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AUDIENCE: OK.
10
00:00:21 --> 00:00:26
I hoped I might have Exam 2
for you today, but it's not
11
00:00:26 --> 00:00:28
quite back from the grader.
12
00:00:28 --> 00:00:33
It's already gone to the second
grader, so it will not be long.
13
00:00:33 --> 00:00:40
And I hope you've had a look
at the MATLAB homework for
14
00:00:40 --> 00:00:43
a variety of possible.
15
00:00:43 --> 00:00:49
I think we've got, there were
some errors in the original
16
00:00:49 --> 00:00:52
statement, location of the
coordinates, but I think
17
00:00:52 --> 00:00:53
they're fixed now.
18
00:00:53 --> 00:00:55
So ready to go on that MATLAB.
19
00:00:55 --> 00:00:58
Don't forget that it's four on
the right-hand side and not
20
00:00:58 --> 00:01:04
one, so if you get an answer
near 1/4 at the center of the
21
00:01:04 --> 00:01:07
circle, that's the reason.
22
00:01:07 --> 00:01:12
Just that factor four
is to remember.
23
00:01:12 --> 00:01:15
I'll talk more about the
MATLAB this afternoon in the
24
00:01:15 --> 00:01:18
review session right here.
25
00:01:18 --> 00:01:22
Just to say, I'm highly
interested in that problem.
26
00:01:22 --> 00:01:28
Not just increasing N, the
number of mesh points in the
27
00:01:28 --> 00:01:34
octagon, but also increasing
the number of sides.
28
00:01:34 --> 00:01:42
So there are two numbers there,
we had N points on a ray,
29
00:01:42 --> 00:01:44
out from the center.
30
00:01:44 --> 00:01:49
But we have M sides
of the polygon.
31
00:01:49 --> 00:01:55
And I'm interested in both
of those, getting big.
32
00:01:55 --> 00:01:57
Growing.
33
00:01:57 --> 00:01:58
I don't know how.
34
00:01:58 --> 00:02:08
And maybe a reasonable balance
is to take, I think N
35
00:02:08 --> 00:02:11
proportional to M is a
pretty good balance.
36
00:02:11 --> 00:02:14
So I'd be very happy;
I mean I'm very happy
37
00:02:14 --> 00:02:15
with whatever you do.
38
00:02:15 --> 00:02:19
But I'm really interested
to know what happens as
39
00:02:19 --> 00:02:22
both of these increase.
40
00:02:22 --> 00:02:25
How close, how quickly do you
approach the eigenvalues
41
00:02:25 --> 00:02:26
of a circle.
42
00:02:26 --> 00:02:29
And you might keep the
two proportional as
43
00:02:29 --> 00:02:31
you increase them.
44
00:02:31 --> 00:02:34
So let me say more about that
this afternoon, because it's a
45
00:02:34 --> 00:02:37
big day today, to
start Fourier.
46
00:02:37 --> 00:02:41
Fourier series, the new
chapter, the new topic.
47
00:02:41 --> 00:02:44
In fact, the final major
topic of the course.
48
00:02:44 --> 00:02:52
So I tried to list here, so
here I'm in Section 4.1, so I'm
49
00:02:52 --> 00:02:54
talking about Fourier series.
50
00:02:54 --> 00:02:58
So Fourier series is for
functions that have period 2pi.
51
00:02:59 --> 00:03:05
It involves things like sin(x),
like cos(x) like e^(ikx), all
52
00:03:05 --> 00:03:11
of those if I increase x by
2pi, I'm back where I started.
53
00:03:11 --> 00:03:15
So that's the sort of functions
that have Fourier series.
54
00:03:15 --> 00:03:21
Then we'll go on to the other
two big forms, crucial
55
00:03:21 --> 00:03:23
forms of the Fourier world.
56
00:03:23 --> 00:03:28
But 4.1 starts with the
classical Fourier series.
57
00:03:28 --> 00:03:34
So I realize, you will have
seen, many of you will have
58
00:03:34 --> 00:03:36
seen Fourier series before.
59
00:03:36 --> 00:03:40
I hope you'll see some
new aspects here.
60
00:03:40 --> 00:03:48
So, let me just get organized.
