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PROFESSOR STRANG: OK, so
I've got Quiz 2 give back

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to with good scores.
 

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So it's Christmas early.
 

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Or, good work on that exam.
 

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That's fine and so we all know
Fourier series is like a

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central topic on the final
third of the course, and I'll

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just keep going on
Fourier series.

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I'll give you homework
on 4.1 and 4.2.

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00:00:51 --> 00:00:56
So 4.1 is what we complete
today, the Fourier series.

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00:00:56 --> 00:01:00
4.2 is the discrete
Fourier series.

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00:01:00 --> 00:01:05
So those are two major,
major topics for this

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part of the course.
 

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This is the periodic one
and 4.2 will be the finite

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one, with the Fourier
matrix showing up.

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OK, so can I pick out, I've
made a list of topics last

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time that were important for
4.1, for Fourier series.

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And I think these are the
remaining entries on the list.

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00:01:33 --> 00:01:41
I did the the Fourier series
for the odd square wave, the

27
00:01:41 --> 00:01:45
minus one stepping
up to plus one.

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00:01:45 --> 00:01:49
And you remember that,
well just let me put

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that example down here.
 

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That was the step function;
easier if I draw it than if I

31
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try to write equations, so it
was minus one up to plus one,

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ending of course period 2pi.
 

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00:02:05 --> 00:02:11
And I guess we called that
S(x), for S signaling step

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function, S signaling
square wave.

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00:02:14 --> 00:02:20
And S the general a signal
that the function is odd.

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00:02:20 --> 00:02:24
And that means that it
goes with sine functions.

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00:02:24 --> 00:02:31
And I think the numbers that we
found from the formula was

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00:02:31 --> 00:02:40
sin(x)/1, sin(3x)/3, sin(5x)/5,
it's just a great

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00:02:40 --> 00:02:45
one to remember.
 

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And that's the example that has
the important, and I don't know

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if I wrote down Gibbs' name.
 

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He was the great physicist at
Yale, well, a hundred years ago

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or more, and did this key idea
that appears all the time, that

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00:03:12 --> 00:03:15
any time you have a step
function then the Fourier

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00:03:15 --> 00:03:23
series it does its best, I if I
take a thousand term, it'll do

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00:03:23 --> 00:03:27
its best but it will overshoot
by an amount that Gibbs found,

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and then it will get really
close and then it will

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overshoot again and
then, symmetrically.

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00:03:36 --> 00:03:39
Or anti-symmetrically,
I should say.

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So that's the Gibbs phenomenon
of great importance.

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I write this one out because
first it's an important

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one to remember.
 

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Second it'll give us a good
example for this important

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equality, that the energy in
the function is the energy

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00:03:59 --> 00:04:00
in the coefficient.
 

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That'll be good.
 

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OK, actually maybe I should do
that one first because the

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delta function has got infinite
energy and we don't learn

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anything from this equation.
 

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So let me jump to the energy
in the function and the

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energy in the coefficient.
 

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So what do I mean by energy?
 

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Well, it's quadratic.
 

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Right?
 

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It's the length squared here.
 

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It's the length squared
of the function.

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So let me compute, maybe I'll
do it on this board underneath

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and leave space for
the delta function.

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The energy in x space is
just, the integral is

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the length squared.
 

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The integral of
S(x) squared. dx.

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It's just what you
would expect.

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We have a function, not a
vector, so we can't sum

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coefficients squared.
 

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Instead we integrate all
the values squared.

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And of course, this is a
number that we can quickly

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compute for that function.
 

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So what does it turn out to be?
 

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Well, what is S(x) squared
for that function?

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00:05:23 --> 00:05:24
One, obviously.
 

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00:05:24 --> 00:05:26
The function is one here.
 

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I'm looking at the original
S(x), not the series.

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00:05:31 --> 00:05:34
The function is one there
and minus one there.

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When I squared those S(x)
squared is one everywhere so

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I'm integrating one everywhere
from minus pi to pi.

86
00:05:42 --> 00:05:43
So I get the answer 2pi.
 

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So that's a case where the
energy in the physical space,

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00:05:50 --> 00:05:53
the x space, was totally
easy to compute.

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00:05:53 --> 00:05:56
Now, what about energy,
what is this equality?

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This really neat easy
to remember equality?

91
00:06:03 --> 00:06:09
I'm just going to find it by
taking this thing squared.

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00:06:09 --> 00:06:13
What's the integral of
the right-hand side?

93
00:06:13 --> 00:06:20
The two are equal, so
suppose I just fire away?

94
00:06:20 --> 00:06:23
I integrate the square of
that infinite series.

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00:06:23 --> 00:06:28
You're going to say, well
that's going to take a while.

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00:06:28 --> 00:06:32
But what's going to be good.
 

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00:06:32 --> 00:06:36
The key point, the first point
in last time's lecture, the

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00:06:36 --> 00:06:40
first point in every discussion
of Fourier series,

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00:06:40 --> 00:06:43
is orthogonality.
 

100
00:06:43 --> 00:06:47
Sines times other sines
integrated are zero.

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00:06:47 --> 00:06:49
So a whole lot of
terms will go.

102
00:06:49 --> 00:06:52
So I take that
thing, I square it.

103
00:06:52 --> 00:06:54
So let me let me
do that one here.

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The interval from
minus pi to pi.

105
00:06:56 --> 00:07:02
May I take out the
(4/pi) squared?

106
00:07:02 --> 00:07:07
Just so it's not confusing.
 

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00:07:07 --> 00:07:16
Now, this is the sin(x)/1,
sin(3x)/3, sin(kx)/k,

108
00:07:16 --> 00:07:17
and so on.
 

109
00:07:17 --> 00:07:21
All squared. dx, and
so what do I get?

110
00:07:21 --> 00:07:27
The (4/pi) squared.
 

111
00:07:27 --> 00:07:30
And now I've got a
whole lot of terms.

112
00:07:30 --> 00:07:32
But the thing is,
I can do this.

113
00:07:32 --> 00:07:35
Because when I square this,
I'll have a lot of terms like

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00:07:35 --> 00:07:40
sin(x), sin(3x), and when I
integrate those I get zero.

115
00:07:40 --> 00:07:44
So the only ones that I don't
get zero are when sin(x)

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00:07:44 --> 00:07:46
integrates against itself.
 

117
00:07:46 --> 00:07:49
And sin(3x) against itself.
 

118
00:07:49 --> 00:07:51
So when sin(x) integrates
against itself,

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00:07:51 --> 00:07:53
that's sine squared.
 

120
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Its integral is, you remember
the integral of sine squared,

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which is, its average
value is 1/2.

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00:08:03 --> 00:08:04
We're over an integral.
 

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00:08:04 --> 00:08:08
I think I'm going to get
pi, for sine squared.

124
00:08:08 --> 00:08:11
Because sine squared, we could
do that calculation separately.

125
00:08:11 --> 00:08:13
It's just a standard integral.
 

126
00:08:13 --> 00:08:16
The integral of sine
squared is pi.

127
00:08:16 --> 00:08:21
Actually, yeah it just uses the
fact that sine squared x is the

128
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same as whatever it
is the same as.

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00:08:25 --> 00:08:33
Is it 1-cos(2x) or something?
 

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00:08:33 --> 00:08:35
Over two.
 

131
00:08:35 --> 00:08:37
Or plus, who cares?
 

132
00:08:37 --> 00:08:42
Because the integral of
whichever, plus or minus,

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00:08:42 --> 00:08:46
let me, well I suppose
for history's sake we

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00:08:46 --> 00:08:47
should get it right.
 

