WEBVTT
00:01:03.000 --> 00:01:09.000
Okay, that's,
so to speak,
00:01:05.000 --> 00:01:11.000
the text for today.
The Fourier series,
00:01:09.000 --> 00:01:15.000
and the Fourier expansion for f
of t,
00:01:14.000 --> 00:01:20.000
so f of t, if it looks like
this should be periodic,
00:01:19.000 --> 00:01:25.000
and two pi should be a period.
Sometimes people rather
00:01:25.000 --> 00:01:31.000
sloppily say periodic with
period two pi,
00:01:29.000 --> 00:01:35.000
but that's a little ambiguous.
So, this period could also be
00:01:37.000 --> 00:01:43.000
pi or a half pi or something
like that as well.
00:01:42.000 --> 00:01:48.000
The an's and bn's are
calculated according to these
00:01:47.000 --> 00:01:53.000
formulas.
Now, we're going to need in
00:01:51.000 --> 00:01:57.000
just a minute a consequence of
those formulas,
00:01:56.000 --> 00:02:02.000
which, it's not subtle,
but because there are formulas
00:02:01.000 --> 00:02:07.000
for an and bn,
it follows that once you know f
00:02:06.000 --> 00:02:12.000
of t,
the an's and bn's are
00:02:10.000 --> 00:02:16.000
determined.
Or, to put it another way,
00:02:15.000 --> 00:02:21.000
a function cannot have two
different Fourier series.
00:02:20.000 --> 00:02:26.000
Or, to put it yet another way,
if f of t,
00:02:24.000 --> 00:02:30.000
if two functions are equal,
you'll see why I write it in
00:02:30.000 --> 00:02:36.000
this rather peculiar form.
Then, the Fourier series for f
00:02:35.000 --> 00:02:41.000
is the same as the Fourier
series for g.
00:02:40.000 --> 00:02:46.000
And, the reason is because if f
is equal to g,
00:02:44.000 --> 00:02:50.000
then this integral with an f
there is the same as the
00:02:49.000 --> 00:02:55.000
integral with a g there.
And therefore,
00:02:52.000 --> 00:02:58.000
the an's come out to be the
same.
00:02:55.000 --> 00:03:01.000
In the same way,
the bn's come out to be the
00:02:58.000 --> 00:03:04.000
same.
So, the Fourier series are the
00:03:01.000 --> 00:03:07.000
same, coefficient by
coefficient, for f and g.
00:03:05.000 --> 00:03:11.000
Now, my ultimate goal-- let's
all put down the argument since
00:03:10.000 --> 00:03:16.000
there are formulas,
since we have formulas for an
00:03:14.000 --> 00:03:20.000
and bn.
Now, a consequence of that is,
00:03:19.000 --> 00:03:25.000
well, let me first say,
what I'm aiming at is you will
00:03:23.000 --> 00:03:29.000
be amazed at how long it's going
to take me to get to this.
00:03:29.000 --> 00:03:35.000
I just want to calculate the
Fourier series for some rather
00:03:33.000 --> 00:03:39.000
simple periodic function.
It's going to look like this.
00:03:38.000 --> 00:03:44.000
So, here's pi,
and here's negative pi.
00:03:41.000 --> 00:03:47.000
So, the function which just
looks like t in between those
00:03:45.000 --> 00:03:51.000
two, so, it goes up to,
it's a function,
00:03:49.000 --> 00:03:55.000
t, more or less,
goes up to pi here,
00:03:51.000 --> 00:03:57.000
minus pi there.
But, of course,
00:03:54.000 --> 00:04:00.000
it's got to be periodic of
period two pi.
00:03:59.000 --> 00:04:05.000
Well, then, it just repeats
itself after that.
00:04:02.000 --> 00:04:08.000
After this, it just does that,
and so on.
00:04:04.000 --> 00:04:10.000
It's a little ambiguous what
happens at these endpoints.
00:04:08.000 --> 00:04:14.000
Well, let's not worry about
that for the moment,
00:04:12.000 --> 00:04:18.000
and frankly,
it won't really matter because
00:04:15.000 --> 00:04:21.000
the integrals don't care about
what happens in individual
00:04:19.000 --> 00:04:25.000
points.
So, there's my f of t.
00:04:21.000 --> 00:04:27.000
Now, I, of course,
could start doing it right
00:04:24.000 --> 00:04:30.000
away.
But, you will quickly find,
00:04:27.000 --> 00:04:33.000
if you start doing these
problems and hacking around with
00:04:30.000 --> 00:04:36.000
them, that the calculations seem
really quite long.
00:04:34.000 --> 00:04:40.000
And therefore,
in the first half of the
00:04:37.000 --> 00:04:43.000
period, the first half of the
period I want to show you how to
00:04:41.000 --> 00:04:47.000
shorten the calculations.
And in the second half of the
00:04:47.000 --> 00:04:53.000
period, after we've done that
and calculated this thing
00:04:50.000 --> 00:04:56.000
successfully,
I hope, I want to show you how
00:04:54.000 --> 00:05:00.000
to remove various restrictions
on these functions,
00:04:57.000 --> 00:05:03.000
how to extend the range of
Fourier series.
00:05:01.000 --> 00:05:07.000
Well, one obvious thing,
for example,
00:05:03.000 --> 00:05:09.000
is suppose the function isn't
periodic of period two pi.
00:05:06.000 --> 00:05:12.000
Suppose it has some other
period.
00:05:08.000 --> 00:05:14.000
Does that mean there's no
formula?
00:05:10.000 --> 00:05:16.000
Well, of course not.
There's a formula.
00:05:13.000 --> 00:05:19.000
But, we need to know what it
is, particularly in the
00:05:16.000 --> 00:05:22.000
applications,
the period is rarely two pi.
00:05:19.000 --> 00:05:25.000
It's normally one,
or something like that.
00:05:21.000 --> 00:05:27.000
But, let's first of all,
I'm sure what you will
00:05:24.000 --> 00:05:30.000
appreciate is how the
calculations can get shortened.
00:05:29.000 --> 00:05:35.000
Now, the main way of shortening
them is by using evenness and
00:05:35.000 --> 00:05:41.000
oddness.
And, what I claim is this,
00:05:39.000 --> 00:05:45.000
that if f of t is an
even function,
00:05:44.000 --> 00:05:50.000
remember what that means,
that f of negative t is equal
00:05:51.000 --> 00:05:57.000
to f of t.
Cosine is a good example,
00:05:57.000 --> 00:06:03.000
of course, cosine nt;
are all these
00:06:02.000 --> 00:06:08.000
functions are even functions.
If f of t is even,
00:06:07.000 --> 00:06:13.000
then its Fourier series
contains only the cosine terms.
00:06:16.000 --> 00:06:22.000
In other words,
half the calculations you don't
00:06:21.000 --> 00:06:27.000
have to do if you start with an
even function.
00:06:26.000 --> 00:06:32.000
That's what I mean by
shortening the work.
00:06:31.000 --> 00:06:37.000
There are no odd terms,
or let's put it positively.
00:06:37.000 --> 00:06:43.000
All the bn's are zero.
Now, one way of doing this
00:06:42.000 --> 00:06:48.000
would be to say,
well, y to the bn zero,
00:06:44.000 --> 00:06:50.000
well, we've got formulas,
and fool around with the
00:06:47.000 --> 00:06:53.000
formula for the bn,
and think about a little bit,
00:06:50.000 --> 00:06:56.000
and finally decide that that
has to come out to be zero.
