minus pi and pi

00:34:06.000 --> 00:34:12.000
because you integrate it,

00:34:09.000 --> 00:34:15.000
and it turns out to be zero.
You integrate it to a cosine,

00:34:13.000 --> 00:34:19.000
which has the same value of
both ends.

00:34:16.000 --> 00:34:22.000
Well, that was a lot of
talking.

00:34:18.000 --> 00:34:24.000
If this proof is too abstract
for you, I won't ask you to

00:34:22.000 --> 00:34:28.000
reproduce it on an exam.
You can go with the proofs

00:34:25.000 --> 00:34:31.000
using trigonometric identities,
and/or complex exponentials.

00:34:31.000 --> 00:34:37.000
But, you ought to know at least
one of those,

00:34:34.000 --> 00:34:40.000
and for the problem set I'm
asking you to fool around a

00:34:39.000 --> 00:34:45.000
little with at least two of
them.

00:34:41.000 --> 00:34:47.000
Okay, now, what has this got to
do with the problem we started

00:34:47.000 --> 00:34:53.000
with originally?
The problem is to explain this

00:34:50.000 --> 00:34:56.000
blue series.
So, our problem is,

00:34:53.000 --> 00:34:59.000
how, from this,
am I going to get the terms of

00:34:57.000 --> 00:35:03.000
this blue series?
So, given f of t,

00:35:02.000 --> 00:35:08.000
two pi s a period.
Find the an and the bn.

00:35:06.000 --> 00:35:12.000
Okay, let's focus on the an.
The bn is the same.

00:35:11.000 --> 00:35:17.000
Once you know how to do one,
you know how to do the other.

00:35:16.000 --> 00:35:22.000
So, here's the idea.
Again, it goes back to the

00:35:21.000 --> 00:35:27.000
something you learned at the
very beginning of 18.02,

00:35:26.000 --> 00:35:32.000
but I don't think it took.
But maybe some of you will

00:35:32.000 --> 00:35:38.000
recognize it.
So, what I'm going to do is

00:35:36.000 --> 00:35:42.000
write it.
Here's the term we're looking

00:35:40.000 --> 00:35:46.000
for here, this one.
Okay, and there are others.

00:35:45.000 --> 00:35:51.000
It's an infinite series that
goes on forever.

00:35:50.000 --> 00:35:56.000
And now, to make the argument,
I've got to put it one more

00:35:56.000 --> 00:36:02.000
term here.
So, I'm going to put in ak

00:36:00.000 --> 00:36:06.000
cosine kt.
I don't mean to imply that that

00:36:07.000 --> 00:36:13.000
k could be more than n,
in which case I should have

00:36:11.000 --> 00:36:17.000
written it here.
I could have also used equally

00:36:16.000 --> 00:36:22.000
well bk sine kt
here, and I could have put it

00:36:22.000 --> 00:36:28.000
there.
This is just some other term.

00:36:25.000 --> 00:36:31.000
This is the an,
and this is the one we want.

00:36:30.000 --> 00:36:36.000
And, this is some other term.
Okay, all right,

00:36:35.000 --> 00:36:41.000
now, what you do is,
to get the an,

00:36:38.000 --> 00:36:44.000
what you do is you multiply
everything through by,

00:36:42.000 --> 00:36:48.000
you focus on the one you want,
so it's dot,

00:36:46.000 --> 00:36:52.000
dot, dot, dot,
dot, and you multiply by cosine

00:36:50.000 --> 00:36:56.000
nt.
So, it's ak cosine kt times

00:36:54.000 --> 00:37:00.000
cosine nt.

00:36:57.000 --> 00:37:03.000
Of course, that gets
multiplied, too.

00:37:02.000 --> 00:37:08.000
But, the one we want also gets
multiplied, an.

00:37:06.000 --> 00:37:12.000
And, it becomes,
when I multiply by cosine nt,

00:37:11.000 --> 00:37:17.000
cosine squared nt,

00:37:16.000 --> 00:37:22.000
and now, I hope you can see
what's going to happen.

00:37:21.000 --> 00:37:27.000
Now, oops, I didn't multiply
the f of t,

00:37:26.000 --> 00:37:32.000
sorry.
It's the oldest trick in the

00:37:30.000 --> 00:37:36.000
book.
I now integrate everything from

00:37:35.000 --> 00:37:41.000
minus, so I don't endlessly
recopy.

00:37:38.000 --> 00:37:44.000
I'll integrate by putting it up
in yellow chalk,

00:37:42.000 --> 00:37:48.000
and you are left to your own
devices.

00:37:46.000 --> 00:37:52.000
This is definitely a colored
pen type of course.

00:37:50.000 --> 00:37:56.000
Okay, so, you want to integrate
from minus pi to pi?