61
00:03:48 --> 00:03:53
It's nice to have some examples
that just involve sine.
62
00:03:53 --> 00:03:57
And since the sine is an odd
function, that means it's sort
63
00:03:57 --> 00:04:01
of anti-symmetric across zero,
those are the functions
64
00:04:01 --> 00:04:03
that will have only sine.
65
00:04:03 --> 00:04:05
That will have a
sine expansion.
66
00:04:05 --> 00:04:07
Cosines are the opposite.
67
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Cosines are symmetric
across zero.
68
00:04:10 --> 00:04:12
Like a constant,
or like cos(x).
69
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Zero comes right at
the symmetric point.
70
00:04:15 --> 00:04:18
So those will have
only cosines.
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00:04:18 --> 00:04:22
And a lot of examples fit in
one or the other of those,
72
00:04:22 --> 00:04:24
and it's easy to see them.
73
00:04:24 --> 00:04:28
The general function, of
course, is a combination
74
00:04:28 --> 00:04:30
odd and even.
75
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It has cosines and it has
sines, it's just the
76
00:04:33 --> 00:04:35
some of the two pieces.
77
00:04:35 --> 00:04:41
So, this is the standard
Fourier series, which I
78
00:04:41 --> 00:04:45
couldn't get onto one line,
but it has all the cosines
79
00:04:45 --> 00:04:49
including this slightly
different cos(0),
80
00:04:49 --> 00:04:51
and all the sines.
81
00:04:51 --> 00:04:57
But because this one has these
three different pieces, the
82
00:04:57 --> 00:05:02
constant term, the other
cosines, all the sines, three
83
00:05:02 --> 00:05:07
slightly different formulas,
it's actually nicest of all,
84
00:05:07 --> 00:05:10
to use this final form.
85
00:05:10 --> 00:05:12
Because there's
just one formula.
86
00:05:12 --> 00:05:13
There's just one kind.
87
00:05:13 --> 00:05:18
And I'll call its coefficient
c_k, and now they multiply
88
00:05:18 --> 00:05:22
e^(ikx), so we have to
get used to e^(ikx).
89
00:05:24 --> 00:05:29
We may be more familiar with
cos, sin(kx) and cos(kx),
90
00:05:29 --> 00:05:34
but everybody knows e^(ikx)
is a combination of them.
91
00:05:34 --> 00:05:38
And if we let k go from minus
infinity to infinity, so
92
00:05:38 --> 00:05:40
we've got all the terms.
93
00:05:40 --> 00:05:48
Including e^(-i3x), and
e^(+i3x), those would
94
00:05:48 --> 00:05:51
combine to give cosines
and sines of 3x.
95
00:05:52 --> 00:05:54
We get one nice formula.
96
00:05:54 --> 00:05:57
There's just one
formula for the C's.
97
00:05:57 --> 00:06:01
So that's one good reason to
look at the complex form.
98
00:06:01 --> 00:06:05
Even if our function
is actually real.
99
00:06:05 --> 00:06:09
That form is kind of neat, and
the second good reason, the
100
00:06:09 --> 00:06:13
really important reason, is
then when we go to the discrete
101
00:06:13 --> 00:06:18
Fourier transform, the DFT,
everybody writes that
102
00:06:18 --> 00:06:20
with complex numbers.
103
00:06:20 --> 00:06:25
So it's good to see complex
numbers first and then we
104
00:06:25 --> 00:06:30
can just translate
the formulas from.
105
00:06:30 --> 00:06:33
And these are also almost
always written with
106
00:06:33 --> 00:06:34
complex numbers.
107
00:06:34 --> 00:06:39
So this is the way to see it.
108
00:06:39 --> 00:06:44
OK, so what do we do
about Fourier series?
109
00:06:44 --> 00:06:46
What do we have to know
how to do and what
110
00:06:46 --> 00:06:47
should we understand?
111
00:06:47 --> 00:06:53
Well, if you've met Fourier
series you may have met the
112
00:06:53 --> 00:06:56
formula for these coefficients.
113
00:06:56 --> 00:06:58
That's sort of like step one.
114
00:06:58 --> 00:07:01
If I'm given the function,
whatever the function might be,
115
00:07:01 --> 00:07:02
might be a delta function.
