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AUDIENCE: OK.
 

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I hoped I might have Exam 2
for you today, but it's not

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quite back from the grader.
 

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00:00:28 --> 00:00:33
It's already gone to the second
grader, so it will not be long.

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And I hope you've had a look
at the MATLAB homework for

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a variety of possible.
 

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I think we've got, there were
some errors in the original

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statement, location of the
coordinates, but I think

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they're fixed now.
 

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So ready to go on that MATLAB.
 

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Don't forget that it's four on
the right-hand side and not

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one, so if you get an answer
near 1/4 at the center of the

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circle, that's the reason.
 

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Just that factor four
is to remember.

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I'll talk more about the
MATLAB this afternoon in the

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review session right here.
 

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Just to say, I'm highly
interested in that problem.

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Not just increasing N, the
number of mesh points in the

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octagon, but also increasing
the number of sides.

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So there are two numbers there,
we had N points on a ray,

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out from the center.
 

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But we have M sides
of the polygon.

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And I'm interested in both
of those, getting big.

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Growing.
 

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I don't know how.
 

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And maybe a reasonable balance
is to take, I think N

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proportional to M is a
pretty good balance.

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So I'd be very happy;
I mean I'm very happy

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with whatever you do.
 

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But I'm really interested
to know what happens as

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both of these increase.
 

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How close, how quickly do you
approach the eigenvalues

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of a circle.
 

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And you might keep the
two proportional as

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you increase them.
 

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So let me say more about that
this afternoon, because it's a

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big day today, to
start Fourier.

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Fourier series, the new
chapter, the new topic.

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In fact, the final major
topic of the course.

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So I tried to list here, so
here I'm in Section 4.1, so I'm

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talking about Fourier series.
 

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So Fourier series is for
functions that have period 2pi.

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It involves things like sin(x),
like cos(x) like e^(ikx), all

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of those if I increase x by
2pi, I'm back where I started.

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So that's the sort of functions
that have Fourier series.

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Then we'll go on to the other
two big forms, crucial

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forms of the Fourier world.
 

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But 4.1 starts with the
classical Fourier series.

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So I realize, you will have
seen, many of you will have

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seen Fourier series before.
 

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I hope you'll see some
new aspects here.

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So, let me just get organized.
 

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It's nice to have some examples
that just involve sine.

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And since the sine is an odd
function, that means it's sort

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of anti-symmetric across zero,
those are the functions

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that will have only sine.
 

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That will have a
sine expansion.

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Cosines are the opposite.
 

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Cosines are symmetric
across zero.

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Like a constant,
or like cos(x).

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Zero comes right at
the symmetric point.

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So those will have
only cosines.

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And a lot of examples fit in
one or the other of those,

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and it's easy to see them.
 

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The general function, of
course, is a combination

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odd and even.
 

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It has cosines and it has
sines, it's just the

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some of the two pieces.
 

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So, this is the standard
Fourier series, which I

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couldn't get onto one line,
but it has all the cosines

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including this slightly
different cos(0),

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and all the sines.
 

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00:04:51 --> 00:04:57
But because this one has these
three different pieces, the

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constant term, the other
cosines, all the sines, three

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slightly different formulas,
it's actually nicest of all,

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to use this final form.
 

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Because there's
just one formula.

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There's just one kind.
 

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And I'll call its coefficient
c_k, and now they multiply

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e^(ikx), so we have to
get used to e^(ikx).

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We may be more familiar with
cos, sin(kx) and cos(kx),

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but everybody knows e^(ikx)
is a combination of them.

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00:05:34 --> 00:05:38
And if we let k go from minus
infinity to infinity, so

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we've got all the terms.
 

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Including e^(-i3x), and
e^(+i3x), those would

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combine to give cosines
and sines of 3x.

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We get one nice formula.
 

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There's just one
formula for the C's.

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00:05:57 --> 00:06:01
So that's one good reason to
look at the complex form.

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Even if our function
is actually real.

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That form is kind of neat, and
the second good reason, the

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really important reason, is
then when we go to the discrete

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Fourier transform, the DFT,
everybody writes that

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with complex numbers.
 

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So it's good to see complex
numbers first and then we

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can just translate
the formulas from.

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00:06:30 --> 00:06:33
And these are also almost
always written with

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complex numbers.
 

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So this is the way to see it.
 

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OK, so what do we do
about Fourier series?

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What do we have to know
how to do and what

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00:06:46 --> 00:06:47
should we understand?
 

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00:06:47 --> 00:06:53
Well, if you've met Fourier
series you may have met the

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formula for these coefficients.
 

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00:06:56 --> 00:06:58
That's sort of like step one.
 

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00:06:58 --> 00:07:01
If I'm given the function,
whatever the function might be,

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00:07:01 --> 00:07:02
might be a delta function.
 

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Interesting case, always.
 

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00:07:04 --> 00:07:07
Always interesting.
 

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00:07:07 --> 00:07:08
Always crazy right?
 

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00:07:08 --> 00:07:13
But it's always interesting,
the delta function.

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The coefficients
can be computed.

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The coefficients, you'll see,
I'll repeat those formulas.

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They involve integrals.
 

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What I want to say right
now is that this isn't a

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course in integration.
 

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So I'm not interested in doing
more and more complicated

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integrals and finding
Fourier coefficients

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of weird functions.
 

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No way.
 

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I want to understand the
simple, straight, the

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00:07:45 --> 00:07:47
important examples.
 

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And here's a point that's
highly interesting.

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In practice, in computing
practice, we're close to

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00:07:56 --> 00:07:57
computing practice here.
 

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00:07:57 --> 00:07:59
In everything we do.
 

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00:07:59 --> 00:08:03
I mean, this is really
constantly used.

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00:08:03 --> 00:08:07
And one important question
is, is the Fourier series

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00:08:07 --> 00:08:09
quickly convergent?
 

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00:08:09 --> 00:08:11
Because if we're going to
compute, we don't want to

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compute a thousand terms.
 

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00:08:14 --> 00:08:19
Hopefully ten terms, 20 terms
would give us good accuracy.

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00:08:19 --> 00:08:24
So that question comes down to
how quickly does those a's

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00:08:24 --> 00:08:27
and b's and c's go to zero?
 

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00:08:27 --> 00:08:28
That's highly important.
 

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00:08:28 --> 00:08:32
And you'll connect this decay
rate, we'll connect this with

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00:08:32 --> 00:08:34
the smoothness of the function.
 

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00:08:34 --> 00:08:38
Oh, I can tell you
even at a start.

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00:08:38 --> 00:08:42
OK, so I just want to
emphasize this point.

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00:08:42 --> 00:08:50
We'll see it over and over that
like for a delta function,

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00:08:50 --> 00:08:56
which is not smooth at all,
we'll see no decay at all.

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In the coefficients.
 

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They're constant.
 

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00:09:03 --> 00:09:06
They don't decrease as we go to
higher and higher frequencies.

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00:09:06 --> 00:09:13
I think of k here, I'll use
the word frequency for k.

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So high frequency means high k,
far off the Fourier series, and

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the question is, are the
coefficients staying up there

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00:09:22 --> 00:09:24
big, and we have to
worry about them.

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Or do they get very small?
 

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So a delta function
is a key example and

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00:09:29 --> 00:09:32
then a step function.
 

