1
00:01:03,000 --> 00:01:09,000
Okay, that's,
so to speak,
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00:01:05,000 --> 00:01:11,000
the text for today.
The Fourier series,
3
00:01:09,000 --> 00:01:15,000
and the Fourier expansion for f
of t,
4
00:01:14,000 --> 00:01:20,000
so f of t, if it looks like
this should be periodic,
5
00:01:19,000 --> 00:01:25,000
and two pi should be a period.
Sometimes people rather
6
00:01:25,000 --> 00:01:31,000
sloppily say periodic with
period two pi,
7
00:01:29,000 --> 00:01:35,000
but that's a little ambiguous.
So, this period could also be
8
00:01:37,000 --> 00:01:43,000
pi or a half pi or something
like that as well.
9
00:01:42,000 --> 00:01:48,000
The an's and bn's are
calculated according to these
10
00:01:47,000 --> 00:01:53,000
formulas.
Now, we're going to need in
11
00:01:51,000 --> 00:01:57,000
just a minute a consequence of
those formulas,
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00:01:56,000 --> 00:02:02,000
which, it's not subtle,
but because there are formulas
13
00:02:01,000 --> 00:02:07,000
for an and bn,
it follows that once you know f
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00:02:06,000 --> 00:02:12,000
of t,
the an's and bn's are
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00:02:10,000 --> 00:02:16,000
determined.
Or, to put it another way,
16
00:02:15,000 --> 00:02:21,000
a function cannot have two
different Fourier series.
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00:02:20,000 --> 00:02:26,000
Or, to put it yet another way,
if f of t,
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00:02:24,000 --> 00:02:30,000
if two functions are equal,
you'll see why I write it in
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00:02:30,000 --> 00:02:36,000
this rather peculiar form.
Then, the Fourier series for f
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00:02:35,000 --> 00:02:41,000
is the same as the Fourier
series for g.
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00:02:40,000 --> 00:02:46,000
And, the reason is because if f
is equal to g,
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00:02:44,000 --> 00:02:50,000
then this integral with an f
there is the same as the
23
00:02:49,000 --> 00:02:55,000
integral with a g there.
And therefore,
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00:02:52,000 --> 00:02:58,000
the an's come out to be the
same.
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00:02:55,000 --> 00:03:01,000
In the same way,
the bn's come out to be the
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00:02:58,000 --> 00:03:04,000
same.
So, the Fourier series are the
27
00:03:01,000 --> 00:03:07,000
same, coefficient by
coefficient, for f and g.
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00:03:05,000 --> 00:03:11,000
Now, my ultimate goal-- let's
all put down the argument since
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00:03:10,000 --> 00:03:16,000
there are formulas,
since we have formulas for an
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00:03:14,000 --> 00:03:20,000
and bn.
Now, a consequence of that is,
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00:03:19,000 --> 00:03:25,000
well, let me first say,
what I'm aiming at is you will
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00:03:23,000 --> 00:03:29,000
be amazed at how long it's going
to take me to get to this.
33
00:03:29,000 --> 00:03:35,000
I just want to calculate the
Fourier series for some rather
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00:03:33,000 --> 00:03:39,000
simple periodic function.
It's going to look like this.
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00:03:38,000 --> 00:03:44,000
So, here's pi,
and here's negative pi.
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00:03:41,000 --> 00:03:47,000
So, the function which just
looks like t in between those
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00:03:45,000 --> 00:03:51,000
two, so, it goes up to,
it's a function,
38
00:03:49,000 --> 00:03:55,000
t, more or less,
goes up to pi here,
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00:03:51,000 --> 00:03:57,000
minus pi there.
But, of course,
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00:03:54,000 --> 00:04:00,000
it's got to be periodic of
period two pi.
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00:03:59,000 --> 00:04:05,000
Well, then, it just repeats
itself after that.
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00:04:02,000 --> 00:04:08,000
After this, it just does that,
and so on.
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00:04:04,000 --> 00:04:10,000
It's a little ambiguous what
happens at these endpoints.
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00:04:08,000 --> 00:04:14,000
Well, let's not worry about
that for the moment,
45
00:04:12,000 --> 00:04:18,000
and frankly,
it won't really matter because
46
00:04:15,000 --> 00:04:21,000
the integrals don't care about
what happens in individual
47
00:04:19,000 --> 00:04:25,000
points.
So, there's my f of t.
48
00:04:21,000 --> 00:04:27,000
Now, I, of course,
could start doing it right
49
00:04:24,000 --> 00:04:30,000
away.
But, you will quickly find,
50
00:04:27,000 --> 00:04:33,000
if you start doing these
problems and hacking around with
51
00:04:30,000 --> 00:04:36,000
them, that the calculations seem
really quite long.
52
00:04:34,000 --> 00:04:40,000
And therefore,
in the first half of the
53
00:04:37,000 --> 00:04:43,000
period, the first half of the
period I want to show you how to
54
00:04:41,000 --> 00:04:47,000
shorten the calculations.
And in the second half of the
55
00:04:47,000 --> 00:04:53,000
period, after we've done that
and calculated this thing
56
00:04:50,000 --> 00:04:56,000
successfully,
I hope, I want to show you how
57
00:04:54,000 --> 00:05:00,000
to remove various restrictions
on these functions,
58
00:04:57,000 --> 00:05:03,000
how to extend the range of
Fourier series.
59
00:05:01,000 --> 00:05:07,000
Well, one obvious thing,
for example,
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00:05:03,000 --> 00:05:09,000
is suppose the function isn't
periodic of period two pi.
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00:05:06,000 --> 00:05:12,000
Suppose it has some other
period.
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00:05:08,000 --> 00:05:14,000
Does that mean there's no
formula?
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00:05:10,000 --> 00:05:16,000
Well, of course not.
There's a formula.
64
00:05:13,000 --> 00:05:19,000
But, we need to know what it
is, particularly in the
65
00:05:16,000 --> 00:05:22,000
applications,
the period is rarely two pi.
66
00:05:19,000 --> 00:05:25,000
It's normally one,
or something like that.
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00:05:21,000 --> 00:05:27,000
But, let's first of all,
I'm sure what you will
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00:05:24,000 --> 00:05:30,000
appreciate is how the
calculations can get shortened.
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00:05:29,000 --> 00:05:35,000
Now, the main way of shortening
them is by using evenness and
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00:05:35,000 --> 00:05:41,000
oddness.
And, what I claim is this,
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00:05:39,000 --> 00:05:45,000
that if f of t is an
even function,
72
00:05:44,000 --> 00:05:50,000
remember what that means,
that f of negative t is equal
73
00:05:51,000 --> 00:05:57,000
to f of t.
Cosine is a good example,
74
00:05:57,000 --> 00:06:03,000
of course, cosine nt;
are all these
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00:06:02,000 --> 00:06:08,000
functions are even functions.
If f of t is even,
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00:06:07,000 --> 00:06:13,000
then its Fourier series
contains only the cosine terms.
77
00:06:16,000 --> 00:06:22,000
In other words,
half the calculations you don't
78
00:06:21,000 --> 00:06:27,000
have to do if you start with an
even function.
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00:06:26,000 --> 00:06:32,000
That's what I mean by
shortening the work.
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00:06:31,000 --> 00:06:37,000
There are no odd terms,
or let's put it positively.
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00:06:37,000 --> 00:06:43,000
All the bn's are zero.
Now, one way of doing this
82
00:06:42,000 --> 00:06:48,000
would be to say,
well, y to the bn zero,
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00:06:44,000 --> 00:06:50,000
well, we've got formulas,
and fool around with the
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00:06:47,000 --> 00:06:53,000
formula for the bn,
and think about a little bit,
85
00:06:50,000 --> 00:06:56,000
and finally decide that that
has to come out to be zero.
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00:06:53,000 --> 00:06:59,000
That's not a bad way,
and it would remind you of some
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00:06:56,000 --> 00:07:02,000
basic facts about integration,
about integrals.