116
00:07:02 --> 00:07:04
Interesting case, always.
117
00:07:04 --> 00:07:07
Always interesting.
118
00:07:07 --> 00:07:08
Always crazy right?
119
00:07:08 --> 00:07:13
But it's always interesting,
the delta function.
120
00:07:13 --> 00:07:16
The coefficients
can be computed.
121
00:07:16 --> 00:07:21
The coefficients, you'll see,
I'll repeat those formulas.
122
00:07:21 --> 00:07:26
They involve integrals.
123
00:07:26 --> 00:07:29
What I want to say right
now is that this isn't a
124
00:07:29 --> 00:07:31
course in integration.
125
00:07:31 --> 00:07:35
So I'm not interested in doing
more and more complicated
126
00:07:35 --> 00:07:39
integrals and finding
Fourier coefficients
127
00:07:39 --> 00:07:40
of weird functions.
128
00:07:40 --> 00:07:41
No way.
129
00:07:41 --> 00:07:45
I want to understand the
simple, straight, the
130
00:07:45 --> 00:07:47
important examples.
131
00:07:47 --> 00:07:52
And here's a point that's
highly interesting.
132
00:07:52 --> 00:07:56
In practice, in computing
practice, we're close to
133
00:07:56 --> 00:07:57
computing practice here.
134
00:07:57 --> 00:07:59
In everything we do.
135
00:07:59 --> 00:08:03
I mean, this is really
constantly used.
136
00:08:03 --> 00:08:07
And one important question
is, is the Fourier series
137
00:08:07 --> 00:08:09
quickly convergent?
138
00:08:09 --> 00:08:11
Because if we're going to
compute, we don't want to
139
00:08:11 --> 00:08:14
compute a thousand terms.
140
00:08:14 --> 00:08:19
Hopefully ten terms, 20 terms
would give us good accuracy.
141
00:08:19 --> 00:08:24
So that question comes down to
how quickly does those a's
142
00:08:24 --> 00:08:27
and b's and c's go to zero?
143
00:08:27 --> 00:08:28
That's highly important.
144
00:08:28 --> 00:08:32
And you'll connect this decay
rate, we'll connect this with
145
00:08:32 --> 00:08:34
the smoothness of the function.
146
00:08:34 --> 00:08:38
Oh, I can tell you
even at a start.
147
00:08:38 --> 00:08:42
OK, so I just want to
emphasize this point.
148
00:08:42 --> 00:08:50
We'll see it over and over that
like for a delta function,
149
00:08:50 --> 00:08:56
which is not smooth at all,
we'll see no decay at all.
150
00:08:56 --> 00:08:58
In the coefficients.
151
00:08:58 --> 00:09:03
They're constant.
152
00:09:03 --> 00:09:06
They don't decrease as we go to
higher and higher frequencies.
153
00:09:06 --> 00:09:13
I think of k here, I'll use
the word frequency for k.
154
00:09:13 --> 00:09:18
So high frequency means high k,
far off the Fourier series, and
155
00:09:18 --> 00:09:22
the question is, are the
coefficients staying up there
156
00:09:22 --> 00:09:24
big, and we have to
worry about them.
157
00:09:24 --> 00:09:26
Or do they get very small?
158
00:09:26 --> 00:09:29
So a delta function
is a key example and
159
00:09:29 --> 00:09:32
then a step function.
160
00:09:32 --> 00:09:34
So what will be the
deal with those?
161
00:09:34 --> 00:09:38
If I have a function that's
a step function, I'll have
162
00:09:38 --> 00:09:41
decay at rate is 1/k.
163
00:09:41 --> 00:09:44
164
00:09:44 --> 00:09:46
So they do go to zero.
165
00:09:46 --> 00:09:53
The thousandth coefficient will
be roughly of size 1/1000.
166
00:09:53 --> 00:09:54
That's not fast.
167
00:09:54 --> 00:10:02
That's not really fast
enough to compute with.
168
00:10:02 --> 00:10:08
Well, we meet step functions,
I mean, functions with jumps.
169
00:10:08 --> 00:10:11
And we'll see that their
Fourier series, the
170
00:10:11 --> 00:10:16
coefficients do go to
zero but not very fast.