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So what will be the
deal with those?

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If I have a function that's
a step function, I'll have

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decay at rate is 1/k.
 

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00:09:41 --> 00:09:44
 
 

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So they do go to zero.
 

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The thousandth coefficient will
be roughly of size 1/1000.

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That's not fast.
 

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00:09:54 --> 00:10:02
That's not really fast
enough to compute with.

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00:10:02 --> 00:10:08
Well, we meet step functions,
I mean, functions with jumps.

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00:10:08 --> 00:10:11
And we'll see that their
Fourier series, the

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coefficients do go to
zero but not very fast.

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And we get something
highly interesting.

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So when we do these examples,
so I've sort of moved on to

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examples, so these are
two basic examples.

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What would be the next example?
 

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00:10:30 --> 00:10:32
Step function.
 

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Well, yeah, or
maybe a hat next.

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A hat function would
be, you see what I'm

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doing at each step?
 

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I'm integrating.
 

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A hat function might be the
next, yeah, a ramp, exactly.

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00:10:44 --> 00:10:48
Hat function, which is
a ramp with a corner.

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Now, so that's one
integral better.

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You want to guess the decay
rate on that one? k squared.

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00:10:55 --> 00:10:58
Now we're getting better.
 

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That's a faster follow-up.
 

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00:11:00 --> 00:11:00
1/k^2.
 

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00:11:02 --> 00:11:04
And then we integrate
again, we'd get 1/k^3.

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Then one more integral, 1/k^4
would be a cubic spline with,

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you remember the cubic
spline is continuous.

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Its derivative is continuous,
that gives us a 1/k^3.

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Its second derivative is
continuous, that gives us a

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1/k^4, and then you really can
compute with that, if you

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00:11:25 --> 00:11:27
have such a function.
 

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So, point, pay attention
to decay rate.

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That, and the connection
to smoothness.

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So examples, we'll start
right off with these guys.

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And then we'll see the
rules for the derivative.

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Oh yeah, rules for
the derivative.

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The beauty of Fourier
series is, well, actually

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00:11:53 --> 00:11:54
you can see this.
 

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You can see the rule.
 

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Let me just show you
the rule for this.

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So the rule for derivatives,
the whole point about

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Fourier is, it connects
perfectly with calculus.

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With taking derivatives.
 

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So suppose I have F(x) equals,
I'll use this form, the

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00:12:16 --> 00:12:18
sum of c_k*e^(ikx).
 

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00:12:18 --> 00:12:22
 
 

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And now I take its
derivative. dF/dx.

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00:12:25 --> 00:12:29
What do you think is the
derivative, what's the Fourier

211
00:12:29 --> 00:12:33
series for the derivative?
 

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Suppose I have the Fourier
series for some function, and

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00:12:36 --> 00:12:38
then I take Fourier series
for the derivative.

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So I'm kind of going
the backwards way.

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Less smooth.
 

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00:12:43 --> 00:12:48
I'm going from, the derivative
of the step function involves

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delta functions, so I'm
going less smooth as

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I take derivatives.
 

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It's so easy, it jumps at you.
 

220
00:13:00 --> 00:13:01
What's the rule?
 

221
00:13:01 --> 00:13:05
Just take the derivative of
every term, so I'll have the

222
00:13:05 --> 00:13:10
sum of, now what happens
when I take the derivative?

223
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Everybody see what happens when
I take the derivative of that

224
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typical term in the
Fourier series?

225
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What happens?
 

226
00:13:19 --> 00:13:22
The derivative brings
down a factor, ik.

227
00:13:23 --> 00:13:32
With k being the thing that, so
it's ik times what we have.

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So these are the Fourier
coefficients of the derivative.

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00:13:38 --> 00:13:42
And that again makes exactly
the same point about the

230
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decay rate or the opposite,
the non decay rate.

231
00:13:46 --> 00:13:50
As I take the derivative you
got a rougher function, right?

232
00:13:50 --> 00:13:54
Derivative of a step function
is a delta, derivative of a

233
00:13:54 --> 00:13:57
hat would have some steps.
 

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00:13:57 --> 00:14:02
We're going less smooth as
we take more derivatives.

235
00:14:02 --> 00:14:06
And every time we do it,
we see, you understand

236
00:14:06 --> 00:14:07
the decay rate now?
 

237
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Because the derivative just
brings a factor ik, so its high

238
00:14:14 --> 00:14:18
frequencies are more present.
 

239
00:14:18 --> 00:14:20
Have larger coefficients.
 

240
00:14:20 --> 00:14:22
So and of course, the
second derivative would

241
00:14:22 --> 00:14:24
bring down (ik)^2.
 

242
00:14:24 --> 00:14:27
 
 

243
00:14:27 --> 00:14:35
So that our equations, for
example, let me just do

244
00:14:35 --> 00:14:38
an application here.
 

245
00:14:38 --> 00:14:41
Without pushing it.
 

246
00:14:41 --> 00:14:45
Our application, we started
this course with equations

247
00:14:45 --> 00:14:46
like -u''(x)=delta(x-a).
 

248
00:14:46 --> 00:14:51
 
 

249
00:14:51 --> 00:14:52
Right?
 

250
00:14:52 --> 00:14:55
If we wanted to apply to
a differential equation,

251
00:14:55 --> 00:14:56
how would I do it?
 

252
00:14:56 --> 00:15:00
I would take the Fourier
series of both sides.

253
00:15:00 --> 00:15:03
I would look at, I'd jump
into what people would

254
00:15:03 --> 00:15:05
call the frequency domain.
 

255
00:15:05 --> 00:15:10
So this is a differential
equation written as usual

256
00:15:10 --> 00:15:13
in the physical domain.
 

257
00:15:13 --> 00:15:17
And with physical
variable x position.

258
00:15:17 --> 00:15:18
Or it could be time.
 

259
00:15:18 --> 00:15:22
And now let me take
Fourier transforms.

260
00:15:22 --> 00:15:23
So what would happen here?
 

261
00:15:23 --> 00:15:27
If I take the Fourier transform
of this, well, we'll

262
00:15:27 --> 00:15:30
soon see, right?
 

263
00:15:30 --> 00:15:33
We get Fourier coefficients
of the deltas.

264
00:15:33 --> 00:15:34
Of the delta function.
 

265
00:15:34 --> 00:15:38
That's a key example,
and you see why.

266
00:15:38 --> 00:15:40
Over here, what will we get?
 

267
00:15:40 --> 00:15:43
And now I'm taking two
derivatives, so I

268
00:15:43 --> 00:15:45
bring down ik twice.
 

269
00:15:45 --> 00:15:46
So I'm looking.
 

270
00:15:46 --> 00:15:51
Here it would be the sum
of whatever the delta's

271
00:15:51 --> 00:15:52
coefficients are.
 

272
00:15:52 --> 00:15:54
Shall we call those d?
 

273
00:15:54 --> 00:15:59
The alphabet's coming
out right. d for delta.

274
00:15:59 --> 00:16:03
So the right side has
coefficients, d_k.

275
00:16:03 --> 00:16:05
And what about the left side?
 

276
00:16:05 --> 00:16:10
What are the coefficients if
the solution u has coefficients

277
00:16:10 --> 00:16:15
c_k, so let's call this u now.
 