88
00:07:00,000 --> 00:07:06,000
Instead of doing that,
I'm going to apply my little
89
00:07:04,000 --> 00:07:10,000
principle that if two functions
are the same,
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00:07:08,000 --> 00:07:14,000
then their Fourier series have
to be the same.
91
00:07:12,000 --> 00:07:18,000
So, the argument I'm going to
give is this,
92
00:07:16,000 --> 00:07:22,000
so, I'm going to try to prove
this statement now.
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00:07:20,000 --> 00:07:26,000
And, I'm going to use the facts
on the first board to do it.
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00:07:25,000 --> 00:07:31,000
So, what is f of minus t?
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00:07:30,000 --> 00:07:36,000
Well, if that's equal to f of
t, then in terms of the
96
00:07:35,000 --> 00:07:41,000
Fourier series,
how do I get the Fourier series
97
00:07:39,000 --> 00:07:45,000
for f of minus t?
Well, I take the Fourier series
98
00:07:44,000 --> 00:07:50,000
for f of t, and substitute t
equals minus t.
99
00:07:48,000 --> 00:07:54,000
Now, what happens when I do
that?
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00:07:51,000 --> 00:07:57,000
So, the Fourier series for this
looks like a zero over two
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00:07:56,000 --> 00:08:02,000
plus summation what?
Well, the an cosine nt,
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00:08:02,000 --> 00:08:08,000
that does not change
because when I change t to
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00:08:06,000 --> 00:08:12,000
negative t,
the cosine nt does
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00:08:11,000 --> 00:08:17,000
not change, stays the same
because it's an even function.
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00:08:15,000 --> 00:08:21,000
What happens to the sine term?
Well, the sine of negative nt
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00:08:20,000 --> 00:08:26,000
is equal to minus the
sine of nt.
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00:08:25,000 --> 00:08:31,000
So, the other terms,
the sine terms change sign.
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00:08:30,000 --> 00:08:36,000
So, all that's the result of
substituting t for negative t
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00:08:34,000 --> 00:08:40,000
and f of t.
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00:08:36,000 --> 00:08:42,000
On the other hand,
what's f of t itself?
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00:08:40,000 --> 00:08:46,000
Well, f of t itself is what
happened before that.
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00:08:43,000 --> 00:08:49,000
Now it's got a plus sign
because nothing was done to the
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00:08:48,000 --> 00:08:54,000
series.
Well, if the function is even,
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00:08:50,000 --> 00:08:56,000
then those two right hand sides
are the same function.
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00:08:54,000 --> 00:09:00,000
In other words,
they're like my f of t equals g
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00:08:58,000 --> 00:09:04,000
of t. And therefore,
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00:09:02,000 --> 00:09:08,000
the Fourier series on the left
must be the same.
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00:09:06,000 --> 00:09:12,000
In other words,
if these are equal,
119
00:09:09,000 --> 00:09:15,000
therefore, these have to be
equal, too.
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00:09:13,000 --> 00:09:19,000
Now, there's no problem with
the cosine terms.
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00:09:17,000 --> 00:09:23,000
They look the same.
On the other hand,
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00:09:20,000 --> 00:09:26,000
the sine terms have changed
sign.
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00:09:23,000 --> 00:09:29,000
Therefore, it must be the case
that bn is always equal to
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00:09:28,000 --> 00:09:34,000
negative bn for all n.
That's the only way this series
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00:09:34,000 --> 00:09:40,000
can be the same as that one.
Now, if bn is equal to negative
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00:09:39,000 --> 00:09:45,000
bn,
that implies that bn is zero.
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00:09:43,000 --> 00:09:49,000
Zero is the only number which
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00:09:46,000 --> 00:09:52,000
is equal to its negative.
And so, by this argument,
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00:09:51,000 --> 00:09:57,000
in other words,
using the uniqueness of Fourier
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00:09:54,000 --> 00:10:00,000
series, we conclude that if the
function is even,
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00:09:59,000 --> 00:10:05,000
then its Fourier series can
only have cosine terms in it.
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00:10:05,000 --> 00:10:11,000
Now, you say,
hey, that's obvious.
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The cosine, that's just a point
of logic.
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00:10:09,000 --> 00:10:15,000
But, this is a mathematics
course, after all.
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00:10:12,000 --> 00:10:18,000
It's not just about
calculation.
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00:10:14,000 --> 00:10:20,000
Many of you would say,
yeah, of course that's obvious
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00:10:18,000 --> 00:10:24,000
because cosines are even,
and the sines are odd.
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00:10:21,000 --> 00:10:27,000
I say, yeah,
and so why does that make it
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00:10:24,000 --> 00:10:30,000
true?
Well, the cosine's even.
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00:10:25,000 --> 00:10:31,000
Plus t into minus t,
and what you are proving
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00:10:29,000 --> 00:10:35,000
is the converse.
The converse is obvious.
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00:10:33,000 --> 00:10:39,000
Yeah, obvious,
I don't care.
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00:10:35,000 --> 00:10:41,000
If the right-hand side is the
sum of the functions,
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00:10:39,000 --> 00:10:45,000
well, so is the left.
But I'm saying it the other way
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00:10:43,000 --> 00:10:49,000
around.
If the left is an even
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00:10:45,000 --> 00:10:51,000
function, why does the
right-hand side have to have
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00:10:49,000 --> 00:10:55,000
only even terms in it?
And, this is the argument which
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00:10:53,000 --> 00:10:59,000
makes that true.
Now, there is a further
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00:10:56,000 --> 00:11:02,000
simplification because if you've
got an even function,
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00:11:00,000 --> 00:11:06,000
oh, by the way,
of course the same thing is
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00:11:03,000 --> 00:11:09,000
true for the odd,
I ought to put that down,
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00:11:06,000 --> 00:11:12,000
and so also,
if f of t is odd,
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00:11:09,000 --> 00:11:15,000
then I think one of these
proofs is enough.
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00:11:14,000 --> 00:11:20,000
The other you can supply
yourself.
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00:11:17,000 --> 00:11:23,000
That will imply that all the
an's are zero,
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00:11:20,000 --> 00:11:26,000
even including this first one,
a zero,
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00:11:25,000 --> 00:11:31,000
and by the same reasoning.
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00:11:37,000 --> 00:11:43,000
So, an even function uses only
cosines for its Fourier
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00:11:41,000 --> 00:11:47,000
expansion.
An odd function uses only
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00:11:44,000 --> 00:11:50,000
sines.
Good.
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00:11:45,000 --> 00:11:51,000
But, we still have to,
suppose we got an even
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00:11:49,000 --> 00:11:55,000
function.
We've still got to calculate
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00:11:53,000 --> 00:11:59,000
this integral.
Well, even that can be
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00:11:56,000 --> 00:12:02,000
simplified.
So, the second stage of the
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00:11:59,000 --> 00:12:05,000
simplification,
again, assuming that we have an
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00:12:04,000 --> 00:12:10,000
even or odd function,
and by the way,
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00:12:07,000 --> 00:12:13,000
[LAUGHTER].
Totally unauthorized.
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00:12:26,000 --> 00:12:32,000
So, if f of t is even,
what we'd like to do now is
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00:12:34,000 --> 00:12:40,000
simplify the integral a little.
And, there is an easy way to do
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00:12:43,000 --> 00:12:49,000
that, because,
look, if f of t is an even
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00:12:49,000 --> 00:12:55,000
function, then so is f of t
cosine nt,
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00:12:57,000 --> 00:13:03,000
is also even.
Imagine, we could make little
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00:13:02,000 --> 00:13:08,000
rules about an even function
times an even function is an
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00:13:06,000 --> 00:13:12,000
even function.
There are general rules of that
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00:13:09,000 --> 00:13:15,000
type, and some of you know them,
and they are very useful.
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00:13:13,000 --> 00:13:19,000
But, let's just do it ad hoc
here.
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00:13:15,000 --> 00:13:21,000
If I change t to negative
t here,
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00:13:18,000 --> 00:13:24,000
I don't change the function
because it's even.
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00:13:21,000 --> 00:13:27,000
And, I don't change the cosine
because that's even.