00:06:53.000 --> 00:06:59.000
That's not a bad way,
and it would remind you of some
00:06:56.000 --> 00:07:02.000
basic facts about integration,
about integrals.
00:07:00.000 --> 00:07:06.000
Instead of doing that,
I'm going to apply my little
00:07:04.000 --> 00:07:10.000
principle that if two functions
are the same,
00:07:08.000 --> 00:07:14.000
then their Fourier series have
to be the same.
00:07:12.000 --> 00:07:18.000
So, the argument I'm going to
give is this,
00:07:16.000 --> 00:07:22.000
so, I'm going to try to prove
this statement now.
00:07:20.000 --> 00:07:26.000
And, I'm going to use the facts
on the first board to do it.
00:07:25.000 --> 00:07:31.000
So, what is f of minus t?
00:07:30.000 --> 00:07:36.000
Well, if that's equal to f of
t, then in terms of the
00:07:35.000 --> 00:07:41.000
Fourier series,
how do I get the Fourier series
00:07:39.000 --> 00:07:45.000
for f of minus t?
Well, I take the Fourier series
00:07:44.000 --> 00:07:50.000
for f of t, and substitute t
equals minus t.
00:07:48.000 --> 00:07:54.000
Now, what happens when I do
that?
00:07:51.000 --> 00:07:57.000
So, the Fourier series for this
looks like a zero over two
00:07:56.000 --> 00:08:02.000
plus summation what?
Well, the an cosine nt,
00:08:02.000 --> 00:08:08.000
that does not change
because when I change t to
00:08:06.000 --> 00:08:12.000
negative t,
the cosine nt does
00:08:11.000 --> 00:08:17.000
not change, stays the same
because it's an even function.
00:08:15.000 --> 00:08:21.000
What happens to the sine term?
Well, the sine of negative nt
00:08:20.000 --> 00:08:26.000
is equal to minus the
sine of nt.
00:08:25.000 --> 00:08:31.000
So, the other terms,
the sine terms change sign.
00:08:30.000 --> 00:08:36.000
So, all that's the result of
substituting t for negative t
00:08:34.000 --> 00:08:40.000
and f of t.
00:08:36.000 --> 00:08:42.000
On the other hand,
what's f of t itself?
00:08:40.000 --> 00:08:46.000
Well, f of t itself is what
happened before that.
00:08:43.000 --> 00:08:49.000
Now it's got a plus sign
because nothing was done to the
00:08:48.000 --> 00:08:54.000
series.
Well, if the function is even,
00:08:50.000 --> 00:08:56.000
then those two right hand sides
are the same function.
00:08:54.000 --> 00:09:00.000
In other words,
they're like my f of t equals g
00:08:58.000 --> 00:09:04.000
of t. And therefore,
00:09:02.000 --> 00:09:08.000
the Fourier series on the left
must be the same.
00:09:06.000 --> 00:09:12.000
In other words,
if these are equal,
00:09:09.000 --> 00:09:15.000
therefore, these have to be
equal, too.
00:09:13.000 --> 00:09:19.000
Now, there's no problem with
the cosine terms.
00:09:17.000 --> 00:09:23.000
They look the same.
On the other hand,
00:09:20.000 --> 00:09:26.000
the sine terms have changed
sign.
00:09:23.000 --> 00:09:29.000
Therefore, it must be the case
that bn is always equal to
00:09:28.000 --> 00:09:34.000
negative bn for all n.
That's the only way this series
00:09:34.000 --> 00:09:40.000
can be the same as that one.
Now, if bn is equal to negative
00:09:39.000 --> 00:09:45.000
bn,
that implies that bn is zero.
00:09:43.000 --> 00:09:49.000
Zero is the only number which
00:09:46.000 --> 00:09:52.000
is equal to its negative.
And so, by this argument,
00:09:51.000 --> 00:09:57.000
in other words,
using the uniqueness of Fourier
00:09:54.000 --> 00:10:00.000
series, we conclude that if the
function is even,
00:09:59.000 --> 00:10:05.000
then its Fourier series can
only have cosine terms in it.
00:10:05.000 --> 00:10:11.000
Now, you say,
hey, that's obvious.
00:10:07.000 --> 00:10:13.000
The cosine, that's just a point
of logic.
00:10:09.000 --> 00:10:15.000
But, this is a mathematics
course, after all.
00:10:12.000 --> 00:10:18.000
It's not just about
calculation.
00:10:14.000 --> 00:10:20.000
Many of you would say,
yeah, of course that's obvious
00:10:18.000 --> 00:10:24.000
because cosines are even,
and the sines are odd.
00:10:21.000 --> 00:10:27.000
I say, yeah,
and so why does that make it
00:10:24.000 --> 00:10:30.000
true?
Well, the cosine's even.
00:10:25.000 --> 00:10:31.000
Plus t into minus t,
and what you are proving
00:10:29.000 --> 00:10:35.000
is the converse.
The converse is obvious.
00:10:33.000 --> 00:10:39.000
Yeah, obvious,
I don't care.
00:10:35.000 --> 00:10:41.000
If the right-hand side is the
sum of the functions,
00:10:39.000 --> 00:10:45.000
well, so is the left.
But I'm saying it the other way
00:10:43.000 --> 00:10:49.000
around.
If the left is an even
00:10:45.000 --> 00:10:51.000
function, why does the
right-hand side have to have
00:10:49.000 --> 00:10:55.000
only even terms in it?
And, this is the argument which
00:10:53.000 --> 00:10:59.000
makes that true.
Now, there is a further
00:10:56.000 --> 00:11:02.000
simplification because if you've
got an even function,
00:11:00.000 --> 00:11:06.000
oh, by the way,
of course the same thing is
00:11:03.000 --> 00:11:09.000
true for the odd,
I ought to put that down,
00:11:06.000 --> 00:11:12.000
and so also,
if f of t is odd,
00:11:09.000 --> 00:11:15.000
then I think one of these
proofs is enough.
00:11:14.000 --> 00:11:20.000
The other you can supply
yourself.
00:11:17.000 --> 00:11:23.000
That will imply that all the
an's are zero,
00:11:20.000 --> 00:11:26.000
even including this first one,
a zero,
00:11:25.000 --> 00:11:31.000
and by the same reasoning.
00:11:37.000 --> 00:11:43.000
So, an even function uses only
cosines for its Fourier
00:11:41.000 --> 00:11:47.000
expansion.
An odd function uses only
00:11:44.000 --> 00:11:50.000
sines.
Good.
00:11:45.000 --> 00:11:51.000
But, we still have to,
suppose we got an even
00:11:49.000 --> 00:11:55.000
function.
We've still got to calculate
00:11:53.000 --> 00:11:59.000
this integral.
Well, even that can be
00:11:56.000 --> 00:12:02.000
simplified.
So, the second stage of the
00:11:59.000 --> 00:12:05.000
simplification,
again, assuming that we have an
00:12:04.000 --> 00:12:10.000
even or odd function,
and by the way,
00:12:07.000 --> 00:12:13.000
[LAUGHTER].
Totally unauthorized.
00:12:26.000 --> 00:12:32.000
So, if f of t is even,
what we'd like to do now is
00:12:34.000 --> 00:12:40.000
simplify the integral a little.