278
00:16:15 --> 00:16:18
Has coefficients c_k, then
what happens to the second

279
00:16:18 --> 00:16:23
derivative? ik, ik again,
that's i squared k

280
00:16:23 --> 00:16:25
squared, the minus sign.
 

281
00:16:25 --> 00:16:29
So we would have the sum
of k squared c_k*e^(ikx).

282
00:16:29 --> 00:16:33
 
 

283
00:16:33 --> 00:16:37
This is if u itself has
coefficient c_k, then -u''

284
00:16:37 --> 00:16:39
has these coefficients.
 

285
00:16:39 --> 00:16:41
So what's up?
 

286
00:16:41 --> 00:16:43
How would we use that?
 

287
00:16:43 --> 00:16:44
It's going to be easy.
 

288
00:16:44 --> 00:16:48
We'll just match terms.
 

289
00:16:48 --> 00:16:49
Right?
 

290
00:16:49 --> 00:16:52
I can see, what's my
formula, what should c_k

291
00:16:52 --> 00:16:54
be if I know the d_k?
 

292
00:16:54 --> 00:16:58
I'm given the right-hand side.
 

293
00:16:58 --> 00:17:01
We're just doing what's
constantly happening,

294
00:17:01 --> 00:17:03
this three step process.
 

295
00:17:03 --> 00:17:04
You're given the right side.
 

296
00:17:04 --> 00:17:09
Step one, expand it in
Fourier series now.

297
00:17:09 --> 00:17:13
Step two, match the two sides.
 

298
00:17:13 --> 00:17:14
So what's the formula for c_k?
 

299
00:17:16 --> 00:17:19
In this application, which
by the way I had no

300
00:17:19 --> 00:17:20
intention to do this.
 

301
00:17:20 --> 00:17:24
But it jumped into my head and
I thought why not just do it.

302
00:17:24 --> 00:17:28
What would be the
formula for c_k?

303
00:17:30 --> 00:17:35
It'll be d_k divided
by? k squared.

304
00:17:35 --> 00:17:37
You're just matching terms.
 

305
00:17:37 --> 00:17:43
Just the way, when we expanded
things in eigenvectors, we'd

306
00:17:43 --> 00:17:46
match the coefficients of the
eigenvectors, and that involved

307
00:17:46 --> 00:17:52
just the simple step, here
it's d_k over k squared.

308
00:17:52 --> 00:17:53
Good.
 

309
00:17:53 --> 00:17:55
And then what's the final step?
 

310
00:17:55 --> 00:17:59
The final step is, now you
know the right coefficients,

311
00:17:59 --> 00:18:01
add them back up.
 

312
00:18:01 --> 00:18:03
Add the thing back
up, like here.

313
00:18:03 --> 00:18:10
Only I'm temporarily calling
it u, to find the solution.

314
00:18:10 --> 00:18:11
Right?
 

315
00:18:11 --> 00:18:13
Three steps.
 

316
00:18:13 --> 00:18:16
Go into the frequency domain.
 

317
00:18:16 --> 00:18:21
Write the right-hand side
as a Fourier series.

318
00:18:21 --> 00:18:28
Second quick step is look at
the equation for each separate

319
00:18:28 --> 00:18:30
Fourier coefficient.
 

320
00:18:30 --> 00:18:34
Match the coefficients
of these eigenvectors.

321
00:18:34 --> 00:18:35
Eigenfunctions.
 

322
00:18:35 --> 00:18:38
And that's this
quick middle step.

323
00:18:38 --> 00:18:42
And then you've got the answer,
but you're still in Fourier

324
00:18:42 --> 00:18:44
space, you're still
in frequency space.

325
00:18:44 --> 00:18:48
So you have to use these,
put them back to get the

326
00:18:48 --> 00:18:51
answer in physical space.
 

327
00:18:51 --> 00:18:51
Right?
 

328
00:18:51 --> 00:18:53
That's the pattern.
 

329
00:18:53 --> 00:18:54
Over and over.
 

330
00:18:54 --> 00:18:59
So that's sort of the general
plan of applying Fourier.

331
00:18:59 --> 00:19:02
And when does it work?
 

332
00:19:02 --> 00:19:03
When does it work?
 

333
00:19:03 --> 00:19:07
Because, I mean it's
fantastic when it works.

334
00:19:07 --> 00:19:13
So what is it about this
problem that made it work?

335
00:19:13 --> 00:19:15
What is Fourier happy?
 

336
00:19:15 --> 00:19:18
You know, when does he raise
his hand, say yes I can

337
00:19:18 --> 00:19:20
solve that problem?
 

338
00:19:20 --> 00:19:26
OK, what do I need here
for this plan to work?

339
00:19:26 --> 00:19:29
I certainly don't need always
just -u'', Fourier could

340
00:19:29 --> 00:19:31
do better than that.
 

341
00:19:31 --> 00:19:37
But what's the requirement for
Fourier to work perfectly?

342
00:19:37 --> 00:19:40
Well, linear equation, right?
 

343
00:19:40 --> 00:19:42
If we didn't have linear
equations we couldn't do

344
00:19:42 --> 00:19:45
all this adding and
matching and stuff.

345
00:19:45 --> 00:19:47
So linear equations.
 

346
00:19:47 --> 00:19:51
Well, OK.
 

347
00:19:51 --> 00:19:54
Now, what other
linear equations?

348
00:19:54 --> 00:19:56
Could I have a c(x) in here?
 

349
00:19:56 --> 00:20:01
My familiar c(x), variable
material property

350
00:20:01 --> 00:20:03
inside this equation?
 

351
00:20:03 --> 00:20:04
No.
 

352
00:20:04 --> 00:20:05
Well, not easily, anyway.
 

353
00:20:05 --> 00:20:09
That would really mess things
up if there's a variable

354
00:20:09 --> 00:20:14
coefficient in here then
it's going to have its

355
00:20:14 --> 00:20:15
own Fourier series.
 

356
00:20:15 --> 00:20:18
We're going to be
multiplying Fourier series.

357
00:20:18 --> 00:20:22
That comes later and
it's not so clean.

358
00:20:22 --> 00:20:25
So we want, it works
perfectly when it's

359
00:20:25 --> 00:20:28
constant coefficients.
 

360
00:20:28 --> 00:20:33
Constant coefficients in the
differential equations.

361
00:20:33 --> 00:20:36
And then one more thing.
 

362
00:20:36 --> 00:20:38
Very important other thing.
 

363
00:20:38 --> 00:20:39
The boundary conditions.
 

364
00:20:39 --> 00:20:42
Everybody remembers now, it's a
part of the message of this

365
00:20:42 --> 00:20:47
course is that boundary
conditions are often

366
00:20:47 --> 00:20:48
a source of trouble.
 

367
00:20:48 --> 00:20:51
They're part of the problem,
you have to deal with them.

368
00:20:51 --> 00:20:56
Now, what boundary conditions
do we think about here?

369
00:20:56 --> 00:21:02
Well, fixed-fixed was
where we started.

370
00:21:02 --> 00:21:04
So if we had fixed-fixed
boundary conditions

371
00:21:04 --> 00:21:06
what would I expect?
 

372
00:21:06 --> 00:21:12
Then things would give me
a sine series, possibly.