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00:13:24,000 --> 00:13:30,000
So, if I change t to negative
t, I don't change the function.
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00:13:28,000 --> 00:13:34,000
Either factor that function,
and therefore I don't change
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00:13:32,000 --> 00:13:38,000
the product of those two things
either.
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00:13:36,000 --> 00:13:42,000
So, it's also even.
Now, what about an even
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00:13:41,000 --> 00:13:47,000
function when you integrate it?
Here's a typical looking even
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00:13:48,000 --> 00:13:54,000
function, let's say,
something like,
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00:13:52,000 --> 00:13:58,000
I don't know,
wiggle, wiggle,
187
00:13:56,000 --> 00:14:02,000
again.
Here's our better even
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00:13:59,000 --> 00:14:05,000
function.
All right, so,
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00:14:02,000 --> 00:14:08,000
minus pi to pi,
even, even though the t-axis is
190
00:14:08,000 --> 00:14:14,000
somewhat curvy.
So, there is an even function.
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00:14:14,000 --> 00:14:20,000
The point is that if you
integrate an even function from
192
00:14:17,000 --> 00:14:23,000
negative pi to pi,
I think you all know even from
193
00:14:21,000 --> 00:14:27,000
calculus you were taught to do
this simplification.
194
00:14:24,000 --> 00:14:30,000
Don't do that.
Instead, integrate from zero to
195
00:14:27,000 --> 00:14:33,000
pi, and double the answer.
Why should you do that?
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00:14:31,000 --> 00:14:37,000
The answer is because it's
always nice to have zero as one
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00:14:35,000 --> 00:14:41,000
of the limits of integration.
I trust to your experience,
198
00:14:39,000 --> 00:14:45,000
I don't have to sell that.
Minus pi is a particularly
199
00:14:43,000 --> 00:14:49,000
unpleasant lower limit of
integration because you are sure
200
00:14:47,000 --> 00:14:53,000
to get in trouble with negative
signs.
201
00:14:50,000 --> 00:14:56,000
There are bound to be at least
three negative signs floating
202
00:14:54,000 --> 00:15:00,000
around.
And, if you miss one of them,
203
00:14:57,000 --> 00:15:03,000
you'll get the wrong signs of
answer.
204
00:15:01,000 --> 00:15:07,000
The answer will have the wrong
sign.
205
00:15:03,000 --> 00:15:09,000
So, the way the formula from
this simplifies is that an,
206
00:15:08,000 --> 00:15:14,000
instead of integrating from
negative pi to pi,
207
00:15:12,000 --> 00:15:18,000
I can integrate only from zero
to pi, and double the answer.
208
00:15:17,000 --> 00:15:23,000
So, our better formula is this.
If the function is even,
209
00:15:22,000 --> 00:15:28,000
this is the formula you should
use: zero to pi,
210
00:15:26,000 --> 00:15:32,000
f of t cosine nt dt.
211
00:15:31,000 --> 00:15:37,000
Of course, I don't have to tell
212
00:15:35,000 --> 00:15:41,000
you what bn should be because bn
will be zero.
213
00:15:39,000 --> 00:15:45,000
And, in the same way,
if f is odd,
214
00:15:42,000 --> 00:15:48,000
the same reasoning shows that
bn-- of course,
215
00:15:45,000 --> 00:15:51,000
an will be zero this time.
But it will be bn that will be
216
00:15:50,000 --> 00:15:56,000
two over pi times the integral
from zero to pi of f of t sine
217
00:15:55,000 --> 00:16:01,000
nt dt.
218
00:16:00,000 --> 00:16:06,000
Maybe we'd better just a word
219
00:16:03,000 --> 00:16:09,000
about that since,
why is that so?
220
00:16:06,000 --> 00:16:12,000
If it's odd,
doesn't that mean things become
221
00:16:08,000 --> 00:16:14,000
zero?
If you integrate an odd
222
00:16:10,000 --> 00:16:16,000
function like that,
the integral over minus pi to
223
00:16:14,000 --> 00:16:20,000
pi, you get zero.
Well, but this is not an odd
224
00:16:17,000 --> 00:16:23,000
function.
This is an odd function,
225
00:16:19,000 --> 00:16:25,000
and this is an odd function.
But the product of two odd
226
00:16:22,000 --> 00:16:28,000
functions is an even function.
Odd times odd is even.
227
00:16:26,000 --> 00:16:32,000
I said I wasn't going to give
you those rules,
228
00:16:29,000 --> 00:16:35,000
but since this is the one which
trips everybody up,
229
00:16:32,000 --> 00:16:38,000
maybe I'd better say it just
justbecause it looks wrong.
230
00:16:38,000 --> 00:16:44,000
Right, this is odd.
That's odd.
231
00:16:40,000 --> 00:16:46,000
Think about it.
If I change t to negative t,
232
00:16:43,000 --> 00:16:49,000
this multiplies by
minus one.
233
00:16:46,000 --> 00:16:52,000
This multiplies by minus one.
And therefore,
234
00:16:49,000 --> 00:16:55,000
the product multiplies by minus
one times minus one.
235
00:16:54,000 --> 00:17:00,000
In other words,
it multiplies by plus one.
236
00:16:57,000 --> 00:17:03,000
Nothing happens,
so it stays the same.
237
00:17:01,000 --> 00:17:07,000
Why does nobody believe this,
even though it's true?
238
00:17:04,000 --> 00:17:10,000
It's because they are thinking
about numbers.
239
00:17:08,000 --> 00:17:14,000
Everybody knows that an odd
number times an odd number is an
240
00:17:12,000 --> 00:17:18,000
odd number.
So, I'm not multiplying numbers
241
00:17:15,000 --> 00:17:21,000
here, which also I'll put them
in boxes to indicate that they
242
00:17:20,000 --> 00:17:26,000
are not numbers.
How's that?
243
00:17:22,000 --> 00:17:28,000
Brand-new invented notation.
The box means caution.
244
00:17:25,000 --> 00:17:31,000
The inside is not a number,
it's the word odd or even.
245
00:17:31,000 --> 00:17:37,000
It's just a symbolic statement
that the product of an odd
246
00:17:35,000 --> 00:17:41,000
function and an odd function is
an even function.
247
00:17:39,000 --> 00:17:45,000
Even times even is even.
What's odd times even?
248
00:17:43,000 --> 00:17:49,000
Yes, it has to get equal time.
Obviously, something must come
249
00:17:47,000 --> 00:17:53,000
out to be odd,
right.
250
00:17:49,000 --> 00:17:55,000
Okay, so, now that we've got
our two simplifications,
251
00:17:53,000 --> 00:17:59,000
we are ready to do this
problem.
252
00:17:56,000 --> 00:18:02,000
Instead of attacking it with
the original formulas,
253
00:18:00,000 --> 00:18:06,000
we are going to think about it
and attack it with our better
254
00:18:04,000 --> 00:18:10,000
formulas.
So, now we are going to
255
00:18:11,000 --> 00:18:17,000
calculate the Fourier series for
f of t.
256
00:18:19,000 --> 00:18:25,000
The first thing I see,
so f of t is our little thing
257
00:18:29,000 --> 00:18:35,000
here.
Well, first of all,
258
00:18:32,000 --> 00:18:38,000
what kind of function is it:
odd, even, or neither?
259
00:18:35,000 --> 00:18:41,000
Most functions are neither,
of course.
260
00:18:38,000 --> 00:18:44,000
But, fortunately in the
applications,
261
00:18:40,000 --> 00:18:46,000
functions tend to be one or the
other.
262
00:18:42,000 --> 00:18:48,000
Or, they can be converted into
one to the other.
263
00:18:46,000 --> 00:18:52,000
Maybe if I get a chance,
I'll show you a little how,
264
00:18:49,000 --> 00:18:55,000
or the recitations will.
So, this function is odd.
265
00:18:52,000 --> 00:18:58,000
Okay, half the work just
disappeared.
266
00:18:55,000 --> 00:19:01,000
I don't have to calculate any
an's.