171
00:10:16 --> 00:10:19
And we get something
highly interesting.
172
00:10:19 --> 00:10:23
So when we do these examples,
so I've sort of moved on to
173
00:10:23 --> 00:10:27
examples, so these are
two basic examples.
174
00:10:27 --> 00:10:30
What would be the next example?
175
00:10:30 --> 00:10:32
Step function.
176
00:10:32 --> 00:10:35
Well, yeah, or
maybe a hat next.
177
00:10:35 --> 00:10:37
A hat function would
be, you see what I'm
178
00:10:37 --> 00:10:38
doing at each step?
179
00:10:38 --> 00:10:40
I'm integrating.
180
00:10:40 --> 00:10:44
A hat function might be the
next, yeah, a ramp, exactly.
181
00:10:44 --> 00:10:48
Hat function, which is
a ramp with a corner.
182
00:10:48 --> 00:10:50
Now, so that's one
integral better.
183
00:10:50 --> 00:10:55
You want to guess the decay
rate on that one? k squared.
184
00:10:55 --> 00:10:58
Now we're getting better.
185
00:10:58 --> 00:11:00
That's a faster follow-up.
186
00:11:00 --> 00:11:00
1/k^2.
187
00:11:02 --> 00:11:04
And then we integrate
again, we'd get 1/k^3.
188
00:11:06 --> 00:11:11
Then one more integral, 1/k^4
would be a cubic spline with,
189
00:11:11 --> 00:11:14
you remember the cubic
spline is continuous.
190
00:11:14 --> 00:11:16
Its derivative is continuous,
that gives us a 1/k^3.
191
00:11:17 --> 00:11:20
Its second derivative is
continuous, that gives us a
192
00:11:20 --> 00:11:25
1/k^4, and then you really can
compute with that, if you
193
00:11:25 --> 00:11:27
have such a function.
194
00:11:27 --> 00:11:31
So, point, pay attention
to decay rate.
195
00:11:31 --> 00:11:37
That, and the connection
to smoothness.
196
00:11:37 --> 00:11:41
So examples, we'll start
right off with these guys.
197
00:11:41 --> 00:11:44
And then we'll see the
rules for the derivative.
198
00:11:44 --> 00:11:48
Oh yeah, rules for
the derivative.
199
00:11:48 --> 00:11:53
The beauty of Fourier
series is, well, actually
200
00:11:53 --> 00:11:54
you can see this.
201
00:11:54 --> 00:11:56
You can see the rule.
202
00:11:56 --> 00:11:58
Let me just show you
the rule for this.
203
00:11:58 --> 00:12:03
So the rule for derivatives,
the whole point about
204
00:12:03 --> 00:12:08
Fourier is, it connects
perfectly with calculus.
205
00:12:08 --> 00:12:10
With taking derivatives.
206
00:12:10 --> 00:12:16
So suppose I have F(x) equals,
I'll use this form, the
207
00:12:16 --> 00:12:18
sum of c_k*e^(ikx).
208
00:12:18 --> 00:12:22
209
00:12:22 --> 00:12:24
And now I take its
derivative. dF/dx.
210
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?
212
00:12:33 --> 00:12:36
Suppose I have the Fourier
series for some function, and
213
00:12:36 --> 00:12:38
then I take Fourier series
for the derivative.
214
00:12:38 --> 00:12:42
So I'm kind of going
the backwards way.
215
00:12:42 --> 00:12:43
Less smooth.
216
00:12:43 --> 00:12:48
I'm going from, the derivative
of the step function involves
217
00:12:48 --> 00:12:53
delta functions, so I'm
going less smooth as
218
00:12:53 --> 00:12:57
I take derivatives.
219
00:12:57 --> 00:13:00
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
00:13:10 --> 00:13:14
Everybody see what happens when
I take the derivative of that
224
00:13:14 --> 00:13:17
typical term in the
Fourier series?
225
00:13:17 --> 00:13:19
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.
228
00:13:32 --> 00:13:38
So these are the Fourier
coefficients of the derivative.
229
00:13:38 --> 00:13:42
And that again makes exactly
the same point about the
230
00:13:42 --> 00:13:46
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.
234
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
00:14:07 --> 00:14:14
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.