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00:08:47 --> 00:08:49
Which is it?
 

136
00:08:49 --> 00:08:51
Is it a minus, so
it looks OK now?

137
00:08:51 --> 00:08:55
OK, alright, if it's
wrong I didn't say.

138
00:08:55 --> 00:08:57
OK, but I'm going to integrate.
 

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00:08:57 --> 00:09:00
So the integral of the cosine
is zero, and the integral of

140
00:09:00 --> 00:09:04
the 1/2 is the part
I'm talking about.

141
00:09:04 --> 00:09:07
That 1/2 is there all the
way from minus pi to pi.

142
00:09:07 --> 00:09:09
So I get a pi.
 

143
00:09:09 --> 00:09:11
From all these sines.
 

144
00:09:11 --> 00:09:14
And now, what are
all the terms?

145
00:09:14 --> 00:09:20
Well, one over one squared,
that just had a coefficient

146
00:09:20 --> 00:09:22
one, but what's the next guy?
 

147
00:09:22 --> 00:09:25
You remember I'm squaring
it, I'm integrating.

148
00:09:25 --> 00:09:29
But I have a 1/3 squared.
 

149
00:09:29 --> 00:09:31
And 1/5 squared.
 

150
00:09:31 --> 00:09:34
And so on.
 

151
00:09:34 --> 00:09:37
And here's a great point.
 

152
00:09:37 --> 00:09:40
These two are equal.
 

153
00:09:40 --> 00:09:44
I've got the same function,
expressed in x space, and

154
00:09:44 --> 00:09:49
here it's expressed in
sine space, you could say.

155
00:09:49 --> 00:09:50
In harmonic space.
 

156
00:09:50 --> 00:09:52
OK, so that's equal.
 

157
00:09:52 --> 00:09:56
And that's going to be
the fact in general.

158
00:09:56 --> 00:10:02
In general that the integral of
S(x) squared, so the general

159
00:10:02 --> 00:10:05
fact will be the integral of -
well, I'll write it down below.

160
00:10:05 --> 00:10:08
But let's just see what
we got for numbers here.

161
00:10:08 --> 00:10:11
So I had pi on both sides.
 

162
00:10:11 --> 00:10:14
And so if I lift that over
there, I get something

163
00:10:14 --> 00:10:16
like - what do I have?
 

164
00:10:16 --> 00:10:22
Pi squared over 16, maybe I
have pi squared over eight.

165
00:10:22 --> 00:10:26
You just get a
remarkable formula.

166
00:10:26 --> 00:10:29
Putting that up there would
be pi squared over 16, and

167
00:10:29 --> 00:10:30
the two makes it an eight.
 

168
00:10:30 --> 00:10:36
And here I have the sum of 1/1
squared plus 1/3 squared.

169
00:10:36 --> 00:10:39
Plus 1/5 squared.
 

170
00:10:39 --> 00:10:46
So, that's an infinite sum that
I would not have known how to

171
00:10:46 --> 00:10:50
do except it appears here.
 

172
00:10:50 --> 00:10:54
The sum one over
all those squares.

173
00:10:54 --> 00:10:58
If I picked another example,
I could get the sum

174
00:10:58 --> 00:11:02
- oh, this was all the
odd numbers squared.

175
00:11:02 --> 00:11:05
If I picked a different
function, I could have got one

176
00:11:05 --> 00:11:11
that also had the sin(2x)/2
and the sin(4x)/4.

177
00:11:12 --> 00:11:15
So this would have been the
sum of all the squares.

178
00:11:15 --> 00:11:18
Do you happen to know what
that comes out to be?

179
00:11:18 --> 00:11:21
I mean, here's a
way to compute pi.

180
00:11:21 --> 00:11:24
We have a formula for pi.
 

181
00:11:24 --> 00:11:29
And we'd have another formula
that involved all the sums.

182
00:11:29 --> 00:11:32
Maybe I have room
for it up here.

183
00:11:32 --> 00:11:37
This would be the sum
of 1/n squared, right?

184
00:11:37 --> 00:11:39
This here I have
only the odd ones.

185
00:11:39 --> 00:11:41
And I get pi squared
over eight.

186
00:11:41 --> 00:11:46
Do you happen to know what
I get for all of them?

187
00:11:46 --> 00:11:49
So I'm also including
1/2 squared, there's a

188
00:11:49 --> 00:11:51
quarter also in here.
 

189
00:11:51 --> 00:11:52
And also a 16.
 

190
00:11:52 --> 00:11:55
And also a 36.
 

191
00:11:55 --> 00:11:57
In this one, and the
answer happens to be

192
00:11:57 --> 00:12:01
pi squared over six.
 

193
00:12:01 --> 00:12:08
Pi squared over six.
 

194
00:12:08 --> 00:12:14
The important point about this
energy equality is not being

195
00:12:14 --> 00:12:21
able to get a few very
remarkable formulas for pi.

196
00:12:21 --> 00:12:26
There's another remarkable
formula in the homework.

197
00:12:26 --> 00:12:27
This is a little famous.
 

198
00:12:27 --> 00:12:29
Do you know what this is?
 

199
00:12:29 --> 00:12:33
This is the famous
Riemann zeta function.

200
00:12:33 --> 00:12:39
The sum of (1/n)^x is
the zeta function at x.

201
00:12:39 --> 00:12:41
Here's the zeta
function at two.

202
00:12:41 --> 00:12:46
So if I could draws a zeta.
 

203
00:12:46 --> 00:12:49
Maybe?
 

204
00:12:49 --> 00:12:52
There's a Greek guy in this
class who could do it

205
00:12:52 --> 00:12:54
properly, but anyway.
 

206
00:12:54 --> 00:12:55
I'll chicken out.
 

207
00:12:55 --> 00:13:01
Zeta of, at the
value two, zeta(2).

208
00:13:02 --> 00:13:05
So we know zeta(2),
we know zeta(4).

209
00:13:06 --> 00:13:12
I don't think we know
zeta(3), I think it's not

210
00:13:12 --> 00:13:18
a special number like
pi squared over six.

211
00:13:18 --> 00:13:25
So the zeta function, the sum
of 1/n to this thing is,

212
00:13:25 --> 00:13:29
actually that's the subject of
a problem that Riemann

213
00:13:29 --> 00:13:30
did not solve.
 

214
00:13:30 --> 00:13:32
There's a problem Riemann did
not solve, and nobody has

215
00:13:32 --> 00:13:34
succeeded to find it.
 

216
00:13:34 --> 00:13:36
To solve it since.
 

217
00:13:36 --> 00:13:40
There's a million-dollar
prize for its solution.

218
00:13:40 --> 00:13:43
My neighbors think I should
be working on this,

219
00:13:43 --> 00:13:45
but I know better.
 

220
00:13:45 --> 00:13:50
It's going to be solved one
day, but it's pretty difficult.

221
00:13:50 --> 00:13:56
And that is so nowhere this
zeta function, where it's zero.

222
00:13:56 --> 00:13:58
Of course, it isn't
zero at two.

223
00:13:58 --> 00:14:01
Because it's pi squared six.
 

224
00:14:01 --> 00:14:06
And actually the conjecture is
that it's zero, all the zeroes

225
00:14:06 --> 00:14:12
are at points, complex
numbers with real part 1/2.

226
00:14:12 --> 00:14:16
So they're on this famous
line, the imaginary line

227
00:14:16 --> 00:14:18
with real part 1/2.
 