267
00:18:57,000 --> 00:19:03,000
They will be zero.
So, I only have to calculate
268
00:19:01,000 --> 00:19:07,000
bn, and I'll calculate them by
my better formula.
269
00:19:04,000 --> 00:19:10,000
So, it's two over pi times the
integral from zero to pi,
270
00:19:08,000 --> 00:19:14,000
and what I have to integrate,
well, now, finally you've got
271
00:19:11,000 --> 00:19:17,000
to integrate something.
From zero to pi,
272
00:19:14,000 --> 00:19:20,000
this is the function,
t.
273
00:19:15,000 --> 00:19:21,000
So, I have to integrate t times
sine of nt dt.
274
00:19:18,000 --> 00:19:24,000
Okay,
275
00:19:22,000 --> 00:19:28,000
so this is why you learned
integration by parts,
276
00:19:25,000 --> 00:19:31,000
one of many reasons why you
learned integration by parts,
277
00:19:29,000 --> 00:19:35,000
so that you wouldn't have to
pull out your little calculators
278
00:19:32,000 --> 00:19:38,000
to do this.
Okay, now, let's do it.
279
00:19:36,000 --> 00:19:42,000
So, it's two over pi.
280
00:19:39,000 --> 00:19:45,000
Let's solve that away so we can
forget about it.
281
00:19:42,000 --> 00:19:48,000
And, what's then left is just
the evaluation of the integral
282
00:19:47,000 --> 00:19:53,000
between limits.
So, if I integrate by parts,
283
00:19:50,000 --> 00:19:56,000
I'll want to differentiate the
t, and integrate the sign,
284
00:19:54,000 --> 00:20:00,000
right?
So, the first step is you don't
285
00:19:57,000 --> 00:20:03,000
do the differentiation.
You only do the integration.
286
00:20:02,000 --> 00:20:08,000
So, that integrates to be
cosine nt over n,
287
00:20:05,000 --> 00:20:11,000
more or less.
The only thing is,
288
00:20:08,000 --> 00:20:14,000
if I differentiate this,
I get negative sine nt
289
00:20:11,000 --> 00:20:17,000
instead of,
so, I want to put a negative
290
00:20:15,000 --> 00:20:21,000
sign in front of all this.
And, I will evaluate that
291
00:20:19,000 --> 00:20:25,000
between the limits,
zero and pi,
292
00:20:21,000 --> 00:20:27,000
and then subtract what you get
by doing both things,
293
00:20:25,000 --> 00:20:31,000
both the differentiation and
the integration.
294
00:20:28,000 --> 00:20:34,000
So, I subtract the integral
from zero to pi.
295
00:20:33,000 --> 00:20:39,000
I now differentiate the t,
and integrate.
296
00:20:35,000 --> 00:20:41,000
Well, I just did the
integration.
297
00:20:37,000 --> 00:20:43,000
That's negative cosine nt over
n.
298
00:20:40,000 --> 00:20:46,000
You see how the negative signs
pile up?
299
00:20:43,000 --> 00:20:49,000
And, if this is negative pi
instead of zero,
300
00:20:45,000 --> 00:20:51,000
it's at that point when it
starts to lose heart.
301
00:20:48,000 --> 00:20:54,000
You see three negative signs,
and then when you substitute,
302
00:20:52,000 --> 00:20:58,000
you're going to have to put in
still something else negative,
303
00:20:56,000 --> 00:21:02,000
and you just have the feeling
you're going to make a mistake.
304
00:21:01,000 --> 00:21:07,000
And, you will.
Okay, now all we have to do is
305
00:21:05,000 --> 00:21:11,000
a little evaluation.
Let's see, at the lower limit I
306
00:21:09,000 --> 00:21:15,000
get zero, here.
Let's right away,
307
00:21:12,000 --> 00:21:18,000
as two over pi.
At the lower limit,
308
00:21:16,000 --> 00:21:22,000
I get zero.
That's nice.
309
00:21:18,000 --> 00:21:24,000
At the upper limit,
I get minus pi over n times the
310
00:21:23,000 --> 00:21:29,000
cosine of n pi.
311
00:21:26,000 --> 00:21:32,000
Now, once and for all,
the cosine of n pi--
312
00:21:31,000 --> 00:21:37,000
If you like to make
separate steps out of
313
00:21:35,000 --> 00:21:41,000
everything, okay,
I'll let you do it this time,
314
00:21:39,000 --> 00:21:45,000
--
-- but in the long run,
315
00:21:43,000 --> 00:21:49,000
it's good to remember that
that's negative one to the n'th
316
00:21:48,000 --> 00:21:54,000
power
The cosine of pi is minus one .
317
00:21:51,000 --> 00:21:57,000
The cosine of two pi is plus
one,
318
00:21:55,000 --> 00:22:01,000
three pi, minus one,
and so on.
319
00:22:00,000 --> 00:22:06,000
So, at the upper limit,
we get minus pi over n,
320
00:22:05,000 --> 00:22:11,000
oh, I didn't finish the
calculation, times the cosine of
321
00:22:11,000 --> 00:22:17,000
n pi,
which is minus one to the n'th
322
00:22:17,000 --> 00:22:23,000
power.
And now, how about the other
323
00:22:22,000 --> 00:22:28,000
guy?
Shall we do in our heads?
324
00:22:26,000 --> 00:22:32,000
Well, I can do it in my head,
but I'm not so sure about your
325
00:22:32,000 --> 00:22:38,000
heads.
Maybe just this once we won't.
326
00:22:37,000 --> 00:22:43,000
What is it?
It's plus sine nt,
327
00:22:41,000 --> 00:22:47,000
right?
So, I combined the two negative
328
00:22:44,000 --> 00:22:50,000
signs to a plus sign by putting
one this way and the other one
329
00:22:50,000 --> 00:22:56,000
that way.
And then, if I integrate that
330
00:22:53,000 --> 00:22:59,000
now, it's sine nt divided by n
squared,
331
00:22:58,000 --> 00:23:04,000
right?
And that's evaluated between
332
00:23:02,000 --> 00:23:08,000
zero and pi.
And of course,
333
00:23:05,000 --> 00:23:11,000
the sign function vanishes at
both ends.
334
00:23:09,000 --> 00:23:15,000
So, that part is simply zero.
And so, the final answer is
335
00:23:14,000 --> 00:23:20,000
that bn is equal to,
well, the pi's cancel.
336
00:23:19,000 --> 00:23:25,000
This minus combines with those
n to make one more.
337
00:23:23,000 --> 00:23:29,000
And so, the answer is two over
n times minus one to the n plus
338
00:23:30,000 --> 00:23:36,000
first power.
339
00:23:35,000 --> 00:23:41,000
And therefore,
the final result is that our
340
00:23:40,000 --> 00:23:46,000
Fourier series,
the Fourier series for f of t,
341
00:23:46,000 --> 00:23:52,000
that funny function
is, the Fourier series is
342
00:23:53,000 --> 00:23:59,000
summation bn,
which is two,
343
00:23:56,000 --> 00:24:02,000
put the two out front because
it's in every term.
344
00:24:04,000 --> 00:24:10,000
There's no reason to repeat it,
minus one to the n plus first
345
00:24:10,000 --> 00:24:16,000
power over n times the sign of
nt.
346
00:24:16,000 --> 00:24:22,000
That's summed from one to
347
00:24:19,000 --> 00:24:25,000
infinity.
Let's stop and take a look at
348
00:24:23,000 --> 00:24:29,000
that for a second.
Does that look right?
349
00:24:28,000 --> 00:24:34,000
Okay, here's our function.
350
00:24:41,000 --> 00:24:47,000
Here's our function.
What's the first term of this?
351
00:24:48,000 --> 00:24:54,000
When n is one,
this is plus one.
352
00:24:54,000 --> 00:25:00,000
So, the first term is sine t.
353
00:25:00,000 --> 00:25:06,000
What's the next term?
When n is two,
354
00:25:04,000 --> 00:25:10,000
this is negative.
So, it's minus one to the third
355
00:25:08,000 --> 00:25:14,000
power.