373
00:21:12 --> 00:21:14
Because those are the
eigenfunctions we're used to.

374
00:21:14 --> 00:21:19
Fixed-fixed, it's sines that
go from zero back to zero.

375
00:21:19 --> 00:21:24
Fixed-free will have
some sines or cosines.

376
00:21:24 --> 00:21:27
Periodic would be
the best of all.

377
00:21:27 --> 00:21:32
Yeah, so we need nice
boundary conditions.

378
00:21:32 --> 00:21:38
So the boundary conditions,
let me just say,

379
00:21:38 --> 00:21:40
periodic would be great.
 

380
00:21:40 --> 00:21:51
Or sometimes a fixed-free,
are familiar ones.

381
00:21:51 --> 00:21:55
At least in simple cases
can be dealt with.

382
00:21:55 --> 00:21:57
OK.
 

383
00:21:57 --> 00:22:04
So now, boy, that board is
already full of formulas.

384
00:22:04 --> 00:22:09
But, let's go back to the
start and say how do we

385
00:22:09 --> 00:22:13
find the coefficients?
 

386
00:22:13 --> 00:22:15
So because that was
the first step.

387
00:22:15 --> 00:22:18
Take the right-hand side,
find its coefficient.

388
00:22:18 --> 00:22:22
If we want to, just as applying
eigenvalues, the first step

389
00:22:22 --> 00:22:25
is always find eigenvalues.
 

390
00:22:25 --> 00:22:29
Here, in applying Fourier,
the first step is always

391
00:22:29 --> 00:22:31
find the coefficients.
 

392
00:22:31 --> 00:22:33
So, how do we do that?
 

393
00:22:33 --> 00:22:36
And at the beginning it
doesn't look too easy, right?

394
00:22:36 --> 00:22:39
Because let me take the
first guy, sin(x).

395
00:22:40 --> 00:22:43
Let me take an example.
 

396
00:22:43 --> 00:22:46
Particular S(x).
 

397
00:22:46 --> 00:22:49
The most important,
interesting function, S(x).

398
00:22:50 --> 00:22:53
I want it to be an odd
function, so that it

399
00:22:53 --> 00:22:55
will have only sine.
 

400
00:22:55 --> 00:22:57
And I should have
two period, 2pi.

401
00:22:58 --> 00:23:01
So let me just graph it.
 

402
00:23:01 --> 00:23:08
So it's going to have
coefficients, and I use b

403
00:23:08 --> 00:23:15
for sine, so it's going
to have b_1*sin(x), and

404
00:23:15 --> 00:23:18
b_2*sin(2x), and so on.
 

405
00:23:18 --> 00:23:23
And so it's got a whole
infinity of coefficients.

406
00:23:23 --> 00:23:24
Right?
 

407
00:23:24 --> 00:23:25
We're in function space.
 

408
00:23:25 --> 00:23:27
We're not dealing
with vectors now.

409
00:23:27 --> 00:23:32
So how is it possible to
find those coefficients?

410
00:23:32 --> 00:23:38
And let me chose a particular
S(x) so I'll put, since it's

411
00:23:38 --> 00:23:45
2pi periodic, if I tell you
what it is over a 2pi interval,

412
00:23:45 --> 00:23:47
just, repeat, repeat, repeat.
 

413
00:23:47 --> 00:23:52
So I'll pick the 2pi interval
to be minus pi to pi here.

414
00:23:52 --> 00:23:57
Just because it's a nice way,
and so that's a 2pi length.

415
00:23:57 --> 00:24:02
There's zero, I want to
function to be odd across zero.

416
00:24:02 --> 00:24:04
And I want it to be simple,
because it's going to be an

417
00:24:04 --> 00:24:07
important example that I
can actually compute.

418
00:24:07 --> 00:24:09
So I'm going to make it a one.
 

419
00:24:09 --> 00:24:13
And a minus one there.
 

420
00:24:13 --> 00:24:15
So, a step function.
 

421
00:24:15 --> 00:24:18
A step function, a square.
 

422
00:24:18 --> 00:24:22
And if I repeat it, of
course, it would go down,

423
00:24:22 --> 00:24:25
up, down, up, so on.
 

424
00:24:25 --> 00:24:30
But we only have to
look over this part.

425
00:24:30 --> 00:24:33
OK.
 

426
00:24:33 --> 00:24:37
Now, well, you might say wait
a minute how are we going to

427
00:24:37 --> 00:24:41
expand this function in sine.
 

428
00:24:41 --> 00:24:46
Well, sines are odd functions.
 

429
00:24:46 --> 00:24:48
Everybody knows what odd means?
 

430
00:24:48 --> 00:24:53
Odd means that S(-x) is -S(x).
 

431
00:24:56 --> 00:25:01
So that's the anti-symmetric
that we see in that graph.

432
00:25:01 --> 00:25:04
We also see a few
problems in this graph.

433
00:25:04 --> 00:25:11
At x=0, what is our sine
series going to give us?

434
00:25:11 --> 00:25:14
If I plug in x=0 on
the right-hand side I

435
00:25:14 --> 00:25:16
get zero, certainly.
 

436
00:25:16 --> 00:25:21
So this sine series
is going to do that.

437
00:25:21 --> 00:25:24
And actually Fourier
series tend to do this.

438
00:25:24 --> 00:25:26
In the middle of a jump
it'll pick the middle

439
00:25:26 --> 00:25:27
point of a jump.
 

440
00:25:27 --> 00:25:31
Fourier series generally, it's
the best possible, will pick

441
00:25:31 --> 00:25:33
the middle point of the jump.
 

442
00:25:33 --> 00:25:34
And what about at x=pi?
 

443
00:25:36 --> 00:25:39
At the end of the interval?
 

444
00:25:39 --> 00:25:41
What does my series
add up at x=pi?

445
00:25:42 --> 00:25:47
Zero again, because sin(pi),
sin(2pi), all zero.

446
00:25:47 --> 00:25:49
And that'll be in the
middle of that jump.

447
00:25:49 --> 00:25:52
So it's pretty good.
 

448
00:25:52 --> 00:25:57
But now what I'm hoping is that
my sine series is going to

449
00:25:57 --> 00:26:04
somehow get real fast up to
one, and level out at one.

450
00:26:04 --> 00:26:06
We're asking a lot.
 

451
00:26:06 --> 00:26:12
In fact, when Fourier proposed
this idea, Fourier series,

452
00:26:12 --> 00:26:17
there was a lot of doubters.
 

453
00:26:17 --> 00:26:23
Was it really possible to
represent other functions,

454
00:26:23 --> 00:26:27
maybe even including a step
function, in terms of

455
00:26:27 --> 00:26:31
sines or maybe cosines?
 

456
00:26:31 --> 00:26:33
And Fourier said
yes, go with it.

457
00:26:33 --> 00:26:34
So let's do it.
 

458
00:26:34 --> 00:26:42
OK, so and he turned out
to be incredibly right.

459
00:26:42 --> 00:26:43
How do I find b_2?
 

460
00:26:44 --> 00:26:48
Do you remember how to, I don't
want to know the formula.

461
00:26:48 --> 00:26:50
I want to know why.
 

462
00:26:50 --> 00:26:55
What's the step to find
the coefficient b_2?

463
00:26:56 --> 00:27:02
Well, the step is,
the key point.