And, there is an easy way to do
00:12:43.000 --> 00:12:49.000
that, because,
look, if f of t is an even
00:12:49.000 --> 00:12:55.000
function, then so is f of t
cosine nt,
00:12:57.000 --> 00:13:03.000
is also even.
Imagine, we could make little
00:13:02.000 --> 00:13:08.000
rules about an even function
times an even function is an
00:13:06.000 --> 00:13:12.000
even function.
There are general rules of that
00:13:09.000 --> 00:13:15.000
type, and some of you know them,
and they are very useful.
00:13:13.000 --> 00:13:19.000
But, let's just do it ad hoc
here.
00:13:15.000 --> 00:13:21.000
If I change t to negative
t here,
00:13:18.000 --> 00:13:24.000
I don't change the function
because it's even.
00:13:21.000 --> 00:13:27.000
And, I don't change the cosine
because that's even.
00:13:24.000 --> 00:13:30.000
So, if I change t to negative
t, I don't change the function.
00:13:28.000 --> 00:13:34.000
Either factor that function,
and therefore I don't change
00:13:32.000 --> 00:13:38.000
the product of those two things
either.
00:13:36.000 --> 00:13:42.000
So, it's also even.
Now, what about an even
00:13:41.000 --> 00:13:47.000
function when you integrate it?
Here's a typical looking even
00:13:48.000 --> 00:13:54.000
function, let's say,
something like,
00:13:52.000 --> 00:13:58.000
I don't know,
wiggle, wiggle,
00:13:56.000 --> 00:14:02.000
again.
Here's our better even
00:13:59.000 --> 00:14:05.000
function.
All right, so,
00:14:02.000 --> 00:14:08.000
minus pi to pi,
even, even though the t-axis is
00:14:08.000 --> 00:14:14.000
somewhat curvy.
So, there is an even function.
00:14:14.000 --> 00:14:20.000
The point is that if you
integrate an even function from
00:14:17.000 --> 00:14:23.000
negative pi to pi,
I think you all know even from
00:14:21.000 --> 00:14:27.000
calculus you were taught to do
this simplification.
00:14:24.000 --> 00:14:30.000
Don't do that.
Instead, integrate from zero to
00:14:27.000 --> 00:14:33.000
pi, and double the answer.
Why should you do that?
00:14:31.000 --> 00:14:37.000
The answer is because it's
always nice to have zero as one
00:14:35.000 --> 00:14:41.000
of the limits of integration.
I trust to your experience,
00:14:39.000 --> 00:14:45.000
I don't have to sell that.
Minus pi is a particularly
00:14:43.000 --> 00:14:49.000
unpleasant lower limit of
integration because you are sure
00:14:47.000 --> 00:14:53.000
to get in trouble with negative
signs.
00:14:50.000 --> 00:14:56.000
There are bound to be at least
three negative signs floating
00:14:54.000 --> 00:15:00.000
around.
And, if you miss one of them,
00:14:57.000 --> 00:15:03.000
you'll get the wrong signs of
answer.
00:15:01.000 --> 00:15:07.000
The answer will have the wrong
sign.
00:15:03.000 --> 00:15:09.000
So, the way the formula from
this simplifies is that an,
00:15:08.000 --> 00:15:14.000
instead of integrating from
negative pi to pi,
00:15:12.000 --> 00:15:18.000
I can integrate only from zero
to pi, and double the answer.
00:15:17.000 --> 00:15:23.000
So, our better formula is this.
If the function is even,
00:15:22.000 --> 00:15:28.000
this is the formula you should
use: zero to pi,
00:15:26.000 --> 00:15:32.000
f of t cosine nt dt.
00:15:31.000 --> 00:15:37.000
Of course, I don't have to tell
00:15:35.000 --> 00:15:41.000
you what bn should be because bn
will be zero.
00:15:39.000 --> 00:15:45.000
And, in the same way,
if f is odd,
00:15:42.000 --> 00:15:48.000
the same reasoning shows that
bn-- of course,
00:15:45.000 --> 00:15:51.000
an will be zero this time.
But it will be bn that will be
00:15:50.000 --> 00:15:56.000
two over pi times the integral
from zero to pi of f of t sine
00:15:55.000 --> 00:16:01.000
nt dt.
00:16:00.000 --> 00:16:06.000
Maybe we'd better just a word
00:16:03.000 --> 00:16:09.000
about that since,
why is that so?
00:16:06.000 --> 00:16:12.000
If it's odd,
doesn't that mean things become
00:16:08.000 --> 00:16:14.000
zero?
If you integrate an odd
00:16:10.000 --> 00:16:16.000
function like that,
the integral over minus pi to
00:16:14.000 --> 00:16:20.000
pi, you get zero.
Well, but this is not an odd
00:16:17.000 --> 00:16:23.000
function.
This is an odd function,
00:16:19.000 --> 00:16:25.000
and this is an odd function.
But the product of two odd
00:16:22.000 --> 00:16:28.000
functions is an even function.
Odd times odd is even.
00:16:26.000 --> 00:16:32.000
I said I wasn't going to give
you those rules,
00:16:29.000 --> 00:16:35.000
but since this is the one which
trips everybody up,
00:16:32.000 --> 00:16:38.000
maybe I'd better say it just
justbecause it looks wrong.
00:16:38.000 --> 00:16:44.000
Right, this is odd.
That's odd.
00:16:40.000 --> 00:16:46.000
Think about it.
If I change t to negative t,
00:16:43.000 --> 00:16:49.000
this multiplies by
minus one.
00:16:46.000 --> 00:16:52.000
This multiplies by minus one.
And therefore,
00:16:49.000 --> 00:16:55.000
the product multiplies by minus
one times minus one.
00:16:54.000 --> 00:17:00.000
In other words,
it multiplies by plus one.
00:16:57.000 --> 00:17:03.000
Nothing happens,
so it stays the same.
00:17:01.000 --> 00:17:07.000
Why does nobody believe this,
even though it's true?
00:17:04.000 --> 00:17:10.000
It's because they are thinking
about numbers.
00:17:08.000 --> 00:17:14.000
Everybody knows that an odd
number times an odd number is an
00:17:12.000 --> 00:17:18.000
odd number.
So, I'm not multiplying numbers
00:17:15.000 --> 00:17:21.000
here, which also I'll put them
in boxes to indicate that they
00:17:20.000 --> 00:17:26.000
are not numbers.
How's that?
00:17:22.000 --> 00:17:28.000
Brand-new invented notation.
The box means caution.
00:17:25.000 --> 00:17:31.000
The inside is not a number,
it's the word odd or even.
00:17:31.000 --> 00:17:37.000
It's just a symbolic statement
that the product of an odd
00:17:35.000 --> 00:17:41.000
function and an odd function is
an even function.
00:17:39.000 --> 00:17:45.000
Even times even is even.
What's odd times even?
00:17:43.000 --> 00:17:49.000
Yes, it has to get equal time.
Obviously, something must come
00:17:47.000 --> 00:17:53.000
out to be odd,
right.
00:17:49.000 --> 00:17:55.000
Okay, so, now that we've got
our two simplifications,
00:17:53.000 --> 00:17:59.000
we are ready to do this
problem.
00:17:56.000 --> 00:18:02.000
Instead of attacking it with
the original formulas,
00:18:00.000 --> 00:18:06.000
we are going to think about it
and attack it with our better
00:18:04.000 --> 00:18:10.000
formulas.
So, now we are going to
00:18:11.000 --> 00:18:17.000
calculate the Fourier series for
f of t.