So, that's negative one over
356
00:25:11,000 --> 00:25:17,000
two.
So, it's minus one half sine
357
00:25:13,000 --> 00:25:19,000
two t,
and then it obviously continues
358
00:25:17,000 --> 00:25:23,000
in the same way plus a third
sign three t.
359
00:25:21,000 --> 00:25:27,000
Now, watch carefully because
what I'm going to say in the
360
00:25:24,000 --> 00:25:30,000
next minute is the heart of
Fourier series.
361
00:25:27,000 --> 00:25:33,000
I've given you that visual to
look at to try to reinforce
362
00:25:31,000 --> 00:25:37,000
this, but it's really very
important, as you go to the
363
00:25:34,000 --> 00:25:40,000
terminal yourself and do that
work, simple as it is,
364
00:25:38,000 --> 00:25:44,000
and pay attention now.
Now, if you think
365
00:25:42,000 --> 00:25:48,000
old-fashioned,
i.e.
366
00:25:43,000 --> 00:25:49,000
if you think taylor series,
you're not going to believe
367
00:25:47,000 --> 00:25:53,000
this because you will say,
well, let's see,
368
00:25:50,000 --> 00:25:56,000
these go on and on.
Obviously, it's the first term
369
00:25:53,000 --> 00:25:59,000
that's the important one.
That's two sine t.
370
00:25:57,000 --> 00:26:03,000
Now, the derivative,
two sine t, sine t would
371
00:26:00,000 --> 00:26:06,000
exactly follow the pink curve.
Sine t would look like this.
372
00:26:06,000 --> 00:26:12,000
Two sine t goes up
with the wrong angle.
373
00:26:10,000 --> 00:26:16,000
The first term,
in other words,
374
00:26:12,000 --> 00:26:18,000
does this.
It's going off with the wrong
375
00:26:15,000 --> 00:26:21,000
slope.
Now, that's the whole point of
376
00:26:18,000 --> 00:26:24,000
Fourier series.
Fourier series is not trying to
377
00:26:22,000 --> 00:26:28,000
approximate the function at zero
at the central starting point
378
00:26:27,000 --> 00:26:33,000
the way Taylor series do.
Fourier series tries to treat
379
00:26:31,000 --> 00:26:37,000
the whole interval,
and approximate the function
380
00:26:35,000 --> 00:26:41,000
nicely over the entire interval,
in this case,
381
00:26:38,000 --> 00:26:44,000
minus pi to pi,
as well as possible.
382
00:26:40,000 --> 00:26:46,000
Taylor series concentrates at
this point, does it the best it
383
00:26:44,000 --> 00:26:50,000
can at this point.
Then it tries,
384
00:26:46,000 --> 00:26:52,000
with the next term,
to do a little better,
385
00:26:49,000 --> 00:26:55,000
and then a little better.
The whole philosophy is
386
00:26:52,000 --> 00:26:58,000
entirely different.
Taylor series are used for
387
00:26:55,000 --> 00:27:01,000
analyzing what a function of
looks like which you stick close
388
00:26:59,000 --> 00:27:05,000
to the base point.
Fourier series analyze what a
389
00:27:04,000 --> 00:27:10,000
function looks like over the
whole interval.
390
00:27:07,000 --> 00:27:13,000
And, to do that,
you should therefore aim to,
391
00:27:10,000 --> 00:27:16,000
so the first approximation is
going to look like that,
392
00:27:14,000 --> 00:27:20,000
going to have entirely the
wrong slope.
393
00:27:16,000 --> 00:27:22,000
But, the next one will subtract
off something which sort of
394
00:27:21,000 --> 00:27:27,000
helps to fix it up.
I can't draw this.
395
00:27:23,000 --> 00:27:29,000
That's why I'm sending you to
the visual because the visual
396
00:27:27,000 --> 00:27:33,000
draws them beautifully.
And, it shows you how each
397
00:27:31,000 --> 00:27:37,000
successive term corrects the
Fourier series,
398
00:27:34,000 --> 00:27:40,000
and makes the sum a little
closer to what you started with.
399
00:27:40,000 --> 00:27:46,000
So, the next guy would,
let's see, so it's 2t.
400
00:27:44,000 --> 00:27:50,000
So, I'm subtracting off,
probably I'm just guessing,
401
00:27:50,000 --> 00:27:56,000
but I don't dare draw this.
I haven't prepared to draw it,
402
00:27:56,000 --> 00:28:02,000
and I know I'll get it wrong.
So, okay, your exercise.
403
00:28:02,000 --> 00:28:08,000
But, it'll look better.
It'll go, maybe,
404
00:28:07,000 --> 00:28:13,000
something like,
let's see, it has to end up...
405
00:28:12,000 --> 00:28:18,000
some of it gets subtracted
off...
406
00:28:17,000 --> 00:28:23,000
I don't know what it looks
like.
407
00:28:20,000 --> 00:28:26,000
When you use the visual at the
computer terminal,
408
00:28:25,000 --> 00:28:31,000
I've asked you to use it three
times on a variety of functions.
409
00:28:32,000 --> 00:28:38,000
I think this is maybe even one
of them.
410
00:28:34,000 --> 00:28:40,000
Notice that you can set the
parameter, you can set the
411
00:28:38,000 --> 00:28:44,000
coefficients independently.
In other words,
412
00:28:41,000 --> 00:28:47,000
you can go back and correct
your works, improving the
413
00:28:45,000 --> 00:28:51,000
earlier coefficients,
and it won't affect anything
414
00:28:48,000 --> 00:28:54,000
you did before.
But, the most vivid way to do
415
00:28:51,000 --> 00:28:57,000
it is to try to get,
visually, by moving the slider,
416
00:28:55,000 --> 00:29:01,000
to try to get the very best
value for the first coefficient
417
00:28:59,000 --> 00:29:05,000
you can, and look at the curve.
Then get the very best value
418
00:29:05,000 --> 00:29:11,000
for the second coefficient and
see how that improves the
419
00:29:09,000 --> 00:29:15,000
approximation,
and the third,
420
00:29:11,000 --> 00:29:17,000
and so on.
And, the point is,
421
00:29:13,000 --> 00:29:19,000
watch the approximations
approaching the function nicely
422
00:29:18,000 --> 00:29:24,000
over the whole interval instead
of concentrating all their
423
00:29:22,000 --> 00:29:28,000
goodness at the origin the way a
Taylor series would.
424
00:29:26,000 --> 00:29:32,000
Now, there is still one
mathematical point left.
425
00:29:30,000 --> 00:29:36,000
It's that equality sign,
which is wrong.
426
00:29:35,000 --> 00:29:41,000
Why is it wrong?
Well, what I'm saying is that
427
00:29:38,000 --> 00:29:44,000
if I add that the series,
it adds up to f of t.
428
00:29:43,000 --> 00:29:49,000
Now, it almost does but not
quite.
429
00:29:46,000 --> 00:29:52,000
And, I'd better give you the
rule, the theorem.
430
00:29:50,000 --> 00:29:56,000
Of all the theorems in this
course that aren't being proved,
431
00:29:56,000 --> 00:30:02,000
this is the one that would be
most outside the scope of this
432
00:30:01,000 --> 00:30:07,000
course, the one which I would
most like to prove,
433
00:30:05,000 --> 00:30:11,000
in fact, just because I'm a
mathematician but wouldn't dare.
434
00:30:12,000 --> 00:30:18,000
The theorem tells you when a
Fourier series converges to the
435
00:30:17,000 --> 00:30:23,000
function you started with.
And, the essence of it is this.
436
00:30:22,000 --> 00:30:28,000
If f is continuous,
is a continuous function,
437
00:30:26,000 --> 00:30:32,000
let's give the point,
it's confusing just to keep
438
00:30:30,000 --> 00:30:36,000
calling it t.
If you like,
439
00:30:32,000 --> 00:30:38,000
call it t, but I think it would
be better to call it t zero
440
00:30:37,000 --> 00:30:43,000
just to indicate I'm
looking at a specific point.