228
00:14:18 --> 00:14:25
And that's the most important
problem in pure mathematics.

229
00:14:25 --> 00:14:26
So here we go.
 

230
00:14:26 --> 00:14:31
We got a formula for pi out
of this energy identity.

231
00:14:31 --> 00:14:36
And I'll write it again, once
I have the complex form.

232
00:14:36 --> 00:14:40
OK, but you see where
it comes from.

233
00:14:40 --> 00:14:42
It just comes from
orthogonality.

234
00:14:42 --> 00:14:46
The fact that we could
integrate that square is

235
00:14:46 --> 00:14:47
what made it all work.
 

236
00:14:47 --> 00:14:53
OK, let's do the
delta function.

237
00:14:53 --> 00:14:56
So that's an even function,
the delta, right?

238
00:14:56 --> 00:14:58
Now I'm looking at
the delta function.

239
00:14:58 --> 00:15:01
Minus pi to pi.
 

240
00:15:01 --> 00:15:05
It has the spike at zero
and it's certainly even

241
00:15:05 --> 00:15:07
so we expect cosines.
 

242
00:15:07 --> 00:15:09
And what are the coefficients?
 

243
00:15:09 --> 00:15:15
So it's just an
important one to know.

244
00:15:15 --> 00:15:17
Very important example.
 

245
00:15:17 --> 00:15:18
So what's a_0?
 

246
00:15:18 --> 00:15:25
In general, the coefficient a_0
in the Fourier series is the,

247
00:15:25 --> 00:15:30
if I have a function, delta(x),
S(x), whatever my function, the

248
00:15:30 --> 00:15:36
a_0 coefficient is the average.
a for average, a_0 is

249
00:15:36 --> 00:15:38
the average value.
 

250
00:15:38 --> 00:15:45
So this is 1/2pi, the
integral of minus pi

251
00:15:45 --> 00:15:49
to pi of my function.
 

252
00:15:49 --> 00:15:51
Where did that come from?
 

253
00:15:51 --> 00:15:53
I just integrated.
 

254
00:15:53 --> 00:15:56
I just multiplied
both sides by one.

255
00:15:56 --> 00:15:59
Or by 1/2pi, and integrated.
 

256
00:15:59 --> 00:16:02
And those terms disappeared,
and I was left with a_0,

257
00:16:02 --> 00:16:05
and what's the answer?
 

258
00:16:05 --> 00:16:08
Everybody knows that integral.
 

259
00:16:08 --> 00:16:11
The integral of the delta
function is one, so

260
00:16:11 --> 00:16:12
I just get 1/2pi.
 

261
00:16:13 --> 00:16:13
So 1/2pi.
 

262
00:16:15 --> 00:16:17
OK, now ready for a_1.
 

263
00:16:17 --> 00:16:20
How much of cos(x) do I have?
 

264
00:16:20 --> 00:16:24
Can I just change this formula
to give me a_1 and you can

265
00:16:24 --> 00:16:26
tell me what it gives?
 

266
00:16:26 --> 00:16:28
Well let me do it here.
 

267
00:16:28 --> 00:16:28
Here's a_1.
 

268
00:16:29 --> 00:16:30
What's the formula for a_1?
 

269
00:16:32 --> 00:16:35
It's just like b_1, like
the sine formulas.

270
00:16:35 --> 00:16:39
You have to remember you're
only dividing by pi.

271
00:16:39 --> 00:16:43
Because that average value
was 1/2, as we saw.

272
00:16:43 --> 00:16:46
And then you have the integral
of whatever your function

273
00:16:46 --> 00:16:48
is. delta(x) in this case.
 

274
00:16:48 --> 00:16:51
Times the cos(1x).
 

275
00:16:52 --> 00:16:56
If we're looking for a_1 we've
multiplied both sides by

276
00:16:56 --> 00:16:58
cos(x) and integrated.
 

277
00:16:58 --> 00:17:02
And what answer do we get?
 

278
00:17:02 --> 00:17:03
For a_1?
 

279
00:17:05 --> 00:17:09
What's the integral of
delta(x) times cos(x)?

280
00:17:11 --> 00:17:14
dx, so I should put in a dx.
 

281
00:17:14 --> 00:17:15
And the answer is?
 

282
00:17:15 --> 00:17:16
One, also one.
 

283
00:17:16 --> 00:17:21
The delta function, this spike,
picks out the value of this

284
00:17:21 --> 00:17:24
function at the spike.
 

285
00:17:24 --> 00:17:26
Because of course it doesn't
matter what that function is

286
00:17:26 --> 00:17:28
away from the spike, because
the other factor,

287
00:17:28 --> 00:17:31
delta, is zero.
 

288
00:17:31 --> 00:17:33
Everything is at that spike,
all the action, and this

289
00:17:33 --> 00:17:35
happens to be one at the spike.
 

290
00:17:35 --> 00:17:36
So I get a 1/pi.
 

291
00:17:36 --> 00:17:39
 
 

292
00:17:39 --> 00:17:43
And actually, the same
formula for all these guys.

293
00:17:43 --> 00:17:46
This will be the same with
cos(x) changed to cost(2x).

294
00:17:48 --> 00:17:51
And what will be
the answer now?

295
00:17:51 --> 00:17:53
Again, one.
 

296
00:17:53 --> 00:17:54
Right?
 

297
00:17:54 --> 00:17:57
It's the value of
this integral.

298
00:17:57 --> 00:18:01
The formula for a_2 would have
come by multiplying both sides

299
00:18:01 --> 00:18:05
by cos(2x), integrating.
 

300
00:18:05 --> 00:18:08
And the integral of cos squared
would give me the pi, and I'm

301
00:18:08 --> 00:18:12
dividing by the pi, and I just
need that integral

302
00:18:12 --> 00:18:12
and it's easy.
 

303
00:18:12 --> 00:18:13
It's also 1/pi.
 

304
00:18:15 --> 00:18:16
So all these are 1/pi.
 

305
00:18:16 --> 00:18:19
Let me just put in the formula.
 

306
00:18:19 --> 00:18:22
1/2pi, and all the
rest are 1/pi.

307
00:18:22 --> 00:18:30
 
 

308
00:18:30 --> 00:18:30
cos(3x).
 

309
00:18:31 --> 00:18:33
All the cosines are there.
 

310
00:18:33 --> 00:18:36
All in the same amount.
 

311
00:18:36 --> 00:18:43
And the constant term
is slightly different.

312
00:18:43 --> 00:18:49
OK, that's the formal Fourier
series for the delta function.

313
00:18:49 --> 00:18:55
Formal meaning you can use it
to compute, of course some

314
00:18:55 --> 00:18:59
things will fail, like
what's the energy?

315
00:18:59 --> 00:19:04
If I integrate, I tried to do
energy in x space of the delta

316
00:19:04 --> 00:19:10
function, or energy in k space,
what answer would I get?

317
00:19:10 --> 00:19:13
Right, the integral of
delta squared, its energy

318
00:19:13 --> 00:19:15
is infinite, right?
 

319
00:19:15 --> 00:19:18
The integral of delta, if I
have delta times delta then

320
00:19:18 --> 00:19:20
I'm really in trouble, right?
 

321
00:19:20 --> 00:19:25
Because this delta says, if I
can speak informally, that

322
00:19:25 --> 00:19:29
delta says take the value of
this function at zero, but of

323
00:19:29 --> 00:19:33
course that's infinite, so
that would be infinite.