00:18:19.000 --> 00:18:25.000
The first thing I see,
so f of t is our little thing
00:18:29.000 --> 00:18:35.000
here.
Well, first of all,
00:18:32.000 --> 00:18:38.000
what kind of function is it:
odd, even, or neither?
00:18:35.000 --> 00:18:41.000
Most functions are neither,
of course.
00:18:38.000 --> 00:18:44.000
But, fortunately in the
applications,
00:18:40.000 --> 00:18:46.000
functions tend to be one or the
other.
00:18:42.000 --> 00:18:48.000
Or, they can be converted into
one to the other.
00:18:46.000 --> 00:18:52.000
Maybe if I get a chance,
I'll show you a little how,
00:18:49.000 --> 00:18:55.000
or the recitations will.
So, this function is odd.
00:18:52.000 --> 00:18:58.000
Okay, half the work just
disappeared.
00:18:55.000 --> 00:19:01.000
I don't have to calculate any
an's.
00:18:57.000 --> 00:19:03.000
They will be zero.
So, I only have to calculate
00:19:01.000 --> 00:19:07.000
bn, and I'll calculate them by
my better formula.
00:19:04.000 --> 00:19:10.000
So, it's two over pi times the
integral from zero to pi,
00:19:08.000 --> 00:19:14.000
and what I have to integrate,
well, now, finally you've got
00:19:11.000 --> 00:19:17.000
to integrate something.
From zero to pi,
00:19:14.000 --> 00:19:20.000
this is the function,
t.
00:19:15.000 --> 00:19:21.000
So, I have to integrate t times
sine of nt dt.
00:19:18.000 --> 00:19:24.000
Okay,
00:19:22.000 --> 00:19:28.000
so this is why you learned
integration by parts,
00:19:25.000 --> 00:19:31.000
one of many reasons why you
learned integration by parts,
00:19:29.000 --> 00:19:35.000
so that you wouldn't have to
pull out your little calculators
00:19:32.000 --> 00:19:38.000
to do this.
Okay, now, let's do it.
00:19:36.000 --> 00:19:42.000
So, it's two over pi.
00:19:39.000 --> 00:19:45.000
Let's solve that away so we can
forget about it.
00:19:42.000 --> 00:19:48.000
And, what's then left is just
the evaluation of the integral
00:19:47.000 --> 00:19:53.000
between limits.
So, if I integrate by parts,
00:19:50.000 --> 00:19:56.000
I'll want to differentiate the
t, and integrate the sign,
00:19:54.000 --> 00:20:00.000
right?
So, the first step is you don't
00:19:57.000 --> 00:20:03.000
do the differentiation.
You only do the integration.
00:20:02.000 --> 00:20:08.000
So, that integrates to be
cosine nt over n,
00:20:05.000 --> 00:20:11.000
more or less.
The only thing is,
00:20:08.000 --> 00:20:14.000
if I differentiate this,
I get negative sine nt
00:20:11.000 --> 00:20:17.000
instead of,
so, I want to put a negative
00:20:15.000 --> 00:20:21.000
sign in front of all this.
And, I will evaluate that
00:20:19.000 --> 00:20:25.000
between the limits,
zero and pi,
00:20:21.000 --> 00:20:27.000
and then subtract what you get
by doing both things,
00:20:25.000 --> 00:20:31.000
both the differentiation and
the integration.
00:20:28.000 --> 00:20:34.000
So, I subtract the integral
from zero to pi.
00:20:33.000 --> 00:20:39.000
I now differentiate the t,
and integrate.
00:20:35.000 --> 00:20:41.000
Well, I just did the
integration.
00:20:37.000 --> 00:20:43.000
That's negative cosine nt over
n.
00:20:40.000 --> 00:20:46.000
You see how the negative signs
pile up?
00:20:43.000 --> 00:20:49.000
And, if this is negative pi
instead of zero,
00:20:45.000 --> 00:20:51.000
it's at that point when it
starts to lose heart.
00:20:48.000 --> 00:20:54.000
You see three negative signs,
and then when you substitute,
00:20:52.000 --> 00:20:58.000
you're going to have to put in
still something else negative,
00:20:56.000 --> 00:21:02.000
and you just have the feeling
you're going to make a mistake.
00:21:01.000 --> 00:21:07.000
And, you will.
Okay, now all we have to do is
00:21:05.000 --> 00:21:11.000
a little evaluation.
Let's see, at the lower limit I
00:21:09.000 --> 00:21:15.000
get zero, here.
Let's right away,
00:21:12.000 --> 00:21:18.000
as two over pi.
At the lower limit,
00:21:16.000 --> 00:21:22.000
I get zero.
That's nice.
00:21:18.000 --> 00:21:24.000
At the upper limit,
I get minus pi over n times the
00:21:23.000 --> 00:21:29.000
cosine of n pi.
00:21:26.000 --> 00:21:32.000
Now, once and for all,
the cosine of n pi--
00:21:31.000 --> 00:21:37.000
If you like to make
separate steps out of
00:21:35.000 --> 00:21:41.000
everything, okay,
I'll let you do it this time,
00:21:39.000 --> 00:21:45.000
--
-- but in the long run,
00:21:43.000 --> 00:21:49.000
it's good to remember that
that's negative one to the n'th
00:21:48.000 --> 00:21:54.000
power
The cosine of pi is minus one .
00:21:51.000 --> 00:21:57.000
The cosine of two pi is plus
one,
00:21:55.000 --> 00:22:01.000
three pi, minus one,
and so on.
00:22:00.000 --> 00:22:06.000
So, at the upper limit,
we get minus pi over n,
00:22:05.000 --> 00:22:11.000
oh, I didn't finish the
calculation, times the cosine of
00:22:11.000 --> 00:22:17.000
n pi,
which is minus one to the n'th
00:22:17.000 --> 00:22:23.000
power.
And now, how about the other
00:22:22.000 --> 00:22:28.000
guy?
Shall we do in our heads?
00:22:26.000 --> 00:22:32.000
Well, I can do it in my head,
but I'm not so sure about your
00:22:32.000 --> 00:22:38.000
heads.
Maybe just this once we won't.
00:22:37.000 --> 00:22:43.000
What is it?
It's plus sine nt,
00:22:41.000 --> 00:22:47.000
right?
So, I combined the two negative
00:22:44.000 --> 00:22:50.000
signs to a plus sign by putting
one this way and the other one
00:22:50.000 --> 00:22:56.000
that way.
And then, if I integrate that
00:22:53.000 --> 00:22:59.000
now, it's sine nt divided by n
squared,
00:22:58.000 --> 00:23:04.000
right?
And that's evaluated between
00:23:02.000 --> 00:23:08.000
zero and pi.
And of course,
00:23:05.000 --> 00:23:11.000
the sign function vanishes at
both ends.
00:23:09.000 --> 00:23:15.000
So, that part is simply zero.
And so, the final answer is
00:23:14.000 --> 00:23:20.000
that bn is equal to,
well, the pi's cancel.
00:23:19.000 --> 00:23:25.000
This minus combines with those
n to make one more.
00:23:23.000 --> 00:23:29.000
And so, the answer is two over
n times minus one to the n plus
00:23:30.000 --> 00:23:36.000
first power.