441
00:30:44,000 --> 00:30:50,000
So, if the function is
continuous there,
442
00:30:48,000 --> 00:30:54,000
the value of f of t is
equal to, the Fourier series
443
00:30:54,000 --> 00:31:00,000
converges, and it's equal to its
Fourier series,
444
00:30:59,000 --> 00:31:05,000
the sum of the Fourier series
at t zero.
445
00:31:05,000 --> 00:31:11,000
And, the fact that I can even
use the word sum means that the
446
00:31:09,000 --> 00:31:15,000
Fourier series converges.
In other words,
447
00:31:12,000 --> 00:31:18,000
when you add up all these guys,
you don't go to infinity or get
448
00:31:16,000 --> 00:31:22,000
something which just oscillates
around crazily.
449
00:31:20,000 --> 00:31:26,000
They really do add up to
something.
450
00:31:22,000 --> 00:31:28,000
Now, if f is not continuous at
t zero,
451
00:31:26,000 --> 00:31:32,000
this emphatically will not be
the case.
452
00:31:28,000 --> 00:31:34,000
It will definitely not,
but by far, the kinds of
453
00:31:32,000 --> 00:31:38,000
discontinuities which occur in
the applications are ones like
454
00:31:36,000 --> 00:31:42,000
in this picture,
where the discontinuities are
455
00:31:39,000 --> 00:31:45,000
jump discontinuities.
They are almost always jump
456
00:31:44,000 --> 00:31:50,000
discontinuities.
And, in that case,
457
00:31:47,000 --> 00:31:53,000
in other words,
they are isolated.
458
00:31:49,000 --> 00:31:55,000
The function looks good here
and here, but there's a break.
459
00:31:53,000 --> 00:31:59,000
Typically, electrical engineers
just don't leave a gap because
460
00:31:57,000 --> 00:32:03,000
they like, I don't know why.
But electrical engineer,
461
00:32:02,000 --> 00:32:08,000
and others of his or her ilk
would draw that function like
462
00:32:09,000 --> 00:32:15,000
this, like a rip saw tooth.
Even those vertical lines have
463
00:32:16,000 --> 00:32:22,000
no meaning whatever,
but they make people look
464
00:32:21,000 --> 00:32:27,000
happier.
So, if f has a jump
465
00:32:24,000 --> 00:32:30,000
discontinuity at t zero,
and as I said,
466
00:32:30,000 --> 00:32:36,000
that's the most important kind,
then f of t,
467
00:32:36,000 --> 00:32:42,000
then the Fourier series adds up
to, converges to,
468
00:32:42,000 --> 00:32:48,000
it converges,
and it converges to the mid
469
00:32:46,000 --> 00:32:52,000
point of the jump.
Let me just write it out in
470
00:32:52,000 --> 00:32:58,000
words like that,
the midpoint of the jump.
471
00:32:55,000 --> 00:33:01,000
That's the way we'll be using
it in this course.
472
00:32:58,000 --> 00:33:04,000
There's a notation for this,
and it's in your book.
473
00:33:02,000 --> 00:33:08,000
But, those of you who would be
interested in such things would
474
00:33:07,000 --> 00:33:13,000
know it anyway.
So, let's just call it the
475
00:33:11,000 --> 00:33:17,000
midpoint of the jump.
So, if I ask you,
476
00:33:14,000 --> 00:33:20,000
to what does this converge?
In other words,
477
00:33:18,000 --> 00:33:24,000
this series,
what this shows is that the
478
00:33:22,000 --> 00:33:28,000
series, I'll write it out in the
abbreviated form,
479
00:33:26,000 --> 00:33:32,000
summation minus one to the n
plus one over n sine nt,
480
00:33:32,000 --> 00:33:38,000
what's the sum of the series?
481
00:33:39,000 --> 00:33:45,000
What is it?
Let's call this not
482
00:33:41,000 --> 00:33:47,000
little f of t.
Let's call it capital F of t.
483
00:33:44,000 --> 00:33:50,000
I want to know,
484
00:33:46,000 --> 00:33:52,000
what's the graph of capital F
of t?
485
00:33:48,000 --> 00:33:54,000
Well, the initial thing is to
say, well, it must be the same
486
00:33:53,000 --> 00:33:59,000
as the graph of the function you
started with.
487
00:33:56,000 --> 00:34:02,000
And, my answer is almost,
but not quite.
488
00:34:00,000 --> 00:34:06,000
In fact, what will its graph
look like?
489
00:34:04,000 --> 00:34:10,000
Well, regardless of what
definition I made for the
490
00:34:09,000 --> 00:34:15,000
endpoints of those pink lines,
this function will converge to
491
00:34:16,000 --> 00:34:22,000
the following.
From here to here,
492
00:34:19,000 --> 00:34:25,000
I'll draw it.
I won't put in minus pi's.
493
00:34:23,000 --> 00:34:29,000
I'll leave that to your
imagination.
494
00:34:27,000 --> 00:34:33,000
So, there's a hole at the end
here.
495
00:34:33,000 --> 00:34:39,000
In other words,
the end of the line is not
496
00:34:36,000 --> 00:34:42,000
included.
And, the end of this line,
497
00:34:39,000 --> 00:34:45,000
regardless of whether it was
included to start with or not,
498
00:34:43,000 --> 00:34:49,000
it's not now.
And here, similarly,
499
00:34:46,000 --> 00:34:52,000
I start it here with a hole,
and then go down parallel to
500
00:34:51,000 --> 00:34:57,000
the function,
t, slope one.
501
00:34:53,000 --> 00:34:59,000
And now, how do I fill in,
so the missing places,
502
00:34:57,000 --> 00:35:03,000
this is the point,
pi.
503
00:35:00,000 --> 00:35:06,000
This is the point,
negative pi,
504
00:35:02,000 --> 00:35:08,000
and there are similar points as
I go out.
505
00:35:05,000 --> 00:35:11,000
Well, since the function is
continuous here,
506
00:35:08,000 --> 00:35:14,000
the Fourier series will
converge to this orange line.
507
00:35:12,000 --> 00:35:18,000
But here, there's a jump
discontinuity,
508
00:35:14,000 --> 00:35:20,000
and therefore,
the Fourier series,
509
00:35:17,000 --> 00:35:23,000
this function converges to the
midpoint of the jump,
510
00:35:20,000 --> 00:35:26,000
in other words,
to here.
511
00:35:22,000 --> 00:35:28,000
This function,
in other words,
512
00:35:24,000 --> 00:35:30,000
converges to this very
discontinuous looking function,
513
00:35:28,000 --> 00:35:34,000
and rather odd how these points
are, I say, but in this case,
514
00:35:33,000 --> 00:35:39,000
I can prove to you that it
converges here by calculating
515
00:35:37,000 --> 00:35:43,000
it.
Look, this is the point,
516
00:35:41,000 --> 00:35:47,000
pi.
What happens when you plug in t
517
00:35:44,000 --> 00:35:50,000
equals pi?
You get everyone of these terms
518
00:35:50,000 --> 00:35:56,000
is zero, and therefore the sum
is zero.
519
00:35:53,000 --> 00:35:59,000
So, it certainly converges,
and it converges to zero.
520
00:36:00,000 --> 00:36:06,000
Now, that's a general theorem.
It's rather difficult to prove.
521
00:36:04,000 --> 00:36:10,000
You would have to take,
again, an analysis course.
522
00:36:07,000 --> 00:36:13,000
But, I don't even get to it in
the analysis course which I
523
00:36:12,000 --> 00:36:18,000
teach.
If I had another semester I'd
524
00:36:14,000 --> 00:36:20,000
get to it, but I can't get
everything.
525
00:36:17,000 --> 00:36:23,000
Anyway, we're not going to get
to it this semester to your
526
00:36:21,000 --> 00:36:27,000
infinite relief.
But, you should know the
527
00:36:24,000 --> 00:36:30,000
theorem anyway.
People will expect you to know
528
00:36:27,000 --> 00:36:33,000
it.