324
00:19:33 --> 00:19:39
That would be the
energy in x space.

325
00:19:39 --> 00:19:42
What about the
energy in k space?

326
00:19:42 --> 00:19:47
Well, let's think, what's
the energy in k space?

327
00:19:47 --> 00:19:50
I'm going to do the squares of
the coefficients, you remember

328
00:19:50 --> 00:19:54
that's what I had down here?
 

329
00:19:54 --> 00:19:56
And I'll have it
again in a moment.

330
00:19:56 --> 00:20:00
It's the sum of the squares
of the coefficients, fixed

331
00:20:00 --> 00:20:03
up by factors of pi.
 

332
00:20:03 --> 00:20:07
And here, all the
coefficients are constants.

333
00:20:07 --> 00:20:09
So that sum is infinite.
 

334
00:20:09 --> 00:20:13
Again, it's the sum of squares,
of constant, constant,

335
00:20:13 --> 00:20:16
constant, constants, and
that series doesn't

336
00:20:16 --> 00:20:17
converge, it flows up.
 

337
00:20:17 --> 00:20:20
So the energy is infinite.
 

338
00:20:20 --> 00:20:21
That's OK.
 

339
00:20:21 --> 00:20:25
The key is that formula.
 

340
00:20:25 --> 00:20:29
OK, there's the
formula to remember.

341
00:20:29 --> 00:20:31
There's the formula.
 

342
00:20:31 --> 00:20:32
On forever.
 

343
00:20:32 --> 00:20:40
Every frequency is in
here to the same amount.

344
00:20:40 --> 00:20:42
OK, good.
 

345
00:20:42 --> 00:20:44
That's the delta
function example.

346
00:20:44 --> 00:20:51
OK, ready to go to complex?
 

347
00:20:51 --> 00:20:56
Complex is no big deal because
we know, actually, you can

348
00:20:56 --> 00:21:00
tell me the complex series
for the delta function.

349
00:21:00 --> 00:21:02
I'll write it right underneath.
 

350
00:21:02 --> 00:21:07
The complex series for the
delta function, just turn

351
00:21:07 --> 00:21:15
these guys into e^(i*theta),
e^(i*x)'s, and e^(i*2x)'s,

352
00:21:15 --> 00:21:16
and e^(-i*2x)'s.
 

353
00:21:18 --> 00:21:21
Just term by term,
just to see it.

354
00:21:21 --> 00:21:24
To see it clearly for
this great example.

355
00:21:24 --> 00:21:31
So what does cos(2x)
look like in terms of

356
00:21:31 --> 00:21:34
complex exponentials?
 

357
00:21:34 --> 00:21:43
Everybody knows cos(x)
is the same as e^(ix),

358
00:21:43 --> 00:21:47
and e^(-ix), right?
 

359
00:21:47 --> 00:21:51
Divided by two, because this
is cos(x)+i*sin(x), this is

360
00:21:51 --> 00:21:56
cos(x)-i*sin(x), when I add
them I get two cos(x)'s.

361
00:21:56 --> 00:21:58
So I must divide by two.
 

362
00:21:58 --> 00:22:02
Let me divide by
two all the way.

363
00:22:02 --> 00:22:05
So I'm dividing
this 1/pi by two.

364
00:22:05 --> 00:22:07
You'll see it's so much nicer.
 

365
00:22:07 --> 00:22:11
So 1/2pi times the one,
that's our first guy.

366
00:22:11 --> 00:22:15
And now I've got the next guy.
 

367
00:22:15 --> 00:22:15
Right?
 

368
00:22:15 --> 00:22:18
Because I need to divide by
the two to get the cosine,

369
00:22:18 --> 00:22:20
and there's my two.
 

370
00:22:20 --> 00:22:22
OK, what's the next?
 

371
00:22:22 --> 00:22:25
What do I have next?
 

372
00:22:25 --> 00:22:27
From this guy.
 

373
00:22:27 --> 00:22:30
1/pi, still there.
 

374
00:22:30 --> 00:22:35
What's cos(2x), if I want
to write it in terms of

375
00:22:35 --> 00:22:37
complex exponentials?
 

376
00:22:37 --> 00:22:40
So this guy now,
I'm ready for him.

377
00:22:40 --> 00:22:44
Is e^(i*2x).
 

378
00:22:44 --> 00:22:48
 
 

379
00:22:48 --> 00:22:50
And e^(-i*2x).
 

380
00:22:50 --> 00:22:54
 
 

381
00:22:54 --> 00:22:57
Divided by two, and
there's my two.

382
00:22:57 --> 00:23:01
Do you see what's happening?
 

383
00:23:01 --> 00:23:04
There's an example to
show you why the complex

384
00:23:04 --> 00:23:05
case is so nice.
 

385
00:23:05 --> 00:23:11
Here we had to remember a
different number for a_0.

386
00:23:12 --> 00:23:17
Here it's just, so the next one
will be e^(i*3x), and

387
00:23:17 --> 00:23:26
e^(-i*3x), so that the delta
function in the complex Fourier

388
00:23:26 --> 00:23:29
series, all the terms
have coefficients one

389
00:23:29 --> 00:23:30
divided by the 2pi.
 

390
00:23:31 --> 00:23:35
That's a great example.
 

391
00:23:35 --> 00:23:40
And of course we see again that
the sum of squares, oh yeah.

392
00:23:40 --> 00:23:48
So let's do the complex
formula, by which I mean I'm

393
00:23:48 --> 00:23:53
taking any function, F, not
necessarily even, not

394
00:23:53 --> 00:23:57
necessarily odd, not
necessarily real.

395
00:23:57 --> 00:24:00
Any function can now be a
complex function, because

396
00:24:00 --> 00:24:03
we're going to use
complex things here.

397
00:24:03 --> 00:24:12
So I'll have all the
complex exponentials.

398
00:24:12 --> 00:24:14
For integer k.
 

399
00:24:14 --> 00:24:19
This k is an integer but
it can go from minus

400
00:24:19 --> 00:24:21
infinity to infinity.
 

401
00:24:21 --> 00:24:23
That's the complex form.
 

402
00:24:23 --> 00:24:32
F(x) is is a series again.
 

403
00:24:32 --> 00:24:37
The beauty is that every
term looks the same.

404
00:24:37 --> 00:24:40
The thing you have to remember
is that k negative k is

405
00:24:40 --> 00:24:42
allowed, as well as positive k.
 

406
00:24:42 --> 00:24:49
You see that we needed the
negative k to get cosines.

407
00:24:49 --> 00:24:52
k was minus 1 there,
k was minus 2 there.

408
00:24:52 --> 00:24:55
And for sines we would
also need them.

409
00:24:55 --> 00:24:59
So cosines go into it,
sines go into it.

410
00:24:59 --> 00:25:01
F(x) could be complex.
 

411
00:25:01 --> 00:25:03
That's the complex series.
 

412
00:25:03 --> 00:25:07
So maybe we could have
started with that series.

413
00:25:07 --> 00:25:12
But we didn't, we
came to it here.

414
00:25:12 --> 00:25:15
But what's the formula
for its coefficient?

415
00:25:15 --> 00:25:19
OK, actually, so the next
half-hour now, we have to

416
00:25:19 --> 00:25:24
think complex. and that
will bring a few changes.

417
00:25:24 --> 00:25:25
So watch for the changes.
 

418
00:25:25 --> 00:25:33
You see we're almost in the
same ballpark, but there are a

419
00:25:33 --> 00:25:35
couple of things to notice.
 