00:37:55.000 --> 00:38:01.000
Good.
Just integrate everything on

00:37:59.000 --> 00:38:05.000
the right hand side,
also, from minus pi to pi.

00:38:05.000 --> 00:38:11.000
Plus, these are the guys just
to indicate that I haven't,

00:38:10.000 --> 00:38:16.000
they are out there,
too.

00:38:13.000 --> 00:38:19.000
And now, what happens?
What's this?

00:38:16.000 --> 00:38:22.000
Zero.
Every term is zero because of

00:38:20.000 --> 00:38:26.000
the orthogonality relations.
They are all of the form,

00:38:25.000 --> 00:38:31.000
a constant times cosine nt
times something different from

00:38:31.000 --> 00:38:37.000
cosine nt, sine kt,

00:38:35.000 --> 00:38:41.000
cosine kt,
or even that constant term.

00:38:42.000 --> 00:38:48.000
All of the other terms are
zero, and the only one which

00:38:46.000 --> 00:38:52.000
survives is this one.
And, what's its value?

00:38:50.000 --> 00:38:56.000
The integral from minus pi to
pi of cosine squared,

00:38:54.000 --> 00:39:00.000
I put that up somewhere.
It's right here,

00:38:57.000 --> 00:39:03.000
down there?
It is pi.

00:39:00.000 --> 00:39:06.000
So, this term turns into an pi,
an, dragged along,

00:39:04.000 --> 00:39:10.000
but this, the integral of the
square of the cosine turns out

00:39:10.000 --> 00:39:16.000
to be pi.
And so, the end result is that

00:39:14.000 --> 00:39:20.000
we get a formula for an.
What is an?

00:39:18.000 --> 00:39:24.000
an is, well,
an times pi,

00:39:20.000 --> 00:39:26.000
all these terms of zero,
and nothing is left but this

00:39:25.000 --> 00:39:31.000
left-hand side.
And therefore,

00:39:28.000 --> 00:39:34.000
an times pi is the integral
from negative pi to pi of f of t

00:39:34.000 --> 00:39:40.000
times cosine nt dt.

00:39:40.000 --> 00:39:46.000
But, that's an times pi.

00:39:45.000 --> 00:39:51.000
Therefore, if I want just an,
I have to divide it by pi.

00:39:50.000 --> 00:39:56.000
And, that's the formula for the
coefficient an.

00:39:54.000 --> 00:40:00.000
The argument is exactly the
same if you want bn,

00:39:57.000 --> 00:40:03.000
but I will write it down for
the sake of completeness,

00:40:02.000 --> 00:40:08.000
as they say,
and to give you a chance to

00:40:05.000 --> 00:40:11.000
digest what I've done,
you know, 30 seconds to digest

00:40:09.000 --> 00:40:15.000
it. Sine nt dt.

00:40:12.000 --> 00:40:18.000
And, that's because the
argument is the same.

00:40:16.000 --> 00:40:22.000
And, the integral of sine
squared nt is also

00:40:20.000 --> 00:40:26.000
pi. So, there's no difference

00:40:22.000 --> 00:40:28.000
there.
Now, there's only one little

00:40:24.000 --> 00:40:30.000
caution.
It have to be a little careful.

00:40:27.000 --> 00:40:33.000
This is n one,
two, and so on,

00:40:29.000 --> 00:40:35.000
and this is also n one,
two, and unfortunately,

00:40:33.000 --> 00:40:39.000
the constant term is a slight
exception.

00:40:35.000 --> 00:40:41.000
We better look at that
specifically because if you

00:40:39.000 --> 00:40:45.000
forget it, you can get them to
gross, gross,

00:40:42.000 --> 00:40:48.000
gross errors.
How about the constant term?

00:40:48.000 --> 00:40:54.000
Suppose I repeat the argument
for that in miniature.

00:40:54.000 --> 00:41:00.000
There is a constant term plus
other stuff, a typical other

00:41:01.000 --> 00:41:07.000
stuff, an cosine,
let's say.

00:41:06.000 --> 00:41:12.000
How am I going to get that
constant term?

00:41:10.000 --> 00:41:16.000
Well, if you think of this as
sort of like a constant times,

00:41:16.000 --> 00:41:22.000
the reason is the constant is
because it's being multiplied by

00:41:22.000 --> 00:41:28.000
cosine zero t.
So, that suggests I should

00:41:27.000 --> 00:41:33.000
multiply by one.
In other words,

00:41:31.000 --> 00:41:37.000
what I should do is simply
integrate this from negative pi

00:41:36.000 --> 00:41:42.000
to pi, f of t dt.