00:23:35.000 --> 00:23:41.000
And therefore,
the final result is that our
00:23:40.000 --> 00:23:46.000
Fourier series,
the Fourier series for f of t,
00:23:46.000 --> 00:23:52.000
that funny function
is, the Fourier series is
00:23:53.000 --> 00:23:59.000
summation bn,
which is two,
00:23:56.000 --> 00:24:02.000
put the two out front because
it's in every term.
00:24:04.000 --> 00:24:10.000
There's no reason to repeat it,
minus one to the n plus first
00:24:10.000 --> 00:24:16.000
power over n times the sign of
nt.
00:24:16.000 --> 00:24:22.000
That's summed from one to
00:24:19.000 --> 00:24:25.000
infinity.
Let's stop and take a look at
00:24:23.000 --> 00:24:29.000
that for a second.
Does that look right?
00:24:28.000 --> 00:24:34.000
Okay, here's our function.
00:24:41.000 --> 00:24:47.000
Here's our function.
What's the first term of this?
00:24:48.000 --> 00:24:54.000
When n is one,
this is plus one.
00:24:54.000 --> 00:25:00.000
So, the first term is sine t.
00:25:00.000 --> 00:25:06.000
What's the next term?
When n is two,
00:25:04.000 --> 00:25:10.000
this is negative.
So, it's minus one to the third
00:25:08.000 --> 00:25:14.000
power.
So, that's negative one over
00:25:11.000 --> 00:25:17.000
two.
So, it's minus one half sine
00:25:13.000 --> 00:25:19.000
two t,
and then it obviously continues
00:25:17.000 --> 00:25:23.000
in the same way plus a third
sign three t.
00:25:21.000 --> 00:25:27.000
Now, watch carefully because
what I'm going to say in the
00:25:24.000 --> 00:25:30.000
next minute is the heart of
Fourier series.
00:25:27.000 --> 00:25:33.000
I've given you that visual to
look at to try to reinforce
00:25:31.000 --> 00:25:37.000
this, but it's really very
important, as you go to the
00:25:34.000 --> 00:25:40.000
terminal yourself and do that
work, simple as it is,
00:25:38.000 --> 00:25:44.000
and pay attention now.
Now, if you think
00:25:42.000 --> 00:25:48.000
old-fashioned,
i.e.
00:25:43.000 --> 00:25:49.000
if you think taylor series,
you're not going to believe
00:25:47.000 --> 00:25:53.000
this because you will say,
well, let's see,
00:25:50.000 --> 00:25:56.000
these go on and on.
Obviously, it's the first term
00:25:53.000 --> 00:25:59.000
that's the important one.
That's two sine t.
00:25:57.000 --> 00:26:03.000
Now, the derivative,
two sine t, sine t would
00:26:00.000 --> 00:26:06.000
exactly follow the pink curve.
Sine t would look like this.
00:26:06.000 --> 00:26:12.000
Two sine t goes up
with the wrong angle.
00:26:10.000 --> 00:26:16.000
The first term,
in other words,
00:26:12.000 --> 00:26:18.000
does this.
It's going off with the wrong
00:26:15.000 --> 00:26:21.000
slope.
Now, that's the whole point of
00:26:18.000 --> 00:26:24.000
Fourier series.
Fourier series is not trying to
00:26:22.000 --> 00:26:28.000
approximate the function at zero
at the central starting point
00:26:27.000 --> 00:26:33.000
the way Taylor series do.
Fourier series tries to treat
00:26:31.000 --> 00:26:37.000
the whole interval,
and approximate the function
00:26:35.000 --> 00:26:41.000
nicely over the entire interval,
in this case,
00:26:38.000 --> 00:26:44.000
minus pi to pi,
as well as possible.
00:26:40.000 --> 00:26:46.000
Taylor series concentrates at
this point, does it the best it
00:26:44.000 --> 00:26:50.000
can at this point.
Then it tries,
00:26:46.000 --> 00:26:52.000
with the next term,
to do a little better,
00:26:49.000 --> 00:26:55.000
and then a little better.
The whole philosophy is
00:26:52.000 --> 00:26:58.000
entirely different.
Taylor series are used for
00:26:55.000 --> 00:27:01.000
analyzing what a function of
looks like which you stick close
00:26:59.000 --> 00:27:05.000
to the base point.
Fourier series analyze what a
00:27:04.000 --> 00:27:10.000
function looks like over the
whole interval.
00:27:07.000 --> 00:27:13.000
And, to do that,
you should therefore aim to,
00:27:10.000 --> 00:27:16.000
so the first approximation is
going to look like that,
00:27:14.000 --> 00:27:20.000
going to have entirely the
wrong slope.
00:27:16.000 --> 00:27:22.000
But, the next one will subtract
off something which sort of
00:27:21.000 --> 00:27:27.000
helps to fix it up.
I can't draw this.
00:27:23.000 --> 00:27:29.000
That's why I'm sending you to
the visual because the visual
00:27:27.000 --> 00:27:33.000
draws them beautifully.
And, it shows you how each
00:27:31.000 --> 00:27:37.000
successive term corrects the
Fourier series,
00:27:34.000 --> 00:27:40.000
and makes the sum a little
closer to what you started with.
00:27:40.000 --> 00:27:46.000
So, the next guy would,
let's see, so it's 2t.
00:27:44.000 --> 00:27:50.000
So, I'm subtracting off,
probably I'm just guessing,
00:27:50.000 --> 00:27:56.000
but I don't dare draw this.
I haven't prepared to draw it,
00:27:56.000 --> 00:28:02.000
and I know I'll get it wrong.
So, okay, your exercise.
00:28:02.000 --> 00:28:08.000
But, it'll look better.
It'll go, maybe,
00:28:07.000 --> 00:28:13.000
something like,
let's see, it has to end up...
00:28:12.000 --> 00:28:18.000
some of it gets subtracted
off...
00:28:17.000 --> 00:28:23.000
I don't know what it looks
like.
00:28:20.000 --> 00:28:26.000
When you use the visual at the
computer terminal,
00:28:25.000 --> 00:28:31.000
I've asked you to use it three
times on a variety of functions.
00:28:32.000 --> 00:28:38.000
I think this is maybe even one
of them.
00:28:34.000 --> 00:28:40.000
Notice that you can set the
parameter, you can set the
00:28:38.000 --> 00:28:44.000
coefficients independently.
In other words,
00:28:41.000 --> 00:28:47.000
you can go back and correct
your works, improving the
00:28:45.000 --> 00:28:51.000
earlier coefficients,
and it won't affect anything
00:28:48.000 --> 00:28:54.000
you did before.
But, the most vivid way to do
00:28:51.000 --> 00:28:57.000
it is to try to get,
visually, by moving the slider,
00:28:55.000 --> 00:29:01.000
to try to get the very best
value for the first coefficient
00:28:59.000 --> 00:29:05.000
you can, and look at the curve.
Then get the very best value
00:29:05.000 --> 00:29:11.000
for the second coefficient and
see how that improves the
00:29:09.000 --> 00:29:15.000
approximation,
and the third,
00:29:11.000 --> 00:29:17.000
and so on.
And, the point is,
00:29:13.000 --> 00:29:19.000
watch the approximations
approaching the function nicely
00:29:18.000 --> 00:29:24.000
over the whole interval instead
of concentrating all their
00:29:22.000 --> 00:29:28.000
goodness at the origin the way a
Taylor series would.
00:29:26.000 --> 00:29:32.000
Now, there is still one
mathematical point left.
00:29:30.000 --> 00:29:36.000
It's that equality sign,
which is wrong.