Well, that was half the period,
529
00:36:32,000 --> 00:36:38,000
and in the remaining half,
you're going to stay a long
530
00:36:39,000 --> 00:36:45,000
time today.
Okay, no, don't panic.
531
00:36:43,000 --> 00:36:49,000
I have to extend the Fourier
series.
532
00:36:47,000 --> 00:36:53,000
Okay, let me give you the hurry
up version indicating the two
533
00:36:54,000 --> 00:37:00,000
ways in which it needs to be
extended.
534
00:37:00,000 --> 00:37:06,000
Extension number one --
535
00:37:14,000 --> 00:37:20,000
The period is not two pi,
but two times,
536
00:37:18,000 --> 00:37:24,000
I'll keep the two just to make
the formulas look as similar as
537
00:37:24,000 --> 00:37:30,000
possible to the old ones.
The period, let's say,
538
00:37:29,000 --> 00:37:35,000
instead of two pi,
is two times L.
539
00:37:34,000 --> 00:37:40,000
Now, I think you know enough
mathematics by this point to
540
00:37:37,000 --> 00:37:43,000
sort of, I hope you can sort of
shrug and say,
541
00:37:40,000 --> 00:37:46,000
well, you know,
isn't that just kind of like
542
00:37:42,000 --> 00:37:48,000
changing the units on the
t-axis?
543
00:37:44,000 --> 00:37:50,000
You're just stretching.
Yeah, right.
544
00:37:46,000 --> 00:37:52,000
All you do is make a change of
variable.
545
00:37:49,000 --> 00:37:55,000
Now, should we make it nicely?
I think I'll give you the final
546
00:37:52,000 --> 00:37:58,000
answer, and then I'll try to
decide while I'm writing it down
547
00:37:56,000 --> 00:38:02,000
how much I'll try to make the
argument.
548
00:38:00,000 --> 00:38:06,000
First of all,
the main thing to get is,
549
00:38:04,000 --> 00:38:10,000
if the period is not pi but L,
what are the natural versions
550
00:38:11,000 --> 00:38:17,000
of the cosine and sine to use?
Use the natural functions.
551
00:38:18,000 --> 00:38:24,000
Natural has no meaning,
but it's psychologically
552
00:38:23,000 --> 00:38:29,000
important.
In other words,
553
00:38:26,000 --> 00:38:32,000
what kind of function should
replace that?
554
00:38:33,000 --> 00:38:39,000
I'll certainly have a t here.
What do I put in front?
555
00:38:37,000 --> 00:38:43,000
I'll keep the n also.
The question is,
556
00:38:40,000 --> 00:38:46,000
what do I fix?
What should I put here in
557
00:38:43,000 --> 00:38:49,000
between in order to make the
thing come out,
558
00:38:47,000 --> 00:38:53,000
so that it has period 2L?
You probably should learn to do
559
00:38:52,000 --> 00:38:58,000
this formally as well as just
sort of psyching it out,
560
00:38:56,000 --> 00:39:02,000
and taking a guess,
or memorizing the answer.
561
00:39:00,000 --> 00:39:06,000
If this is the t-axis,
here is t and L,
562
00:39:03,000 --> 00:39:09,000
zero and L.
What you want to do is make a
563
00:39:08,000 --> 00:39:14,000
change of variable to the u-axis
where the axis is the same.
564
00:39:14,000 --> 00:39:20,000
This is still the point.
But, L, now,
565
00:39:17,000 --> 00:39:23,000
on the u coordinate,
has the name pi.
566
00:39:21,000 --> 00:39:27,000
Now, so I'm just describing a
change of variable on the axis.
567
00:39:26,000 --> 00:39:32,000
What's the one that does this?
Well, when t is L,
568
00:39:31,000 --> 00:39:37,000
u should be pi.
So, t should be L over pi.
569
00:39:37,000 --> 00:39:43,000
When u is pi,
570
00:39:39,000 --> 00:39:45,000
t is L, and vice versa.
How about expressing u in
571
00:39:43,000 --> 00:39:49,000
terms, well, then u is equal to
pi over L times t.
572
00:39:49,000 --> 00:39:55,000
That's the backwards form of
573
00:39:52,000 --> 00:39:58,000
writing it, or the forward form,
depending upon how you like to
574
00:39:58,000 --> 00:40:04,000
think of these things.
Okay, so the cosine should be
575
00:40:05,000 --> 00:40:11,000
pi over L times t,
in order that when t be L,
576
00:40:10,000 --> 00:40:16,000
it should be like cosine of n
pi,
577
00:40:16,000 --> 00:40:22,000
which is what we would have
had.
578
00:40:19,000 --> 00:40:25,000
So, if t is equal to L,
in other words,
579
00:40:25,000 --> 00:40:31,000
where is this from?
What am I trying to say?
580
00:40:32,000 --> 00:40:38,000
That's the function.
This one is probably a little
581
00:40:36,000 --> 00:40:42,000
easier to see.
Where is this one zero?
582
00:40:40,000 --> 00:40:46,000
The sine functions that we used
before was zero at zero pi,
583
00:40:46,000 --> 00:40:52,000
two pi, three pi.
Where is this one zero?
584
00:40:50,000 --> 00:40:56,000
It's zero at zero.
When t is equal to L,
585
00:40:54,000 --> 00:41:00,000
it's zero.
When t is equal to 2L,
586
00:40:58,000 --> 00:41:04,000
so, this is the right thing.
587
00:41:03,000 --> 00:41:09,000
So, it's zero.
It's periodic,
588
00:41:05,000 --> 00:41:11,000
and it's zero plus or minus L
plus or minus 2L.
589
00:41:08,000 --> 00:41:14,000
And, in fact,
formally you can verify that
590
00:41:11,000 --> 00:41:17,000
it's periodic with period 2L.
So, in other words,
591
00:41:15,000 --> 00:41:21,000
we want a Fourier expansion to
use these functions as the
592
00:41:19,000 --> 00:41:25,000
natural analog of what would be
up there.
593
00:41:22,000 --> 00:41:28,000
So, the period of our function
is 2L, and the formula is,
594
00:41:26,000 --> 00:41:32,000
I'll give you the formula.
It's f of t equals
595
00:41:32,000 --> 00:41:38,000
identical summation,
an, except you'll use these as
596
00:41:38,000 --> 00:41:44,000
the natural functions instead of
cosine nt and sine
597
00:41:45,000 --> 00:41:51,000
nt. So, n pi t over L
598
00:41:49,000 --> 00:41:55,000
plus bn, okay,
I'm tired, but I'll put it in
599
00:41:54,000 --> 00:42:00,000
anyway, n pi t over L.
600
00:42:00,000 --> 00:42:06,000
Yeah, but of course,
what about the formulas for an?
601
00:42:04,000 --> 00:42:10,000
Somebody up there is watching
over us.
602
00:42:08,000 --> 00:42:14,000
Here are the formulas.
They are exactly what you would
603
00:42:13,000 --> 00:42:19,000
guess if somebody said produce
the formulas in ten seconds,
604
00:42:19,000 --> 00:42:25,000
and you'd better be right,
and you didn't have time to
605
00:42:24,000 --> 00:42:30,000
calculate.
You say, well,
606
00:42:27,000 --> 00:42:33,000
it must be, let's do the cosine
series.
607
00:42:32,000 --> 00:42:38,000
Okay, let's not do a cosine.
So, it's one over L
608
00:42:36,000 --> 00:42:42,000
times the integral from negative
L, in other words,
609
00:42:41,000 --> 00:42:47,000
wherever you see an L,
wherever you see a pi,
610
00:42:45,000 --> 00:42:51,000
just put an L times the f of t
cosine, and now we'll use our
611
00:42:50,000 --> 00:42:56,000
new function,
not the old one.
612
00:42:52,000 --> 00:42:58,000
I submit that's an easy,
if you know the first formula,
613
00:42:57,000 --> 00:43:03,000
then this would be an easy one
to remember.
614
00:43:01,000 --> 00:43:07,000
All you do is change pi to L
everywhere.