00:41:40.000 --> 00:41:46.000
What's the answer?
Well, this integrated from

00:41:44.000 --> 00:41:50.000
minus pi to pi is how much?
It's c zero times two pi,

00:41:49.000 --> 00:41:55.000
right?
And, the other terms all give

00:41:52.000 --> 00:41:58.000
me zero.
Every other term is zero

00:41:55.000 --> 00:42:01.000
because if you integrate cosine
nt or sine nt

00:42:00.000 --> 00:42:06.000
over a complete
period, you always get zero.

00:42:06.000 --> 00:42:12.000
There is as much area above the
axis or below.

00:42:10.000 --> 00:42:16.000
Or, you can look at two special
cases.

00:42:13.000 --> 00:42:19.000
Anyway, you always get zero.
It's the same thing with sine

00:42:18.000 --> 00:42:24.000
here.
So, the answer is that c zero

00:42:21.000 --> 00:42:27.000
is equal to,
is a little special.

00:42:24.000 --> 00:42:30.000
You don't just put n equals
zero here because then

00:42:30.000 --> 00:42:36.000
you would lose a factor of two.
So, c zero should be one

00:42:36.000 --> 00:42:42.000
over two pi times
this integral.

00:42:40.000 --> 00:42:46.000
Now, there are two kinds of
people in the world,

00:42:44.000 --> 00:42:50.000
the ones who learn two separate
formulas, and the ones who just

00:42:50.000 --> 00:42:56.000
learn two separate notations.
So, what most people do is they

00:42:55.000 --> 00:43:01.000
say, look, I want this to be
always the formula for a zero.

00:43:02.000 --> 00:43:08.000
That means, even when n
is zero, I want this to be the

00:43:07.000 --> 00:43:13.000
formula.
Well, then you are not going to

00:43:10.000 --> 00:43:16.000
get the right leading term.
Instead of getting c zero,

00:43:14.000 --> 00:43:20.000
you're going to get
twice it, and therefore,

00:43:18.000 --> 00:43:24.000
the formula is,
the Fourier series,

00:43:21.000 --> 00:43:27.000
therefore, isn't written this
way.

00:43:24.000 --> 00:43:30.000
It's written-- If you want an a
zero there,

00:43:28.000 --> 00:43:34.000
calculate it by this formula.
Then, you've got to write not c

00:43:34.000 --> 00:43:40.000
zero, but a zero over two.

00:43:37.000 --> 00:43:43.000
I think you will be happiest if
I have to give you advice.

00:43:41.000 --> 00:43:47.000
I think you'll be happiest
remembering a single formula for

00:43:45.000 --> 00:43:51.000
the an's and bn's,
in which case you have to

00:43:48.000 --> 00:43:54.000
remember that the constant
leading term is a zero over two

00:43:52.000 --> 00:43:58.000
if you insist on
using that formula.

00:43:55.000 --> 00:44:01.000
Otherwise, you have to learn a
special formula for the leading

00:43:59.000 --> 00:44:05.000
coefficient, namely one over two
pi instead of one

00:44:03.000 --> 00:44:09.000
over pi.
Well, am I really going to

00:44:08.000 --> 00:44:14.000
calculate a Fourier series in
four minutes?

00:44:11.000 --> 00:44:17.000
Not very likely,
but I'll give it a brave

00:44:14.000 --> 00:44:20.000
college try.
Anyway, you will be doing a

00:44:17.000 --> 00:44:23.000
great deal of it,
and your book has lots and lots

00:44:21.000 --> 00:44:27.000
of examples, too many,
in fact.

00:44:23.000 --> 00:44:29.000
It ruined all the good examples
by calculating them for you.

00:44:28.000 --> 00:44:34.000
But, I will at least outline.
Do you want me to spend three

00:44:34.000 --> 00:44:40.000
minutes outlining a calculation
just so you have something to

00:44:38.000 --> 00:44:44.000
work on in the next boring class
you are in?

00:44:42.000 --> 00:44:48.000
Let's see, so I'll just put a
few key things on the board.

00:44:46.000 --> 00:44:52.000
I would advise you to sit still
for this.

00:44:49.000 --> 00:44:55.000
Otherwise you're going to hack
it, and take twice as long as

00:44:54.000 --> 00:45:00.000
you should, even though I knew
you've been up to 3:00 in the

00:44:58.000 --> 00:45:04.000
morning doing your problem set.
Cheer up.

00:45:03.000 --> 00:45:09.000
I got up at 6:00 to make up the
new one.

00:45:08.000 --> 00:45:14.000
So, we're even.
This should be zero here.

00:45:13.000 --> 00:45:19.000
So, here's minus pi.
Here's pi.