00:29:35.000 --> 00:29:41.000
Why is it wrong?
Well, what I'm saying is that
00:29:38.000 --> 00:29:44.000
if I add that the series,
it adds up to f of t.
00:29:43.000 --> 00:29:49.000
Now, it almost does but not
quite.
00:29:46.000 --> 00:29:52.000
And, I'd better give you the
rule, the theorem.
00:29:50.000 --> 00:29:56.000
Of all the theorems in this
course that aren't being proved,
00:29:56.000 --> 00:30:02.000
this is the one that would be
most outside the scope of this
00:30:01.000 --> 00:30:07.000
course, the one which I would
most like to prove,
00:30:05.000 --> 00:30:11.000
in fact, just because I'm a
mathematician but wouldn't dare.
00:30:12.000 --> 00:30:18.000
The theorem tells you when a
Fourier series converges to the
00:30:17.000 --> 00:30:23.000
function you started with.
And, the essence of it is this.
00:30:22.000 --> 00:30:28.000
If f is continuous,
is a continuous function,
00:30:26.000 --> 00:30:32.000
let's give the point,
it's confusing just to keep
00:30:30.000 --> 00:30:36.000
calling it t.
If you like,
00:30:32.000 --> 00:30:38.000
call it t, but I think it would
be better to call it t zero
00:30:37.000 --> 00:30:43.000
just to indicate I'm
looking at a specific point.
00:30:44.000 --> 00:30:50.000
So, if the function is
continuous there,
00:30:48.000 --> 00:30:54.000
the value of f of t is
equal to, the Fourier series
00:30:54.000 --> 00:31:00.000
converges, and it's equal to its
Fourier series,
00:30:59.000 --> 00:31:05.000
the sum of the Fourier series
at t zero.
00:31:05.000 --> 00:31:11.000
And, the fact that I can even
use the word sum means that the
00:31:09.000 --> 00:31:15.000
Fourier series converges.
In other words,
00:31:12.000 --> 00:31:18.000
when you add up all these guys,
you don't go to infinity or get
00:31:16.000 --> 00:31:22.000
something which just oscillates
around crazily.
00:31:20.000 --> 00:31:26.000
They really do add up to
something.
00:31:22.000 --> 00:31:28.000
Now, if f is not continuous at
t zero,
00:31:26.000 --> 00:31:32.000
this emphatically will not be
the case.
00:31:28.000 --> 00:31:34.000
It will definitely not,
but by far, the kinds of
00:31:32.000 --> 00:31:38.000
discontinuities which occur in
the applications are ones like
00:31:36.000 --> 00:31:42.000
in this picture,
where the discontinuities are
00:31:39.000 --> 00:31:45.000
jump discontinuities.
They are almost always jump
00:31:44.000 --> 00:31:50.000
discontinuities.
And, in that case,
00:31:47.000 --> 00:31:53.000
in other words,
they are isolated.
00:31:49.000 --> 00:31:55.000
The function looks good here
and here, but there's a break.
00:31:53.000 --> 00:31:59.000
Typically, electrical engineers
just don't leave a gap because
00:31:57.000 --> 00:32:03.000
they like, I don't know why.
But electrical engineer,
00:32:02.000 --> 00:32:08.000
and others of his or her ilk
would draw that function like
00:32:09.000 --> 00:32:15.000
this, like a rip saw tooth.
Even those vertical lines have
00:32:16.000 --> 00:32:22.000
no meaning whatever,
but they make people look
00:32:21.000 --> 00:32:27.000
happier.
So, if f has a jump
00:32:24.000 --> 00:32:30.000
discontinuity at t zero,
and as I said,
00:32:30.000 --> 00:32:36.000
that's the most important kind,
then f of t,
00:32:36.000 --> 00:32:42.000
then the Fourier series adds up
to, converges to,
00:32:42.000 --> 00:32:48.000
it converges,
and it converges to the mid
00:32:46.000 --> 00:32:52.000
point of the jump.
Let me just write it out in
00:32:52.000 --> 00:32:58.000
words like that,
the midpoint of the jump.
00:32:55.000 --> 00:33:01.000
That's the way we'll be using
it in this course.
00:32:58.000 --> 00:33:04.000
There's a notation for this,
and it's in your book.
00:33:02.000 --> 00:33:08.000
But, those of you who would be
interested in such things would
00:33:07.000 --> 00:33:13.000
know it anyway.
So, let's just call it the
00:33:11.000 --> 00:33:17.000
midpoint of the jump.
So, if I ask you,
00:33:14.000 --> 00:33:20.000
to what does this converge?
In other words,
00:33:18.000 --> 00:33:24.000
this series,
what this shows is that the
00:33:22.000 --> 00:33:28.000
series, I'll write it out in the
abbreviated form,
00:33:26.000 --> 00:33:32.000
summation minus one to the n
plus one over n sine nt,
00:33:32.000 --> 00:33:38.000
what's the sum of the series?
00:33:39.000 --> 00:33:45.000
What is it?
Let's call this not
00:33:41.000 --> 00:33:47.000
little f of t.
Let's call it capital F of t.
00:33:44.000 --> 00:33:50.000
I want to know,
00:33:46.000 --> 00:33:52.000
what's the graph of capital F
of t?
00:33:48.000 --> 00:33:54.000
Well, the initial thing is to
say, well, it must be the same
00:33:53.000 --> 00:33:59.000
as the graph of the function you
started with.
00:33:56.000 --> 00:34:02.000
And, my answer is almost,
but not quite.
00:34:00.000 --> 00:34:06.000
In fact, what will its graph
look like?
00:34:04.000 --> 00:34:10.000
Well, regardless of what
definition I made for the
00:34:09.000 --> 00:34:15.000
endpoints of those pink lines,
this function will converge to
00:34:16.000 --> 00:34:22.000
the following.
From here to here,
00:34:19.000 --> 00:34:25.000
I'll draw it.
I won't put in minus pi's.
00:34:23.000 --> 00:34:29.000
I'll leave that to your
imagination.
00:34:27.000 --> 00:34:33.000
So, there's a hole at the end
here.
00:34:33.000 --> 00:34:39.000
In other words,
the end of the line is not
00:34:36.000 --> 00:34:42.000
included.
And, the end of this line,
00:34:39.000 --> 00:34:45.000
regardless of whether it was
included to start with or not,
00:34:43.000 --> 00:34:49.000
it's not now.
And here, similarly,
00:34:46.000 --> 00:34:52.000
I start it here with a hole,
and then go down parallel to
00:34:51.000 --> 00:34:57.000
the function,
t, slope one.
00:34:53.000 --> 00:34:59.000
And now, how do I fill in,
so the missing places,
00:34:57.000 --> 00:35:03.000
this is the point,
pi.
00:35:00.000 --> 00:35:06.000
This is the point,
negative pi,
00:35:02.000 --> 00:35:08.000
and there are similar points as
I go out.
00:35:05.000 --> 00:35:11.000
Well, since the function is
continuous here,
00:35:08.000 --> 00:35:14.000
the Fourier series will
converge to this orange line.
00:35:12.000 --> 00:35:18.000
But here, there's a jump
discontinuity,
00:35:14.000 --> 00:35:20.000
and therefore,
the Fourier series,
00:35:17.000 --> 00:35:23.000
this function converges to the
midpoint of the jump,
00:35:20.000 --> 00:35:26.000
in other words,
to here.