615
00:43:06,000 --> 00:43:12,000
Except, you got to remember
this part.
616
00:43:09,000 --> 00:43:15,000
Make it a function periodic of
period 2L, not 2pi.
617
00:43:15,000 --> 00:43:21,000
And similarly,
bn is similar.
618
00:43:18,000 --> 00:43:24,000
It looks just the same way.
And, how about,
619
00:43:22,000 --> 00:43:28,000
and the same even-odd business
goes, too, so that if f of t,
620
00:43:28,000 --> 00:43:34,000
for example,
is even, and has period 2L,
621
00:43:33,000 --> 00:43:39,000
then the function,
then the best formula for the
622
00:43:38,000 --> 00:43:44,000
an will not be that one.
It will be two over L,
623
00:43:45,000 --> 00:43:51,000
and where you integrate only
from zero to L,
624
00:43:49,000 --> 00:43:55,000
f of t cosine.
625
00:44:01,000 --> 00:44:07,000
So, now, the bn's will be zero,
and you'll just have positive,
626
00:44:07,000 --> 00:44:13,000
etc.
for L.
627
00:44:08,000 --> 00:44:14,000
As I say, this is important
case, particularly if the period
628
00:44:13,000 --> 00:44:19,000
is two, in other words,
if the half period is one
629
00:44:18,000 --> 00:44:24,000
because in the literature,
frequently one is used as the
630
00:44:24,000 --> 00:44:30,000
standard normal reference,
not pi.
631
00:44:27,000 --> 00:44:33,000
Pi is convenient mathematically
because it makes the cosines and
632
00:44:33,000 --> 00:44:39,000
sines look simple.
But, in actual calculation,
633
00:44:39,000 --> 00:44:45,000
it tends to be where L is one.
So, usually you have a pi here.
634
00:44:45,000 --> 00:44:51,000
You don't have just nt.
Well, I should do a
635
00:44:49,000 --> 00:44:55,000
calculation, but instead of
doing that, let me give you the
636
00:44:55,000 --> 00:45:01,000
other extension.
Fortunately,
637
00:44:58,000 --> 00:45:04,000
there are plenty of
calculations in your book.
638
00:45:04,000 --> 00:45:10,000
So, let me give you in the last
couple of minutes the other
639
00:45:10,000 --> 00:45:16,000
extension.
This is going to be a very
640
00:45:14,000 --> 00:45:20,000
important one for us next time.
Typically, in applications,
641
00:45:21,000 --> 00:45:27,000
well, I mean,
the first thing,
642
00:45:24,000 --> 00:45:30,000
periodic functions are nice,
but let's face it.
643
00:45:30,000 --> 00:45:36,000
Most functions aren't periodic,
I have to agree.
644
00:45:37,000 --> 00:45:43,000
So, all this theory is just
about periodic functions?
645
00:45:40,000 --> 00:45:46,000
No.
It's about functions.
646
00:45:42,000 --> 00:45:48,000
Really, it's about functions
where the interval on which you
647
00:45:46,000 --> 00:45:52,000
are interested in them is
finite.
648
00:45:48,000 --> 00:45:54,000
It's a finite interval,
not functions which go to
649
00:45:52,000 --> 00:45:58,000
infinity.
For those, you will have to use
650
00:45:54,000 --> 00:46:00,000
Fourier transforms,
Fourier transforms,
651
00:45:57,000 --> 00:46:03,000
not Fourier series.
But, if you are interested in a
652
00:46:02,000 --> 00:46:08,000
function on a finite interval,
then you can use Fourier series
653
00:46:06,000 --> 00:46:12,000
even though the function isn't
periodic because you can make it
654
00:46:11,000 --> 00:46:17,000
periodic.
So, what you do is,
655
00:46:13,000 --> 00:46:19,000
if f of t is on,
let's take the interval from
656
00:46:17,000 --> 00:46:23,000
zero to L.
That's a sample finite
657
00:46:19,000 --> 00:46:25,000
interval.
I can always change the
658
00:46:22,000 --> 00:46:28,000
variable to make the interval
from zero to L.
659
00:46:25,000 --> 00:46:31,000
I can even make it from zero to
one, but that's a little too
660
00:46:29,000 --> 00:46:35,000
special.
It would be a little awkward.
661
00:46:34,000 --> 00:46:40,000
So, if a function is defined on
a finite interval,
662
00:46:38,000 --> 00:46:44,000
the way to apply the Fourier
series to it is make a periodic
663
00:46:43,000 --> 00:46:49,000
extension.
Now, since I have so little
664
00:46:47,000 --> 00:46:53,000
time, I'm just going to get away
with murder by just drawing
665
00:46:52,000 --> 00:46:58,000
pictures.
So, let me give you a function.
666
00:46:55,000 --> 00:47:01,000
Here's my function defined on
zero to L, colored chalk if you
667
00:47:01,000 --> 00:47:07,000
please.
Let's make it the function t
668
00:47:05,000 --> 00:47:11,000
squared,
and let's make L equal to one.
669
00:47:09,000 --> 00:47:15,000
That function is not periodic.
If I let it go off,
670
00:47:13,000 --> 00:47:19,000
it would just go off to
infinity and never repeat its
671
00:47:17,000 --> 00:47:23,000
values, except on the left-hand
side.
672
00:47:20,000 --> 00:47:26,000
But, I'm not even going to let
it be on the left hand side.
673
00:47:25,000 --> 00:47:31,000
It's only defined from zero to
one as far as I'm concerned.
674
00:47:29,000 --> 00:47:35,000
Okay, that function has an even
periodic extension.
675
00:47:35,000 --> 00:47:41,000
And, its graph looks like this
extended to be an even function.
676
00:47:39,000 --> 00:47:45,000
Okay, now, that means from zero
to negative L,
677
00:47:44,000 --> 00:47:50,000
you've got to make it look
exactly as it looked on the
678
00:47:48,000 --> 00:47:54,000
right-hand side.
Otherwise, it would be even.
679
00:47:51,000 --> 00:47:57,000
And now, what do I do?
Well, now I've got,
680
00:47:54,000 --> 00:48:00,000
from minus L to L.
So, all I'm allowed to do is
681
00:47:59,000 --> 00:48:05,000
keep repeating the values.
In other words,
682
00:48:03,000 --> 00:48:09,000
apply the theory of Fourier
series to this guy,
683
00:48:06,000 --> 00:48:12,000
use a cosine series because
it's an even function,
684
00:48:10,000 --> 00:48:16,000
and then everything you want to
do, you say, okay,
685
00:48:14,000 --> 00:48:20,000
all the rest of this is
garbage.
686
00:48:16,000 --> 00:48:22,000
I only really care about it
from here to here.
687
00:48:20,000 --> 00:48:26,000
And, that's what you will plug
into your differential equation
688
00:48:24,000 --> 00:48:30,000
on the right-hand side,
just that part of it,
689
00:48:28,000 --> 00:48:34,000
just this part of it.
How about the odd extension?
690
00:48:33,000 --> 00:48:39,000
What would that look like?
Okay, the odd extension,
691
00:48:37,000 --> 00:48:43,000
here I start like this.
And now, to extend it to be an
692
00:48:41,000 --> 00:48:47,000
odd function,
I have to make it go down in
693
00:48:44,000 --> 00:48:50,000
exactly the same way it went up.
And, what do I do here?
694
00:48:49,000 --> 00:48:55,000
I have to make it start
repeating its values so it will
695
00:48:53,000 --> 00:48:59,000
look like this.
So, the odd extension is going
696
00:48:57,000 --> 00:49:03,000
to be discontinuous in this
case.
697
00:49:01,000 --> 00:49:07,000
And, what's the Fourier series
going to converge to?
698
00:49:05,000 --> 00:49:11,000
Well, in each case,
to the average,
699
00:49:07,000 --> 00:49:13,000
to the midpoint of the jump,
and the odd extension looks
700
00:49:12,000 --> 00:49:18,000
like this, and this will give me
assigned series.
701
00:49:16,000 --> 00:49:22,000
Okay, you've got lots of
problems to do.