464
00:27:02 --> 00:27:03
Which makes
everything possible.

465
00:27:03 --> 00:27:08
Why don't I identify the
key point without which we

466
00:27:08 --> 00:27:11
would be in real trouble.
 

467
00:27:11 --> 00:27:17
The key point is that all these
sine functions, sin(2x),

468
00:27:17 --> 00:27:22
sin(3x), sin(4x),
are orthogonal.

469
00:27:22 --> 00:27:27
Now, what do I mean by two
functions being orthogonal?

470
00:27:27 --> 00:27:31
Somehow my picture in function
space, so my picture in

471
00:27:31 --> 00:27:38
function space is that here is,
this is the sine x coordinate.

472
00:27:38 --> 00:27:41
And somewhere there's a sin(2x)
coordinate and it's 90 degrees

473
00:27:41 --> 00:27:44
and then there's a sin(3x)
coordinate, and then there's

474
00:27:44 --> 00:27:47
a sine, I don't know
where to point now.

475
00:27:47 --> 00:27:51
But there is a sin(4x), and
we're in infinite dimensions.

476
00:27:51 --> 00:27:56
And the sine vectors are
an orthogonal basis.

477
00:27:56 --> 00:27:58
They're orthogonal
to each other.

478
00:27:58 --> 00:28:00
What does that mean?
 

479
00:28:00 --> 00:28:03
Vectors we take
the dot product.

480
00:28:03 --> 00:28:07
Functions, we take, we don't
use the word dot product

481
00:28:07 --> 00:28:09
as much as inner product.
 

482
00:28:09 --> 00:28:12
So let me take the inner
product of, the whole

483
00:28:12 --> 00:28:13
point is orthogonality.
 

484
00:28:13 --> 00:28:15
Let me write that word down.
 

485
00:28:15 --> 00:28:17
Orthogonal.
 

486
00:28:17 --> 00:28:19
The sines are orthogonal.
 

487
00:28:19 --> 00:28:21
And what does that mean?
 

488
00:28:21 --> 00:28:27
That means that the integral
over our 2pi interval, or any

489
00:28:27 --> 00:28:34
2pi interval, of one sine,
sin(kx), let's say, multiplied

490
00:28:34 --> 00:28:42
by another sine, sin(lx), the x
is, you can guess the answer.

491
00:28:42 --> 00:28:47
And everything is
depending on this answer.

492
00:28:47 --> 00:28:49
And it is?
 

493
00:28:49 --> 00:28:51
Zero.
 

494
00:28:51 --> 00:28:53
It's just terrific.
 

495
00:28:53 --> 00:28:55
If k is different
from l, of course.

496
00:28:55 --> 00:29:00
If k is equal to l then I
have to figure that one out.

497
00:29:00 --> 00:29:01
I'll need that one.
 

498
00:29:01 --> 00:29:08
What is it if sine, if k=l
so I'm integrating sine

499
00:29:08 --> 00:29:12
squared of kx, then it's
certainly not zero.

500
00:29:12 --> 00:29:16
I getting like, the
length squared of the

501
00:29:16 --> 00:29:18
sin(kx) function.
 

502
00:29:18 --> 00:29:25
If k=l, what is it?
 

503
00:29:25 --> 00:29:27
It has some nice formula.
 

504
00:29:27 --> 00:29:28
Very nice.
 

505
00:29:28 --> 00:29:28
Let's see.
 

506
00:29:28 --> 00:29:32
Sine squared, do I need to
think about sine squared kx?

507
00:29:33 --> 00:29:37
Sine squared kx,
what does it do?

508
00:29:37 --> 00:29:39
Well, just graph
sine squared x.

509
00:29:39 --> 00:29:45
What would the graph of
sine squared x look like,

510
00:29:45 --> 00:29:48
from minus pi to pi?
 

511
00:29:48 --> 00:29:51
So it goes up, right?
 

512
00:29:51 --> 00:29:52
Doesn't it go up?
 

513
00:29:52 --> 00:29:54
And then it goes back down.
 

514
00:29:54 --> 00:29:55
OK.
 

515
00:29:55 --> 00:30:00
Sorry, I made that
a little hard.

516
00:30:00 --> 00:30:03
Is that right?
 

517
00:30:03 --> 00:30:05
And then it keeps it up.
 

518
00:30:05 --> 00:30:06
Right.
 

519
00:30:06 --> 00:30:08
So, what's the
integral of that?

520
00:30:08 --> 00:30:12
I'm not seeing quite why.
 

521
00:30:12 --> 00:30:16
The answer is its
average value is 1/2.

522
00:30:16 --> 00:30:23
The integral of sine squared
is 1/2 of the length.

523
00:30:23 --> 00:30:29
The whole interval is of length
2pi, and we're taking the

524
00:30:29 --> 00:30:31
area under sine squared.
 

525
00:30:31 --> 00:30:34
I may have to come back to
it, but the answer would be

526
00:30:34 --> 00:30:36
half of 2pi, which is pi.
 

527
00:30:36 --> 00:30:37
Yeah, yeah.
 

528
00:30:37 --> 00:30:41
So you could say the length
of the sine function

529
00:30:41 --> 00:30:46
is square root of pi.
 

530
00:30:46 --> 00:30:48
So these are integrals.
 

531
00:30:48 --> 00:30:51
You told me the
answer was zero.

532
00:30:51 --> 00:30:55
And I agreed with you, but
we haven't computed it.

533
00:30:55 --> 00:30:57
And nor have we
really got that.

534
00:30:57 --> 00:31:00
So a little bit to fix, still.
 

535
00:31:00 --> 00:31:09
But the crucial fact, I mean,
those are highly important

536
00:31:09 --> 00:31:12
integrals that just
come out beautifully.

537
00:31:12 --> 00:31:16
And beautifully
really means zero.

538
00:31:16 --> 00:31:19
I mean, that's the beautiful
number, right, for an integral.

539
00:31:19 --> 00:31:23
OK, so now how do I use that?
 

540
00:31:23 --> 00:31:24
Again, I'm looking for b_2.
 

541
00:31:26 --> 00:31:32
How do I pick off b_2, using
the fact that sin(2x) times any

542
00:31:32 --> 00:31:37
other sine integrates to zero.
 

543
00:31:37 --> 00:31:38
Ready for the moment?
 

544
00:31:38 --> 00:31:39
To find the coefficient b_2?
 

545
00:31:40 --> 00:31:44
I should, let me start this
sentence and if you finish it.

546
00:31:44 --> 00:31:50
I'll multiply both sides of
this equation by sin(2x).

547
00:31:51 --> 00:31:56
And then I will integrate.
 

548
00:31:56 --> 00:32:00
I'll multiply both sides by
sin(2x), so I take S(x)sin(2x).

549
00:32:00 --> 00:32:04
 
 

550
00:32:04 --> 00:32:07
And on the right hand, I
have b_1*sin(x)sin(2x).

551
00:32:07 --> 00:32:12
 
 

552
00:32:12 --> 00:32:13
And then I have b_2.
 

553
00:32:14 --> 00:32:16
Now, here's the one that's
going to live through

554
00:32:16 --> 00:32:17
the integration.
 