00:35:22.000 --> 00:35:28.000
This function,
in other words,
00:35:24.000 --> 00:35:30.000
converges to this very
discontinuous looking function,
00:35:28.000 --> 00:35:34.000
and rather odd how these points
are, I say, but in this case,
00:35:33.000 --> 00:35:39.000
I can prove to you that it
converges here by calculating
00:35:37.000 --> 00:35:43.000
it.
Look, this is the point,
00:35:41.000 --> 00:35:47.000
pi.
What happens when you plug in t
00:35:44.000 --> 00:35:50.000
equals pi?
You get everyone of these terms
00:35:50.000 --> 00:35:56.000
is zero, and therefore the sum
is zero.
00:35:53.000 --> 00:35:59.000
So, it certainly converges,
and it converges to zero.
00:36:00.000 --> 00:36:06.000
Now, that's a general theorem.
It's rather difficult to prove.
00:36:04.000 --> 00:36:10.000
You would have to take,
again, an analysis course.
00:36:07.000 --> 00:36:13.000
But, I don't even get to it in
the analysis course which I
00:36:12.000 --> 00:36:18.000
teach.
If I had another semester I'd
00:36:14.000 --> 00:36:20.000
get to it, but I can't get
everything.
00:36:17.000 --> 00:36:23.000
Anyway, we're not going to get
to it this semester to your
00:36:21.000 --> 00:36:27.000
infinite relief.
But, you should know the
00:36:24.000 --> 00:36:30.000
theorem anyway.
People will expect you to know
00:36:27.000 --> 00:36:33.000
it.
Well, that was half the period,
00:36:32.000 --> 00:36:38.000
and in the remaining half,
you're going to stay a long
00:36:39.000 --> 00:36:45.000
time today.
Okay, no, don't panic.
00:36:43.000 --> 00:36:49.000
I have to extend the Fourier
series.
00:36:47.000 --> 00:36:53.000
Okay, let me give you the hurry
up version indicating the two
00:36:54.000 --> 00:37:00.000
ways in which it needs to be
extended.
00:37:00.000 --> 00:37:06.000
Extension number one --
00:37:14.000 --> 00:37:20.000
The period is not two pi,
but two times,
00:37:18.000 --> 00:37:24.000
I'll keep the two just to make
the formulas look as similar as
00:37:24.000 --> 00:37:30.000
possible to the old ones.
The period, let's say,
00:37:29.000 --> 00:37:35.000
instead of two pi,
is two times L.
00:37:34.000 --> 00:37:40.000
Now, I think you know enough
mathematics by this point to
00:37:37.000 --> 00:37:43.000
sort of, I hope you can sort of
shrug and say,
00:37:40.000 --> 00:37:46.000
well, you know,
isn't that just kind of like
00:37:42.000 --> 00:37:48.000
changing the units on the
t-axis?
00:37:44.000 --> 00:37:50.000
You're just stretching.
Yeah, right.
00:37:46.000 --> 00:37:52.000
All you do is make a change of
variable.
00:37:49.000 --> 00:37:55.000
Now, should we make it nicely?
I think I'll give you the final
00:37:52.000 --> 00:37:58.000
answer, and then I'll try to
decide while I'm writing it down
00:37:56.000 --> 00:38:02.000
how much I'll try to make the
argument.
00:38:00.000 --> 00:38:06.000
First of all,
the main thing to get is,
00:38:04.000 --> 00:38:10.000
if the period is not pi but L,
what are the natural versions
00:38:11.000 --> 00:38:17.000
of the cosine and sine to use?
Use the natural functions.
00:38:18.000 --> 00:38:24.000
Natural has no meaning,
but it's psychologically
00:38:23.000 --> 00:38:29.000
important.
In other words,
00:38:26.000 --> 00:38:32.000
what kind of function should
replace that?
00:38:33.000 --> 00:38:39.000
I'll certainly have a t here.
What do I put in front?
00:38:37.000 --> 00:38:43.000
I'll keep the n also.
The question is,
00:38:40.000 --> 00:38:46.000
what do I fix?
What should I put here in
00:38:43.000 --> 00:38:49.000
between in order to make the
thing come out,
00:38:47.000 --> 00:38:53.000
so that it has period 2L?
You probably should learn to do
00:38:52.000 --> 00:38:58.000
this formally as well as just
sort of psyching it out,
00:38:56.000 --> 00:39:02.000
and taking a guess,
or memorizing the answer.
00:39:00.000 --> 00:39:06.000
If this is the t-axis,
here is t and L,
00:39:03.000 --> 00:39:09.000
zero and L.
What you want to do is make a
00:39:08.000 --> 00:39:14.000
change of variable to the u-axis
where the axis is the same.
00:39:14.000 --> 00:39:20.000
This is still the point.
But, L, now,
00:39:17.000 --> 00:39:23.000
on the u coordinate,
has the name pi.
00:39:21.000 --> 00:39:27.000
Now, so I'm just describing a
change of variable on the axis.
00:39:26.000 --> 00:39:32.000
What's the one that does this?
Well, when t is L,
00:39:31.000 --> 00:39:37.000
u should be pi.
So, t should be L over pi.
00:39:37.000 --> 00:39:43.000
When u is pi,
00:39:39.000 --> 00:39:45.000
t is L, and vice versa.
How about expressing u in
00:39:43.000 --> 00:39:49.000
terms, well, then u is equal to
pi over L times t.
00:39:49.000 --> 00:39:55.000
That's the backwards form of
00:39:52.000 --> 00:39:58.000
writing it, or the forward form,
depending upon how you like to
00:39:58.000 --> 00:40:04.000
think of these things.
Okay, so the cosine should be
00:40:05.000 --> 00:40:11.000
pi over L times t,
in order that when t be L,
00:40:10.000 --> 00:40:16.000
it should be like cosine of n
pi,
00:40:16.000 --> 00:40:22.000
which is what we would have
had.
00:40:19.000 --> 00:40:25.000
So, if t is equal to L,
in other words,
00:40:25.000 --> 00:40:31.000
where is this from?
What am I trying to say?
00:40:32.000 --> 00:40:38.000
That's the function.
This one is probably a little
00:40:36.000 --> 00:40:42.000
easier to see.
Where is this one zero?
00:40:40.000 --> 00:40:46.000
The sine functions that we used
before was zero at zero pi,
00:40:46.000 --> 00:40:52.000
two pi, three pi.
Where is this one zero?
00:40:50.000 --> 00:40:56.000
It's zero at zero.
When t is equal to L,
00:40:54.000 --> 00:41:00.000
it's zero.
When t is equal to 2L,
00:40:58.000 --> 00:41:04.000
so, this is the right thing.
00:41:03.000 --> 00:41:09.000
So, it's zero.
It's periodic,
00:41:05.000 --> 00:41:11.000
and it's zero plus or minus L
plus or minus 2L.
00:41:08.000 --> 00:41:14.000
And, in fact,
formally you can verify that
00:41:11.000 --> 00:41:17.000
it's periodic with period 2L.
So, in other words,
00:41:15.000 --> 00:41:21.000
we want a Fourier expansion to
use these functions as the
00:41:19.000 --> 00:41:25.000
natural analog of what would be
up there.
00:41:22.000 --> 00:41:28.000
So, the period of our function
is 2L, and the formula is,
00:41:26.000 --> 00:41:32.000
I'll give you the formula.