420
00:25:35 --> 00:25:37
So let me write down
that series again.

421
00:25:37 --> 00:25:41
Minus infinity to infinity of
some coefficient e^(ikx).

422
00:25:43 --> 00:25:46
Now, what is c_k?
 

423
00:25:46 --> 00:25:54
What is the formula for c_k?
 

424
00:25:54 --> 00:25:58
As soon as we answer that
question, you'll see the

425
00:25:58 --> 00:26:02
new aspect for complex.
 

426
00:26:02 --> 00:26:05
How do I find coefficients?
 

427
00:26:05 --> 00:26:07
I multiply by something.
 

428
00:26:07 --> 00:26:11
I integrate, and I
use orthogonality.

429
00:26:11 --> 00:26:14
Same idea, just
repeat after repeat.

430
00:26:14 --> 00:26:18
The question is, what
do I multiply by?

431
00:26:18 --> 00:26:23
If I wanted to know c_3,
suppose I want to know a

432
00:26:23 --> 00:26:26
formula for c_3,
the coefficient.

433
00:26:26 --> 00:26:30
What am I going to multiply by
that's going to give me the

434
00:26:30 --> 00:26:33
orthogonality I need, that all
the other integrals are going

435
00:26:33 --> 00:26:35
to disappear, that's the key?
 

436
00:26:35 --> 00:26:39
I want to multiply by something
so that when I integrate, all

437
00:26:39 --> 00:26:42
the other integrals are
going to disappear.

438
00:26:42 --> 00:26:43
And let's just do it.
 

439
00:26:43 --> 00:26:45
Here, suppose I have e^(i5x).
 

440
00:26:45 --> 00:26:48
 
 

441
00:26:48 --> 00:26:55
And I'm looking for c_3.
 

442
00:26:56 --> 00:26:57
So I'll look at e^(i3x).
 

443
00:26:57 --> 00:27:04
 
 

444
00:27:04 --> 00:27:06
So watch.
 

445
00:27:06 --> 00:27:09
This is the small point
we have to make.

446
00:27:09 --> 00:27:11
So I'm looking for e^(i3x).
 

447
00:27:13 --> 00:27:15
So I'm going to multiply
by something, and I'm

448
00:27:15 --> 00:27:16
going to integrate.
 

449
00:27:16 --> 00:27:21
And what would be the good
thing to multiply by?

450
00:27:21 --> 00:27:25
Well, you would say, if you
were just a real person, you

451
00:27:25 --> 00:27:29
would say multiply by
e^(i3x), integrated.

452
00:27:29 --> 00:27:31
And hope for getting zero.
 

453
00:27:31 --> 00:27:33
You won't get zero.
 

454
00:27:33 --> 00:27:38
If I multiply e^(i3x),
I'm sorry, you might say

455
00:27:38 --> 00:27:39
what am I doing here?
 

456
00:27:39 --> 00:27:42
I'm trying to check
orthogonality.

457
00:27:42 --> 00:27:46
Let me instead of three use kx.
 

458
00:27:46 --> 00:27:49
 
 

459
00:27:49 --> 00:27:52
So that's a typical
complex one.

460
00:27:52 --> 00:27:54
It's any one of these guys.
 

461
00:27:54 --> 00:28:00
And I want to see
what's orthogonality.

462
00:28:00 --> 00:28:02
That's what I'm asking.
 

463
00:28:02 --> 00:28:04
Everything hinged
on orthogonality.

464
00:28:04 --> 00:28:07
We've got to have
orthogonality here.

465
00:28:07 --> 00:28:10
But let me show you what it is.
 

466
00:28:10 --> 00:28:15
The thing you multiply by
to get the c_3 is not

467
00:28:15 --> 00:28:19
e^(i3x), it is? e^(-i3x).
 

468
00:28:21 --> 00:28:23
You take the conjugate.
 

469
00:28:23 --> 00:28:27
You change i to minus i.
 

470
00:28:27 --> 00:28:28
In a complex case.
 

471
00:28:28 --> 00:28:35
So if I take e^(ikx)
times e^(-ilx), and

472
00:28:35 --> 00:28:36
notice that minus.
 

473
00:28:36 --> 00:28:40
I claim that I get zero.
 

474
00:28:40 --> 00:28:43
Except if k is l.
 

475
00:28:43 --> 00:28:46
And when k is l, I
probably get 2pi.

476
00:28:47 --> 00:28:50
If k is l.
 

477
00:28:50 --> 00:28:54
I get zero if k is not l.
 

478
00:28:54 --> 00:28:57
That's the beautiful
orthogonality.

479
00:28:57 --> 00:29:02
I'm not too worried about the
2pi, I'll figure that out.

480
00:29:02 --> 00:29:08
So what I'm saying is, when
you're taking inner products,

481
00:29:08 --> 00:29:11
dot products, and you've got
complex stuff, one of

482
00:29:11 --> 00:29:17
the factors takes a
complex conjugate.

483
00:29:17 --> 00:29:18
Change i to minus i.
 

484
00:29:18 --> 00:29:21
Do you see that we've got a
completely easy integral now?

485
00:29:21 --> 00:29:23
What is the integral
of this guy?

486
00:29:23 --> 00:29:27
How do I see that I
really get zero?

487
00:29:27 --> 00:29:31
So this is the first time I'm
actually doing an integration

488
00:29:31 --> 00:29:34
and seeing orthogonality.
 

489
00:29:34 --> 00:29:39
Anybody likes integrating
these things, because they're

490
00:29:39 --> 00:29:43
so simple to integrate.
 

491
00:29:43 --> 00:29:45
Before I plug in the limits,
what's the integral

492
00:29:45 --> 00:29:46
of this guy?
 

493
00:29:46 --> 00:29:53
How do I rewrite that to
integrate it easily?

494
00:29:53 --> 00:29:55
I put the two
exponents together.

495
00:29:55 --> 00:30:03
This is the same as
e^i(k-l)x, right?

496
00:30:03 --> 00:30:06
The exponentials
follow that rule.

497
00:30:06 --> 00:30:10
If I multiply exponentials,
I combine the exponents.

498
00:30:10 --> 00:30:12
And now I'm ready to integrate.
 

499
00:30:12 --> 00:30:14
And so what is the
integral of e^(i(k-l)x?

500
00:30:14 --> 00:30:18
 
 

501
00:30:18 --> 00:30:22
When I integrate e to the
something x, I get that

502
00:30:22 --> 00:30:29
same thing again divided
by the something.

503
00:30:29 --> 00:30:33
So now I've integrated, and now
I just want to go from zero to

504
00:30:33 --> 00:30:38
2pi, plug in the limits
from x=0 to x=2pi.

505
00:30:40 --> 00:30:44
This is what the integral
is asking me for.

506
00:30:44 --> 00:30:47
So I'm actually doing
the integral here.

507
00:30:47 --> 00:30:52
I put these together into that,
I integrated, which just

508
00:30:52 --> 00:30:56
brought this term down below,
because the derivative will

509
00:30:56 --> 00:31:01
bring that term above.
 

510
00:31:01 --> 00:31:03
Oh, they weren't
meant to change.

511
00:31:03 --> 00:31:07
Actually, it's wrong but right.
 

512
00:31:07 --> 00:31:10
To change those
integration limits.

513
00:31:10 --> 00:31:13
I mean, any 2pi would
work, but thank you.