555
00:32:17 --> 00:32:20
It's going to survive, because
it's the sin(2x) times

556
00:32:20 --> 00:32:26
sin(2x) sin(2x) squared.
 

557
00:32:26 --> 00:32:30
And then comes the b_3 guy,
would be b_3*sin(3x)sin(2x).

558
00:32:37 --> 00:32:40
Everybody sees what I'm doing?
 

559
00:32:40 --> 00:32:44
As we did with the weak form in
differential equations, I'm

560
00:32:44 --> 00:32:47
multiplying through
by these guys.

561
00:32:47 --> 00:32:51
And then I'm integrating
over the interval.

562
00:32:51 --> 00:32:55
And what do I get?
 

563
00:32:55 --> 00:32:57
Integrate everyone dx.
 

564
00:32:59 --> 00:33:02
And what's the result?
 

565
00:33:02 --> 00:33:06
What is that integral?
 

566
00:33:06 --> 00:33:08
Zero.
 

567
00:33:08 --> 00:33:09
It's gone.
 

568
00:33:09 --> 00:33:11
What is this integral,
the integral of

569
00:33:11 --> 00:33:12
sin(3x) times sin(2x)?
 

570
00:33:14 --> 00:33:15
Zero.
 

571
00:33:15 --> 00:33:21
All those sines integrate to
zero, and I have to come

572
00:33:21 --> 00:33:27
back and see it's a simple
trig identity to do it.

573
00:33:27 --> 00:33:29
To see why that's zero.
 

574
00:33:29 --> 00:33:32
Do you see that everything is
disappearing, except b_2.

575
00:33:33 --> 00:33:36
So we finally have the
formula that we want.

576
00:33:36 --> 00:33:42
Let me just with put
these formulas down.

577
00:33:42 --> 00:33:46
So b_k, b_2 or b_k, yeah tell
me the formula for b_k.

578
00:33:47 --> 00:33:50
Let me go back, here.
 

579
00:33:50 --> 00:33:53
What did b_2 come out to be?
 

580
00:33:53 --> 00:33:56
So I have b_2, that's a number.
 

581
00:33:56 --> 00:33:59
It's got this right-hand side.
 

582
00:33:59 --> 00:34:01
That's the integral
that I mentioned.

583
00:34:01 --> 00:34:04
You'd have to compute
that integral.

584
00:34:04 --> 00:34:07
And then what about this stuff?
 

585
00:34:07 --> 00:34:10
This sin(2x) squared?
 

586
00:34:10 --> 00:34:13
I've integrated that.
 

587
00:34:13 --> 00:34:16
And what did I get for that?
 

588
00:34:16 --> 00:34:20
This is b_2, and then
this is some number.

589
00:34:20 --> 00:34:22
And it's pi.
 

590
00:34:22 --> 00:34:26
So this is b_2, and
multiplying, right?

591
00:34:26 --> 00:34:29
That b_2 comes out, and then
I have the integral of

592
00:34:29 --> 00:34:32
sine squared 2x, and
that's what's pi.

593
00:34:32 --> 00:34:37
So that's b_2 times pi here,
and I just divide by the pi.

594
00:34:37 --> 00:34:43
So I divide by pi and I get the
integral from minus pi to pi

595
00:34:43 --> 00:34:52
of my function times my sine.
 

596
00:34:52 --> 00:34:59
That's the model for all
the coefficients of

597
00:34:59 --> 00:35:02
orthogonal series.
 

598
00:35:02 --> 00:35:04
That's the model.
 

599
00:35:04 --> 00:35:11
Cosines, the complete ones,
the complex coefficients.

600
00:35:11 --> 00:35:15
The Legendre series, the Bessel
series, everybody's series

601
00:35:15 --> 00:35:18
will follow this same model.
 

602
00:35:18 --> 00:35:23
Because all those series are
series of orthogonal functions.

603
00:35:23 --> 00:35:26
Everything is hinging
on this orthogonality.

604
00:35:26 --> 00:35:31
The fact that one term
times another gives zero.

605
00:35:31 --> 00:35:34
What that means, really.
 

606
00:35:34 --> 00:35:43
I want to say it with a
picture, too. so let me draw

607
00:35:43 --> 00:35:46
two orthogonal directions.
 

608
00:35:46 --> 00:35:53
I intentionally didn't make
them just x and y axes.

609
00:35:53 --> 00:35:58
This might be the direction
of sin(x), and this might be

610
00:35:58 --> 00:35:59
the direction of sin(2x).
 

611
00:36:00 --> 00:36:05
And then I have a function.
 

612
00:36:05 --> 00:36:08
And I'm trying to find
out how much of sin(2x)

613
00:36:08 --> 00:36:09
has it got in it?
 

614
00:36:09 --> 00:36:12
How much of sin(x) has it got
in it, and then of course

615
00:36:12 --> 00:36:16
there's also a sin(3x) and
all the other sin(kx)'s.

616
00:36:17 --> 00:36:25
The point is, the point of this
90 degree angle there is, that

617
00:36:25 --> 00:36:34
if I can split this S(x),
whatever it might be, I can

618
00:36:34 --> 00:36:39
find its sin(x) piece directly.
 

619
00:36:39 --> 00:36:44
By just projecting it,
it's the projection of my

620
00:36:44 --> 00:36:48
function on that coordinate.
 

621
00:36:48 --> 00:36:51
If you don't like sin(x),
sin(2x), S(x), write

622
00:36:51 --> 00:36:54
v_1, v_2, whatever.
 

623
00:36:54 --> 00:36:56
To think of it as vectors.
 

624
00:36:56 --> 00:36:57
What's the sin 2?
 

625
00:36:57 --> 00:36:59
So that is b_1*sin(x).
 

626
00:37:01 --> 00:37:04
That's the right
amount of sin(x).

627
00:37:04 --> 00:37:11
And the whole point is that
that calculation didn't

628
00:37:11 --> 00:37:14
involve b_2 and b_3
and all the other b's.

629
00:37:14 --> 00:37:19
When I'm projecting onto
orthogonal directions, I

630
00:37:19 --> 00:37:22
can do them one at a time.
 

631
00:37:22 --> 00:37:25
I can do one one-dimensional
projection at a time.

632
00:37:25 --> 00:37:35
This b_ksin(kx) is the, so I'm
just saying this in words,

633
00:37:35 --> 00:37:42
is the projection of my
function onto sin(kx).

634
00:37:44 --> 00:37:49
And the point is, I could
do this and get this

635
00:37:49 --> 00:37:52
answer because of
that 90 degree angle.

636
00:37:52 --> 00:37:54
If I didn't have 90
degrees, do you see that

637
00:37:54 --> 00:37:55
this wouldn't work?
 

638
00:37:55 --> 00:38:02
Suppose my two basis functions
are at some 40 degree angle.

639
00:38:02 --> 00:38:05
Then I take my function.
 

640
00:38:05 --> 00:38:08
Can I project that
onto this guy?

641
00:38:08 --> 00:38:13
And project that onto this guy,
so the projections are there?

642
00:38:13 --> 00:38:14
And there?
 

643
00:38:14 --> 00:38:20
Do they add back to the
function that I started with?

644
00:38:20 --> 00:38:22
The given function?
 

645
00:38:22 --> 00:38:23
No way.
 

646
00:38:23 --> 00:38:26
I mean, these are
much too big, right?

