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
00:14:48.000 --> 00:14:54.000
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
00:15:06.000 --> 00:15:12.000
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
00:16:04.000 --> 00:16:10.000
not different?
Well, you know,
00:16:08.000 --> 00:16:14.000
you talk about two coincident
roots.
00:16:12.000 --> 00:16:18.000
I'm just killing,
doing a little overkill.
00:16:16.000 --> 00:16:22.000
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,
00:16:25.000 --> 00:16:31.000
there's no way to say that,
any two distinct ones are
00:16:31.000 --> 00:16:37.000
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
00:16:43.000 --> 00:16:49.000
category that I'm talking about,
but any two distinct ones are
00:16:47.000 --> 00:16:53.000
orthogonal on the interval for
minus pi to pi.
00:16:51.000 --> 00:16:57.000
So, if I integrate from minus
00:16:53.000 --> 00:16:59.000
pi to pi sine of three t times
cosine of four t dt,
00:16:57.000 --> 00:17:03.000
answer is zero.
00:17:00.000 --> 00:17:06.000
If I integrate sine of 3t times
00:17:05.000 --> 00:17:11.000
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.
00:17:13.000 --> 00:17:19.000
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.
00:17:51.000 --> 00:17:57.000
And the theorem is you always
do.
00:17:54.000 --> 00:18:00.000
Okay, so, why is this?
Well, there are three ways to
00:18:00.000 --> 00:18:06.000
prove this.
It's like many fundamental
00:18:03.000 --> 00:18:09.000
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
00:18:10.000 --> 00:18:16.000
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,
00:18:18.000 --> 00:18:24.000
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.
00:18:34.000 --> 00:18:40.000
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.
00:18:47.000 --> 00:18:53.000
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.
00:18:58.000 --> 00:19:04.000
And, the answer to either one
00:19:02.000 --> 00:19:08.000
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.
00:19:19.000 --> 00:19:25.000
Anyway, so I'm not going to
calculate this out.
00:19:23.000 --> 00:19:29.000
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.