514
00:31:13 --> 00:31:18
You're totally right, I should
have done minus pi to pi.

515
00:31:18 --> 00:31:18
Why do we get zero?
 

516
00:31:18 --> 00:31:20
That's the whole point.
 

517
00:31:20 --> 00:31:25
Here we actually did the
integral, and we can just

518
00:31:25 --> 00:31:27
plug in x=pi and x=-pi.
 

519
00:31:28 --> 00:31:33
Or we could plug in zero and
2pi, or I could plug in any

520
00:31:33 --> 00:31:36
guys that were 2pi apart,
any period of 2pi.

521
00:31:37 --> 00:31:43
Why do we get zero?
 

522
00:31:43 --> 00:31:48
Do you have to do the
plugging in part to see it?

523
00:31:48 --> 00:31:49
You can, certainly.
 

524
00:31:49 --> 00:31:54
But the point is, this
function is periodic.

525
00:31:54 --> 00:31:56
That's a function
that has period 2pi.

526
00:31:56 --> 00:32:01
So it has to be the same at the
lower and the upper limit.

527
00:32:01 --> 00:32:02
That's what it's coming to.
 

528
00:32:02 --> 00:32:05
That's a periodic function.
 

529
00:32:05 --> 00:32:07
It's equal at these two limits.
 

530
00:32:07 --> 00:32:11
And therefore, when I do the
subtraction I take it at the

531
00:32:11 --> 00:32:14
top limit, minus the answer
at the bottom limit,

532
00:32:14 --> 00:32:15
it's the same at both.
 

533
00:32:15 --> 00:32:17
So I get zero.
 

534
00:32:17 --> 00:32:22
So there is the actual
check orthogonality.

535
00:32:22 --> 00:32:27
So the key point was,
orthogonality, or inner

536
00:32:27 --> 00:32:31
product, or complex functions,
one of them has to take

537
00:32:31 --> 00:32:33
the complex conjugate.
 

538
00:32:33 --> 00:32:37
Let me just do for
vectors, too.

539
00:32:37 --> 00:32:41
Complex vectors.
 

540
00:32:41 --> 00:32:44
I may have mentioned it, let
me take the vector 

541
00:32:44 --> 00:32:48
as an extreme example.
 

542
00:32:48 --> 00:32:54
And suppose I wanted to find
the inner product with itself.

543
00:32:54 --> 00:32:56
Which will be the
length squared.

544
00:32:56 --> 00:32:59
What's the inner product of
that vector, that complex

545
00:32:59 --> 00:33:01
vector  with itself?
 

546
00:33:01 --> 00:33:04
Let me just raise it
up so we see it.

547
00:33:04 --> 00:33:06
Focus on this.
 

548
00:33:06 --> 00:33:10
Usually the length squared
would be one squared

549
00:33:10 --> 00:33:12
plus i squared.
 

550
00:33:12 --> 00:33:13
No good.
 

551
00:33:13 --> 00:33:16
Why no good?
 

552
00:33:16 --> 00:33:17
Because it's zero.
 

553
00:33:17 --> 00:33:20
One squared plus i
squared is zero.

554
00:33:20 --> 00:33:26
We want absolute
values squared.

555
00:33:26 --> 00:33:29
Absolute values,
we need squares.

556
00:33:29 --> 00:33:31
We need positive numbers here.
 

557
00:33:31 --> 00:33:35
So the length of this
squared, I would, so let

558
00:33:35 --> 00:33:37
me call that vector v.
 

559
00:33:37 --> 00:33:45
I want to take not v transpose
v, that's the thing that

560
00:33:45 --> 00:33:46
would give me zero.
 

561
00:33:46 --> 00:33:53
If I did , I have to
take the complex conjugate.

562
00:33:53 --> 00:33:56
One of the two factors, and
it's just a matter of

563
00:33:56 --> 00:34:00
convention which one you do,
this is the thing that gives

564
00:34:00 --> 00:34:03
me the length of v squared.
 

565
00:34:03 --> 00:34:05
And it's the same
thing for functions.

566
00:34:05 --> 00:34:07
It's the integral of F(x).
 

567
00:34:08 --> 00:34:12
Do you want me to write it as,
it's the integral of F(x)

568
00:34:12 --> 00:34:16
times its conjugate.
 

569
00:34:16 --> 00:34:21
That gives me the
length of F squared.

570
00:34:21 --> 00:34:27
And of course I can rewrite
that as the integral of F.

571
00:34:27 --> 00:34:33
We have this handy notation
for F times its conjugate.

572
00:34:33 --> 00:34:40
It's like re^(i*theta) times
re^(-i*theta), the product

573
00:34:40 --> 00:34:51
of re^(i*theta), times
re^(-i*theta) is what?

574
00:34:51 --> 00:34:55
This is the complex number,
this is its conjugate, when I

575
00:34:55 --> 00:34:59
multiply, you notice that i
went to minus i when I drop

576
00:34:59 --> 00:35:06
below the real axis, I
just change i to minus i.

577
00:35:06 --> 00:35:11
So those cancel and
I get r squared.

578
00:35:11 --> 00:35:14
The length, the size
of the number.

579
00:35:14 --> 00:35:19
It's just, if you haven't
thought complex just make this

580
00:35:19 --> 00:35:22
change and you're ready to go.
 

581
00:35:22 --> 00:35:28
Yeah just v bar transpose
v, F bar transpose F, if

582
00:35:28 --> 00:35:33
you like, or F squared.
 

583
00:35:33 --> 00:35:36
And when we make the correct
change we still have

584
00:35:36 --> 00:35:39
the orthogonality.
 

585
00:35:39 --> 00:35:41
OK?
 

586
00:35:41 --> 00:35:46
So that's what orthogonality
is, now what's the coefficient?

587
00:35:46 --> 00:35:50
So now, after all that
speech, tell me what do I

588
00:35:50 --> 00:35:51
multiply both sides by?
 

589
00:35:51 --> 00:35:55
Let me start again
and remember here.

590
00:35:55 --> 00:36:03
So my f of x is going to be
sum of c k, e to the i k x.

591
00:36:03 --> 00:36:08
And if I want to get a
coefficient, I'm looking

592
00:36:08 --> 00:36:12
for a formula for c 3.
 

593
00:36:12 --> 00:36:15
So how do I find c 3?
 

594
00:36:15 --> 00:36:17
Let me slow down a second.
 

595
00:36:17 --> 00:36:22
How to find c 3?
 

596
00:36:22 --> 00:36:27
Multiply both sides by what?
 

597
00:36:27 --> 00:36:31
And integrate.
 

598
00:36:31 --> 00:36:33
Minus pi to pi.
 

599
00:36:33 --> 00:36:36
What goes there?
 

600
00:36:36 --> 00:36:41
If I want c 3, I should
multiply both sides by,

601
00:36:41 --> 00:36:45
should I multiply both
sides by e to the i 3 x?

602
00:36:45 --> 00:36:49
Nope. e to the minus i 3 x.
 

603
00:36:49 --> 00:36:52
Multiply both sides by e to the
minus i 3 x, and integrate.

604
00:36:52 --> 00:36:57
e to the minus i 3 x, d x.
 

605
00:36:57 --> 00:37:00
And then on the right
side I get, what?

606
00:37:00 --> 00:37:06
I get c 3, because that'll be
the minus i 3 x times e to the

607
00:37:06 --> 00:37:10
plus i 3 x, I'll be integrating
1 so I get a 2 pi.

608
00:37:10 --> 00:37:14
And all 0's.
 