647
00:38:26 --> 00:38:30
If I add that one to this one
I'm way out here somewhere.

648
00:38:30 --> 00:38:34
But over here, with 90
degrees, these are the two

649
00:38:34 --> 00:38:36
projections, project there.
 

650
00:38:36 --> 00:38:37
Project there.
 

651
00:38:37 --> 00:38:42
Add those two pieces and
I got back exactly.

652
00:38:42 --> 00:38:48
I just want to emphasize the
importance of orthogonality.

653
00:38:48 --> 00:38:52
It breaks the problem down into
one-dimensional projections.

654
00:38:52 --> 00:38:55
So here we go with b_k*sin(kx).
 

655
00:38:56 --> 00:38:59
OK, let me do the
key example now.

656
00:38:59 --> 00:39:01
This example.
 

657
00:39:01 --> 00:39:07
Let me find the coefficients of
that particular function S(x).

658
00:39:08 --> 00:39:13
This is the step function,
the square wave, S(x), let's

659
00:39:13 --> 00:39:15
find its coefficients.
 

660
00:39:15 --> 00:39:17
I'll just use this formula.
 

661
00:39:17 --> 00:39:22
OK, maybe I'll erase so that
I can write the integration

662
00:39:22 --> 00:39:23
right underneath.
 

663
00:39:23 --> 00:39:24
OK.
 

664
00:39:24 --> 00:39:26
Oh, one little point here.
 

665
00:39:26 --> 00:39:30
Well, not so little,
but it's a saving.

666
00:39:30 --> 00:39:35
It's worth noticing.
 

667
00:39:35 --> 00:39:42
The reward for picking off the
odd function is, I think that

668
00:39:42 --> 00:39:46
this integral is the same from
minus pi to zero

669
00:39:46 --> 00:39:48
as zero to a pi.
 

670
00:39:48 --> 00:39:51
In other words, I think that
for an odd function, I get

671
00:39:51 --> 00:39:57
the same answer if I just do
the integral from zero to

672
00:39:57 --> 00:40:03
pi, that I have to do.
 

673
00:40:03 --> 00:40:05
And double it.
 

674
00:40:05 --> 00:40:11
So I think if I just double
it, I don't know if you

675
00:40:11 --> 00:40:14
regard that as a saving.
 

676
00:40:14 --> 00:40:18
In some way, the work
is only half as much.

677
00:40:18 --> 00:40:21
It'll make this particular
example easy, so let

678
00:40:21 --> 00:40:23
me do this example.
 

679
00:40:23 --> 00:40:27
What are the Fourier
coefficients of

680
00:40:27 --> 00:40:29
the square wave?
 

681
00:40:29 --> 00:40:34
OK, so I'll do this integral.
 

682
00:40:34 --> 00:40:39
So from zero to pi,
what is my function?

683
00:40:39 --> 00:40:42
My N from the graph?
 

684
00:40:42 --> 00:40:44
Just one.
 

685
00:40:44 --> 00:40:47
This is going to be
a picnic, right?

686
00:40:47 --> 00:40:50
The function is one here.
 

687
00:40:50 --> 00:40:58
So S(x) is one, so I want 2/pi,
the integral from zero to pi

688
00:40:58 --> 00:41:03
of just sin(kx)dx, right?
 

689
00:41:03 --> 00:41:10
Which is, so I've got 2/pi,
now I integrate sin(kx), I

690
00:41:10 --> 00:41:15
get minus cos(kx), right?
 

691
00:41:15 --> 00:41:18
Between zero and pi.
 

692
00:41:18 --> 00:41:20
And what else?
 

693
00:41:20 --> 00:41:21
What have I forgotten?
 

694
00:41:21 --> 00:41:23
The most important point.
 

695
00:41:23 --> 00:41:27
The integral of sin(kx) k
x is not minus cos(kx).

696
00:41:28 --> 00:41:34
I have to divide by k.
 

697
00:41:34 --> 00:41:36
It's the division by k
that's going to give me

698
00:41:36 --> 00:41:41
the correct decay rate.
 

699
00:41:41 --> 00:41:41
2/(pi*k).
 

700
00:41:42 --> 00:41:45
Alright, now I've got a
little calculation to do.

701
00:41:45 --> 00:41:49
I have to figure out what
is cos(kx) at zero,

702
00:41:49 --> 00:41:51
no problem, it's one.
 

703
00:41:51 --> 00:41:54
And at the other
point, at x=pi.

704
00:41:55 --> 00:41:57
So what am I getting, then?
 

705
00:41:57 --> 00:41:58
I'm getting 2/(pi*k).
 

706
00:41:58 --> 00:42:09
 
 

707
00:42:09 --> 00:42:13
With that minus sign, I'll
evaluate it at x=0, I have

708
00:42:13 --> 00:42:17
one minus whatever I get
at the top. cos(k*pi).

709
00:42:17 --> 00:42:21
 
 

710
00:42:21 --> 00:42:22
That's b_k.
 

711
00:42:22 --> 00:42:24
 
 

712
00:42:24 --> 00:42:32
So there's a typical, well not
typical but very nice, answer.

713
00:42:32 --> 00:42:35
Now let's see what
these numbers are.

714
00:42:35 --> 00:42:39
So let me take a 2/pi out here.
 

715
00:42:39 --> 00:42:44
And then just list
these numbers.

716
00:42:44 --> 00:42:47
So k is one, two, three,
four, five, right?

717
00:42:47 --> 00:42:51
Tell me what these numbers are
for, let me put the k in here

718
00:42:51 --> 00:42:55
because that's part of it.
 

719
00:42:55 --> 00:42:56
So it's a constant, 2/pi.
 

720
00:42:56 --> 00:42:59
 
 

721
00:42:59 --> 00:43:03
At k=1, what do I get?
 

722
00:43:03 --> 00:43:03
At k=1?
 

723
00:43:04 --> 00:43:09
This is the little bit
that needs the patience.

724
00:43:09 --> 00:43:15
At k=1, the cos(pi) is?
 

725
00:43:15 --> 00:43:16
Negative one.
 

726
00:43:16 --> 00:43:20
So I have net minus
minus one, I get a two.

727
00:43:20 --> 00:43:25
I get a two over
a one. k is one.

728
00:43:25 --> 00:43:30
Alright, that is the
coefficient for k=1.

729
00:43:30 --> 00:43:33
Now, what's b_2, the
coefficient for k=2?

730
00:43:33 --> 00:43:37
I have 1-cos(2pi),
what's cos(2pi)?

731
00:43:39 --> 00:43:40
One.
 

732
00:43:40 --> 00:43:43
So they cancel,
so I get a zero.

733
00:43:43 --> 00:43:43
There is no b_2.
 

734
00:43:45 --> 00:43:45
What about b_3?
 

735
00:43:47 --> 00:43:49
So now b_3, I have 1-cos(3pi).
 

736
00:43:49 --> 00:43:51
 
 

737
00:43:51 --> 00:43:52
What's the cos(3pi)?
 

738
00:43:54 --> 00:43:56
It's negative one again.
 

739
00:43:56 --> 00:43:57
Right, same as the cos(pi).
 

740
00:43:58 --> 00:44:01
So that gives me a two, and
now I'm dividing by three.