00:45:17.000 --> 00:45:23.000
Here's one, negative one.
The function starts out like

00:45:24.000 --> 00:45:30.000
that, and now to be periodic,
it then has to continue on in

00:45:31.000 --> 00:45:37.000
the same way.
So, I think that's enough of

00:45:37.000 --> 00:45:43.000
its path through life to
indicate how it runs.

00:45:42.000 --> 00:45:48.000
This is a typical square-away
function, sometimes it's called.

00:45:48.000 --> 00:45:54.000
It's an odd function.
It goes equally above and below

00:45:53.000 --> 00:45:59.000
the axis.
Now, the integrals,

00:45:56.000 --> 00:46:02.000
when you calculate them,
the an is going to be,

00:46:00.000 --> 00:46:06.000
okay, look, the an is going to
turn out to be zero.

00:46:08.000 --> 00:46:14.000
Let me, instead,
and you will get that with a

00:46:11.000 --> 00:46:17.000
little hacking.
I'm much more worried about

00:46:14.000 --> 00:46:20.000
what you'll do with the bn's.
Also, next Monday you'll see

00:46:17.000 --> 00:46:23.000
intuitively that the an is zero,
in which case you won't even

00:46:22.000 --> 00:46:28.000
bother trying to calculate it.
How about the bn,

00:46:25.000 --> 00:46:31.000
though?
Well, you see,

00:46:26.000 --> 00:46:32.000
because the function is
discontinuous,

00:46:29.000 --> 00:46:35.000
so, this is my input.
My f of t is that

00:46:32.000 --> 00:46:38.000
orange discontinuous function.
The bn is going to be,

00:46:37.000 --> 00:46:43.000
I have to break it into two
parts.

00:46:40.000 --> 00:46:46.000
In the first part,
the function is negative one.

00:46:43.000 --> 00:46:49.000
And there, I will be
integrating from minus pi to pi

00:46:47.000 --> 00:46:53.000
of the function,
which is minus one times the

00:46:50.000 --> 00:46:56.000
sine of nt dt.

00:46:54.000 --> 00:47:00.000
And then, there's another part,

00:46:57.000 --> 00:47:03.000
sorry, minus pi to zero.
The other part I integrate from

00:47:02.000 --> 00:47:08.000
zero to pi of what?
Well, f of t is now plus one.

00:47:06.000 --> 00:47:12.000
And so, I simply integrate sine

00:47:10.000 --> 00:47:16.000
nt dt.
Now, each of these is a

00:47:14.000 --> 00:47:20.000
perfectly simple integral.
The only question is how you

00:47:19.000 --> 00:47:25.000
combine them.
So, this is,

00:47:21.000 --> 00:47:27.000
after you calculate it,
it will be (one minus cosine n

00:47:26.000 --> 00:47:32.000
pi) all over n.

00:47:29.000 --> 00:47:35.000
And, this part will turn out to
be (one minus cosine n pi) over

00:47:34.000 --> 00:47:40.000
n also. And therefore,

00:47:40.000 --> 00:47:46.000
the answer will be two minus
two cosine, two over n times,

00:47:48.000 --> 00:47:54.000
right, two minus,
two times (one minus cosine n

00:47:55.000 --> 00:48:01.000
pi) over n.

00:48:01.000 --> 00:48:07.000
No, okay, now,
what's this?

00:48:03.000 --> 00:48:09.000
This is minus one if n is odd.
It's plus one if n is even.

00:48:09.000 --> 00:48:15.000
Now, either you can work with
it this way, or you can combine

00:48:15.000 --> 00:48:21.000
the two of them into a single
expression.

00:48:19.000 --> 00:48:25.000
Its minus one to the nth power
takes care of both of

00:48:26.000 --> 00:48:32.000
them.
But, the way the answer is

00:48:29.000 --> 00:48:35.000
normally expressed,
it would be minus two over n,

00:48:34.000 --> 00:48:40.000
two over n times,
if n is even,

00:48:37.000 --> 00:48:43.000
I get zero.
If n is odd,

00:48:41.000 --> 00:48:47.000
I get two.
So, times two,

00:48:43.000 --> 00:48:49.000
if n is odd,
and zero if n is even.

00:48:46.000 --> 00:48:52.000
So, it's four over n,
or it's zero,

00:48:50.000 --> 00:48:56.000
and the final series is a sum
of those coefficients times the

00:48:55.000 --> 00:49:01.000
appropriate-- cosine or sine?
Sine terms because the cosine

00:49:01.000 --> 00:49:07.000
terms were all coefficients,
all turned out to be zero.

00:49:08.000 --> 00:49:14.000
I'm sorry I didn't have the
chance to do that calculation in

00:49:13.000 --> 00:49:19.000
detail.
But, I think that's enough

00:49:16.000 --> 00:49:22.000
sketch for you to be able to do
the rest of it yourself.