609
00:37:14 --> 00:37:17
From all the other terms.
 

610
00:37:17 --> 00:37:19
Orthogonality doing
its job again.

611
00:37:19 --> 00:37:22
All these 0's are
by orthogonality.

612
00:37:22 --> 00:37:28
By exactly this
integration that we did.

613
00:37:28 --> 00:37:35
If k is 3 and l is 7,
or k is 7 and l is 3,

614
00:37:35 --> 00:37:37
the integral gives 0.
 

615
00:37:37 --> 00:37:39
So what's the formula then?
 

616
00:37:39 --> 00:37:47
This is all gone, divide by
2 pi and you've got it.

617
00:37:47 --> 00:37:48
There you are.
 

618
00:37:48 --> 00:37:51
That's the coefficient.
 

619
00:37:51 --> 00:37:53
OK.
 

620
00:37:53 --> 00:37:58
We have to move to complex
numbers, and now in those 20

621
00:37:58 --> 00:38:04
minutes the only change to make
is conjugate one of them.

622
00:38:04 --> 00:38:11
One of the things when
it can be complex.

623
00:38:11 --> 00:38:16
So that's a complex series.
 

624
00:38:16 --> 00:38:18
What's the energy
inequality now?

625
00:38:18 --> 00:38:25
Let me do the energy equality
in x space and in k space.

626
00:38:25 --> 00:38:31
If I take this, now I'd
like, so that's an

627
00:38:31 --> 00:38:33
equality of two functions.
 

628
00:38:33 --> 00:38:38
Now I'd like to get it,
I want to do energy.

629
00:38:38 --> 00:38:39
This is fantastic.
 

630
00:38:39 --> 00:38:42
And extremely
practical and useful.

631
00:38:42 --> 00:38:45
The fact that the energy
is going to come out

632
00:38:45 --> 00:38:45
beautifully, too.
 

633
00:38:45 --> 00:38:48
So what am I going
to do for energy?

634
00:38:48 --> 00:38:55
I'm going to integrate
F(x) squared. dx.

635
00:38:55 --> 00:38:56
That's the energy.
 

636
00:38:56 --> 00:39:00
That's the energy in x
space, in the function.

637
00:39:00 --> 00:39:03
What do I do now?
 

638
00:39:03 --> 00:39:11
I integrate this series.
c_k*e^(ikx) squared.

639
00:39:11 --> 00:39:13
Oh no, whoa.
 

640
00:39:13 --> 00:39:16
If I put a square there, right?
 

641
00:39:16 --> 00:39:18
I'm fired.
 

642
00:39:18 --> 00:39:18
Right?
 

643
00:39:18 --> 00:39:19
What do I have to do?
 

644
00:39:19 --> 00:39:22
I will put a square there,
but I have to straighten

645
00:39:22 --> 00:39:24
out something.
 

646
00:39:24 --> 00:39:26
What do I do?
 

647
00:39:26 --> 00:39:30
Those curvy lines, which just
meant take the thing and

648
00:39:30 --> 00:39:35
square, should be changed to?
 

649
00:39:35 --> 00:39:36
Straight.
 

650
00:39:36 --> 00:39:38
Just a matter of
getting straight.

651
00:39:38 --> 00:39:39
Straightening this out.
 

652
00:39:39 --> 00:39:41
OK, straight.
 

653
00:39:41 --> 00:39:42
There we are.
 

654
00:39:42 --> 00:39:44
OK.
 

655
00:39:44 --> 00:39:47
Now, that means the thing
times its conjugate.

656
00:39:47 --> 00:39:49
That's the thing
times its conjugate.

657
00:39:49 --> 00:39:51
So what do I get?
 

658
00:39:51 --> 00:39:54
All the terms disappear
except the perfect, the

659
00:39:54 --> 00:39:56
ones that are squared.
 

660
00:39:56 --> 00:39:58
So what do I get here?
 

661
00:39:58 --> 00:40:01
And the ones that are
squared I'm integrating,

662
00:40:01 --> 00:40:02
one, so I get a 2pi.
 

663
00:40:04 --> 00:40:04
So there's a 2pi.
 

664
00:40:05 --> 00:40:08
This whole subject's full of
these 2pi's and 2pi's, it's

665
00:40:08 --> 00:40:11
just part of the deal.
 

666
00:40:11 --> 00:40:15
Now, what's left?
 

667
00:40:15 --> 00:40:18
And now I'm looking only at the
terms where I'm integrating

668
00:40:18 --> 00:40:24
something by its conjugate,
by its own conjugate.

669
00:40:24 --> 00:40:28
And then I'm getting c_k times
its own conjugate, so I'm

670
00:40:28 --> 00:40:30
getting all the terms.
 

671
00:40:30 --> 00:40:34
I'm getting no cross terms,
just the terms that come from

672
00:40:34 --> 00:40:36
that times it's own conjugate.
 

673
00:40:36 --> 00:40:40
Which is c_k squared.
 

674
00:40:40 --> 00:40:46
That's the energy in
the coefficients.

675
00:40:46 --> 00:40:50
That's the energy in k space,
and of course that sum goes

676
00:40:50 --> 00:40:52
minus infinity to infinity.
 

677
00:40:52 --> 00:40:54
I've got them all.
 

678
00:40:54 --> 00:41:00
That's the energy in k space,
here's the energy in x space.

679
00:41:00 --> 00:41:06
You can expect that this
orthogonality is going

680
00:41:06 --> 00:41:11
to give you something
nice for the energy.

681
00:41:11 --> 00:41:13
For the integral of the square.
 

682
00:41:13 --> 00:41:16
Alright, so you saw that.
 

683
00:41:16 --> 00:41:18
And let's see.
 

684
00:41:18 --> 00:41:21
So we saw it for, we
actually got a good number

685
00:41:21 --> 00:41:23
for the square wave.
 

686
00:41:23 --> 00:41:28
We got infinity for
the delta function.

687
00:41:28 --> 00:41:32
I've put on here as a
last topic, to just

688
00:41:32 --> 00:41:36
give the word function.
 

689
00:41:36 --> 00:41:40
Part of what this course
is doing is to speak

690
00:41:40 --> 00:41:49
the language, teach the
language of applied math.

691
00:41:49 --> 00:41:53
So that when you see something,
you see this, you recognize hey

692
00:41:53 --> 00:41:56
I've seen that word before.
 

693
00:41:56 --> 00:42:00
So I want to know, what's the
right space of functions?

694
00:42:00 --> 00:42:07
This is my measure for the
length squared of a function.

695
00:42:07 --> 00:42:11
My square wave is great.
 

696
00:42:11 --> 00:42:14
It's length squared was
whatever it was, pi

697
00:42:14 --> 00:42:16
squared over something.
 

698
00:42:16 --> 00:42:22
And that's its length squared.
 

699
00:42:22 --> 00:42:26
The delta function
gave infinity.

700
00:42:26 --> 00:42:30
So it's not going to be allowed
into the function space.

701
00:42:30 --> 00:42:33
It's a vector of
infinite length.

702
00:42:33 --> 00:42:36
Like the vector <1,
1, 1, 1, 1> forever.

703
00:42:36 --> 00:42:38
Too long.
 

704
00:42:38 --> 00:42:41
Pythagoras fails, because
we sum the squares,

705
00:42:41 --> 00:42:42
we get infinite.
 

706
00:42:42 --> 00:42:49
So what functions, what vectors
should we allow in our space?