741
00:44:01 --> 00:44:08
2/3, alright, let's do two
more. k=4, what do I get?

742
00:44:08 --> 00:44:12
Zero, because the cos(4pi)
has come back to one.

743
00:44:12 --> 00:44:13
So I get a zero.
 

744
00:44:13 --> 00:44:15
And what do I get from k=5?
 

745
00:44:15 --> 00:44:18
 
 

746
00:44:18 --> 00:44:25
1-cos(5pi), which is?
cos(5pi) is back to negative

747
00:44:25 --> 00:44:29
one, so 1--1 is a two.
 

748
00:44:29 --> 00:44:31
You see the pattern.
 

749
00:44:31 --> 00:44:39
And so let me just copy the
famous series for this, S(x).

750
00:44:40 --> 00:44:44
This S(x) is, let's see.
 

751
00:44:44 --> 00:44:48
The twos, I'll make
that 4/pi, right?

752
00:44:48 --> 00:44:50
I'll take out all those twos.
 

753
00:44:50 --> 00:44:51
So I have 4/pi*sin(x).
 

754
00:44:51 --> 00:44:55
 
 

755
00:44:55 --> 00:44:58
I have no sin(2x), forget that.
 

756
00:44:58 --> 00:45:02
Now I do have some sin(3x)'s,
how much do I have?

757
00:45:02 --> 00:45:12
4/pi*sin(3x)'s, But
divide by three, right?

758
00:45:12 --> 00:45:14
And then there's no
4x's, no sin(4x)'s.

759
00:45:16 --> 00:45:19
But then there will be
a 4/pi*sine, what's

760
00:45:19 --> 00:45:22
the next term now?
 

761
00:45:22 --> 00:45:23
Are you with me?
 

762
00:45:23 --> 00:45:27
So this is a typical nice
example, an important example.

763
00:45:27 --> 00:45:29
Sine of what?
 

764
00:45:29 --> 00:45:33
5x divided by five.
 

765
00:45:33 --> 00:45:35
OK.
 

766
00:45:35 --> 00:45:37
That's a great example,
it's worth remembering.

767
00:45:37 --> 00:45:40
Factor the 4/pi out
if you want to.

768
00:45:40 --> 00:45:47
4/pi time sin(x), sin(3x)/3,
sin(5x)/5, it's a beautiful

769
00:45:47 --> 00:45:49
example of an odd function.
 

770
00:45:49 --> 00:45:53
OK, and let's see.
 

771
00:45:53 --> 00:46:01
So what do you think, MATLAB
can draw this graph far

772
00:46:01 --> 00:46:02
better than we can.
 

773
00:46:02 --> 00:46:07
But let me draw enough so
you see what's really

774
00:46:07 --> 00:46:09
interesting here.
 

775
00:46:09 --> 00:46:12
Interesting and famous.
 

776
00:46:12 --> 00:46:15
So the leading term is 4
over pi sine x, that would

777
00:46:15 --> 00:46:18
be something like that.
 

778
00:46:18 --> 00:46:21
That's as close as
sin(x) can get, 4/pi

779
00:46:21 --> 00:46:23
is the optimal number.
 

780
00:46:23 --> 00:46:24
The optimal coefficient.
 

781
00:46:24 --> 00:46:29
The projection, this
4/pi*sin(x) is the best, the

782
00:46:29 --> 00:46:32
closest I can get to one.
 

783
00:46:32 --> 00:46:34
On that integral.
 

784
00:46:34 --> 00:46:35
With just sin(x).
 

785
00:46:36 --> 00:46:40
But now when I put in
sin(3x), I think it'll do

786
00:46:40 --> 00:46:44
something more like this.
 

787
00:46:44 --> 00:46:46
Do you see what's
happening there?

788
00:46:46 --> 00:46:49
That's what I've got with
sin(3x), and of course

789
00:46:49 --> 00:46:50
odd on the other side.
 

790
00:46:50 --> 00:46:52
What do you think it
looks like with sin(5x)?

791
00:46:54 --> 00:46:59
It's just so great you have
to let the computer draw

792
00:46:59 --> 00:47:00
it a couple of times.
 

793
00:47:00 --> 00:47:04
You see, it goes up here.
 

794
00:47:04 --> 00:47:08
And then it's sort of, you
know, it's getting closer.

795
00:47:08 --> 00:47:13
It's going to stay
closer to that.

796
00:47:13 --> 00:47:18
But I don't know if you can see
from my picture, I'm actually

797
00:47:18 --> 00:47:19
proud of that picture.
 

798
00:47:19 --> 00:47:22
It's not as bad as usual.
 

799
00:47:22 --> 00:47:27
And it makes the crucial
point, two crucial points.

800
00:47:27 --> 00:47:31
One is, I am going to get
closer and closer to one.

801
00:47:31 --> 00:47:38
These oscillations, these
ripples, will be smaller.

802
00:47:38 --> 00:47:42
But here is the great fact
and it's a big headache

803
00:47:42 --> 00:47:45
in calculation.
 

804
00:47:45 --> 00:47:51
At the jump, the first
ripple doesn't get smaller.

805
00:47:51 --> 00:47:57
The first ripple gets thinner,
the first ripple gets thinner.

806
00:47:57 --> 00:47:59
You see the ripples moving
over there, but their

807
00:47:59 --> 00:48:01
height doesn't change.
 

808
00:48:01 --> 00:48:04
Do you know whose name is
associated with that,

809
00:48:04 --> 00:48:06
in that phenomenon?
 

810
00:48:06 --> 00:48:07
Gibbs.
 

811
00:48:07 --> 00:48:15
Gibbs noticed that the ripple
height as you add more and more

812
00:48:15 --> 00:48:20
terms, you're closer and closer
to the function over more

813
00:48:20 --> 00:48:23
and more of the interval.
 

814
00:48:23 --> 00:48:25
So the ripples get
squeezed to the left.

815
00:48:25 --> 00:48:29
The area under the ripples
goes to zero, certainly.

816
00:48:29 --> 00:48:32
But the height of the
ripples doesn't.

817
00:48:32 --> 00:48:36
And it doesn't stay constant,
but nearly constant.

818
00:48:36 --> 00:48:39
It approaches a famous number.
 

819
00:48:39 --> 00:48:43
And of course we'll have the
same odd picture down here.

820
00:48:43 --> 00:48:47
And it'll bump up again, the
same thing is happening

821
00:48:47 --> 00:48:49
at every jump.
 

822
00:48:49 --> 00:48:52
In other words, if
you're computing shock.

823
00:48:52 --> 00:48:56
If you're computing air flow
around shocks, with Fourier

824
00:48:56 --> 00:49:00
type method, Gibbs is
going to get you.

825
00:49:00 --> 00:49:02
You'll have to deal with Gibbs.
 

826
00:49:02 --> 00:49:09
Because the shock has
that extra ripple.

827
00:49:09 --> 00:49:12
OK, that's a lot
of Section 4.1.

828
00:49:12 --> 00:49:15
Energy, we didn't get
to, so that'll be the

829
00:49:15 --> 00:49:17
first point on Friday.
 

830
00:49:17 --> 00:49:19
And I'll see you this
afternoon and talk about the

831
00:49:19 --> 00:49:21
MATLAB or anything else.
 

832
00:49:21 --> 00:49:22
OK.