707
00:42:49 --> 00:42:52
So we're going to have a space
that's going to be infinite

708
00:42:52 --> 00:42:55
dimensional because our
coefficients, we've got

709
00:42:55 --> 00:42:57
infinitely many coefficients.
 

710
00:42:57 --> 00:43:01
Our functions have
infinitely many values.

711
00:43:01 --> 00:43:04
So we've moved up from
n dimensional space to

712
00:43:04 --> 00:43:05
infinite dimensional space.
 

713
00:43:05 --> 00:43:13
And everybody calls it after
the guy who, Hilbert space.

714
00:43:13 --> 00:43:17
So I don't know if you've seen
that word before, that name

715
00:43:17 --> 00:43:19
before, Hilbert space.
 

716
00:43:19 --> 00:43:24
It's the space of functions
with finite energy.

717
00:43:24 --> 00:43:26
Finite length.
 

718
00:43:26 --> 00:43:31
So this function is in it, the
delta function is not in it.

719
00:43:31 --> 00:43:38
And the point is that this
space of functions, we've got

720
00:43:38 --> 00:43:42
these guys are a great basis
for the space of functions.

721
00:43:42 --> 00:43:45
The sines and cosines are
another basis for this

722
00:43:45 --> 00:43:46
space of functions.
 

723
00:43:46 --> 00:43:51
We just have a whole lot of
functions, and all the facts

724
00:43:51 --> 00:43:55
of n-dimensional space.
 

725
00:43:55 --> 00:43:59
So what are important facts
about n-dimensional space?

726
00:43:59 --> 00:44:07
One that comes to mind that
involves length is the, a key

727
00:44:07 --> 00:44:17
fact about length is, length
and angle, I could say

728
00:44:17 --> 00:44:21
actually, many people would say
this is the most important

729
00:44:21 --> 00:44:23
inequality in mathematics.
 

730
00:44:23 --> 00:44:36
That the dot product of
two vectors, it's called

731
00:44:36 --> 00:44:39
the Schwarz inequality.
 

732
00:44:39 --> 00:44:48
Several people found it,
independently Schwarz is the

733
00:44:48 --> 00:44:50
single name most often used.
 

734
00:44:50 --> 00:44:55
What do you know about the
dot product of two vectors?

735
00:44:55 --> 00:44:58
Somehow it tells you the
angle between them, right?

736
00:44:58 --> 00:45:06
Somehow the dot product of two
vectors is, if I divide by the

737
00:45:06 --> 00:45:12
length of the vectors, so the
dot product of vectors divided

738
00:45:12 --> 00:45:16
by the length, do you know
what this is, in geometry?

739
00:45:16 --> 00:45:17
It's a cosine.
 

740
00:45:17 --> 00:45:21
It's the cosine of the
angle between them.

741
00:45:21 --> 00:45:25
And cosines are never
larger than one.

742
00:45:25 --> 00:45:28
So this quantity here is never
larger than one; in other

743
00:45:28 --> 00:45:32
words, this is never larger
than the length of one vector

744
00:45:32 --> 00:45:35
times the length of
the other vector.

745
00:45:35 --> 00:45:37
I could do an example.
 

746
00:45:37 --> 00:45:42
Let v be .
 

747
00:45:42 --> 00:45:46
And let w be .
 

748
00:45:46 --> 00:45:49
I don't know how this
is going to work.

749
00:45:49 --> 00:45:51
What's the dot product
of those two vectors?

750
00:45:51 --> 00:45:53
Oh, it's 19.
 

751
00:45:53 --> 00:45:54
Sorry about that.
 

752
00:45:54 --> 00:46:02
Let's change this, I'd
like a nice number here.

753
00:46:02 --> 00:46:06
What do you suggest?
 

754
00:46:06 --> 00:46:08
Make it five somewhere?
 

755
00:46:08 --> 00:46:12
Five wouldn't be bad.
 

756
00:46:12 --> 00:46:19
It wouldn't be too
good either, but.

757
00:46:19 --> 00:46:20
Ah, OK.
 

758
00:46:20 --> 00:46:22
What's the dot
product of those?

759
00:46:22 --> 00:46:23
16.
 

760
00:46:23 --> 00:46:26
And now what am I claiming,
that that's length less

761
00:46:26 --> 00:46:29
than the length of this
vector, which is what?

762
00:46:29 --> 00:46:32
What's the length of ?
 

763
00:46:32 --> 00:46:33
Square root of ten.
 

764
00:46:33 --> 00:46:36
It's good to do these small
ones, just to remember.

765
00:46:36 --> 00:46:38
The length of that vector is
the sum of the square root

766
00:46:38 --> 00:46:39
of the sum of the squares.
 

767
00:46:39 --> 00:46:41
Square root of ten.
 

768
00:46:41 --> 00:46:44
And the length of this guy?
 

769
00:46:44 --> 00:46:48
Is the square root of 26.
 

770
00:46:48 --> 00:46:53
And so I hope, and Schwarz
hopes, that 16 is less

771
00:46:53 --> 00:46:54
than that square root.
 

772
00:46:54 --> 00:46:58
Can we check it?
 

773
00:46:58 --> 00:47:02
Let's square both sides,
that would make it easier.

774
00:47:02 --> 00:47:07
So the right-hand side when I
square both sides will be?

775
00:47:07 --> 00:47:08
260.
 

776
00:47:08 --> 00:47:14
When I square both sides, and
what's the square of 16?

777
00:47:14 --> 00:47:16
256.
 

778
00:47:16 --> 00:47:18
That was close.
 

779
00:47:18 --> 00:47:24
But, it worked.
 

780
00:47:24 --> 00:47:27
I'll admit to you,
oops, not equal.

781
00:47:27 --> 00:47:28
Ah.
 

782
00:47:28 --> 00:47:32
Lesser equal,
Schwarz would say.

783
00:47:32 --> 00:47:36
And it's actually less than
because these vectors are

784
00:47:36 --> 00:47:38
not in the same direction.
 

785
00:47:38 --> 00:47:41
If they were exactly in the
same direction, or opposite

786
00:47:41 --> 00:47:45
directions, the cosine would be
one and we would have equal.

787
00:47:45 --> 00:47:48
But since the angle, you see
the angle between those two

788
00:47:48 --> 00:47:53
vectors is a pretty
small angle.

789
00:47:53 --> 00:47:55
The cosine is quite near one.
 

790
00:47:55 --> 00:47:58
But it's not exactly one.
 

791
00:47:58 --> 00:48:00
So I'm glad somebody
knew 16 squared.

792
00:48:00 --> 00:48:05
Does anybody know 99 squared?
 

793
00:48:05 --> 00:48:09
The reason I ask
that is, or 999.

794
00:48:09 --> 00:48:12
I'll make it sound harder.
 

795
00:48:12 --> 00:48:16
The hope, when I was about 11
or something, I was I was

796
00:48:16 --> 00:48:19
always hoping somebody
would ask me 999 squared.

797
00:48:19 --> 00:48:24
Because I was all ready
with the answer.

798
00:48:24 --> 00:48:25
Nobody ever asked.
 

799
00:48:25 --> 00:48:26
Anyway.
 

800
00:48:26 --> 00:48:28
But you've asked, I think.
 

801
00:48:28 --> 00:48:31
So 998,001.
 

802
00:48:31 --> 00:48:36
And now I I've finally got a
chance to show that I know it.

803
00:48:36 --> 00:48:39
OK, have a great
weekend and see you.

804
00:48:39 --> 00:48:40