It's f of t equals
00:41:32.000 --> 00:41:38.000
identical summation,
an, except you'll use these as
00:41:38.000 --> 00:41:44.000
the natural functions instead of
cosine nt and sine
00:41:45.000 --> 00:41:51.000
nt. So, n pi t over L
00:41:49.000 --> 00:41:55.000
plus bn, okay,
I'm tired, but I'll put it in
00:41:54.000 --> 00:42:00.000
anyway, n pi t over L.
00:42:00.000 --> 00:42:06.000
Yeah, but of course,
what about the formulas for an?
00:42:04.000 --> 00:42:10.000
Somebody up there is watching
over us.
00:42:08.000 --> 00:42:14.000
Here are the formulas.
They are exactly what you would
00:42:13.000 --> 00:42:19.000
guess if somebody said produce
the formulas in ten seconds,
00:42:19.000 --> 00:42:25.000
and you'd better be right,
and you didn't have time to
00:42:24.000 --> 00:42:30.000
calculate.
You say, well,
00:42:27.000 --> 00:42:33.000
it must be, let's do the cosine
series.
00:42:32.000 --> 00:42:38.000
Okay, let's not do a cosine.
So, it's one over L
00:42:36.000 --> 00:42:42.000
times the integral from negative
L, in other words,
00:42:41.000 --> 00:42:47.000
wherever you see an L,
wherever you see a pi,
00:42:45.000 --> 00:42:51.000
just put an L times the f of t
cosine, and now we'll use our
00:42:50.000 --> 00:42:56.000
new function,
not the old one.
00:42:52.000 --> 00:42:58.000
I submit that's an easy,
if you know the first formula,
00:42:57.000 --> 00:43:03.000
then this would be an easy one
to remember.
00:43:01.000 --> 00:43:07.000
All you do is change pi to L
everywhere.
00:43:06.000 --> 00:43:12.000
Except, you got to remember
this part.
00:43:09.000 --> 00:43:15.000
Make it a function periodic of
period 2L, not 2pi.
00:43:15.000 --> 00:43:21.000
And similarly,
bn is similar.
00:43:18.000 --> 00:43:24.000
It looks just the same way.
And, how about,
00:43:22.000 --> 00:43:28.000
and the same even-odd business
goes, too, so that if f of t,
00:43:28.000 --> 00:43:34.000
for example,
is even, and has period 2L,
00:43:33.000 --> 00:43:39.000
then the function,
then the best formula for the
00:43:38.000 --> 00:43:44.000
an will not be that one.
It will be two over L,
00:43:45.000 --> 00:43:51.000
and where you integrate only
from zero to L,
00:43:49.000 --> 00:43:55.000
f of t cosine.
00:44:01.000 --> 00:44:07.000
So, now, the bn's will be zero,
and you'll just have positive,
00:44:07.000 --> 00:44:13.000
etc.
for L.
00:44:08.000 --> 00:44:14.000
As I say, this is important
case, particularly if the period
00:44:13.000 --> 00:44:19.000
is two, in other words,
if the half period is one
00:44:18.000 --> 00:44:24.000
because in the literature,
frequently one is used as the
00:44:24.000 --> 00:44:30.000
standard normal reference,
not pi.
00:44:27.000 --> 00:44:33.000
Pi is convenient mathematically
because it makes the cosines and
00:44:33.000 --> 00:44:39.000
sines look simple.
But, in actual calculation,
00:44:39.000 --> 00:44:45.000
it tends to be where L is one.
So, usually you have a pi here.
00:44:45.000 --> 00:44:51.000
You don't have just nt.
Well, I should do a
00:44:49.000 --> 00:44:55.000
calculation, but instead of
doing that, let me give you the
00:44:55.000 --> 00:45:01.000
other extension.
Fortunately,
00:44:58.000 --> 00:45:04.000
there are plenty of
calculations in your book.
00:45:04.000 --> 00:45:10.000
So, let me give you in the last
couple of minutes the other
00:45:10.000 --> 00:45:16.000
extension.
This is going to be a very
00:45:14.000 --> 00:45:20.000
important one for us next time.
Typically, in applications,
00:45:21.000 --> 00:45:27.000
well, I mean,
the first thing,
00:45:24.000 --> 00:45:30.000
periodic functions are nice,
but let's face it.
00:45:30.000 --> 00:45:36.000
Most functions aren't periodic,
I have to agree.
00:45:37.000 --> 00:45:43.000
So, all this theory is just
about periodic functions?
00:45:40.000 --> 00:45:46.000
No.
It's about functions.
00:45:42.000 --> 00:45:48.000
Really, it's about functions
where the interval on which you
00:45:46.000 --> 00:45:52.000
are interested in them is
finite.
00:45:48.000 --> 00:45:54.000
It's a finite interval,
not functions which go to
00:45:52.000 --> 00:45:58.000
infinity.
For those, you will have to use
00:45:54.000 --> 00:46:00.000
Fourier transforms,
Fourier transforms,
00:45:57.000 --> 00:46:03.000
not Fourier series.
But, if you are interested in a
00:46:02.000 --> 00:46:08.000
function on a finite interval,
then you can use Fourier series
00:46:06.000 --> 00:46:12.000
even though the function isn't
periodic because you can make it
00:46:11.000 --> 00:46:17.000
periodic.
So, what you do is,
00:46:13.000 --> 00:46:19.000
if f of t is on,
let's take the interval from
00:46:17.000 --> 00:46:23.000
zero to L.
That's a sample finite
00:46:19.000 --> 00:46:25.000
interval.
I can always change the
00:46:22.000 --> 00:46:28.000
variable to make the interval
from zero to L.
00:46:25.000 --> 00:46:31.000
I can even make it from zero to
one, but that's a little too
00:46:29.000 --> 00:46:35.000
special.
It would be a little awkward.
00:46:34.000 --> 00:46:40.000
So, if a function is defined on
a finite interval,
00:46:38.000 --> 00:46:44.000
the way to apply the Fourier
series to it is make a periodic
00:46:43.000 --> 00:46:49.000
extension.
Now, since I have so little
00:46:47.000 --> 00:46:53.000
time, I'm just going to get away
with murder by just drawing
00:46:52.000 --> 00:46:58.000
pictures.
So, let me give you a function.
00:46:55.000 --> 00:47:01.000
Here's my function defined on
zero to L, colored chalk if you
00:47:01.000 --> 00:47:07.000
please.
Let's make it the function t
00:47:05.000 --> 00:47:11.000
squared,
and let's make L equal to one.
00:47:09.000 --> 00:47:15.000
That function is not periodic.
If I let it go off,
00:47:13.000 --> 00:47:19.000
it would just go off to
infinity and never repeat its
00:47:17.000 --> 00:47:23.000
values, except on the left-hand
side.
00:47:20.000 --> 00:47:26.000
But, I'm not even going to let
it be on the left hand side.
00:47:25.000 --> 00:47:31.000
It's only defined from zero to
one as far as I'm concerned.
00:47:29.000 --> 00:47:35.000
Okay, that function has an even
periodic extension.
00:47:35.000 --> 00:47:41.000
And, its graph looks like this
extended to be an even function.
00:47:39.000 --> 00:47:45.000
Okay, now, that means from zero
to negative L,
00:47:44.000 --> 00:47:50.000
you've got to make it look
exactly as it looked on the
00:47:48.000 --> 00:47:54.000
right-hand side.
Otherwise, it would be even.
00:47:51.000 --> 00:47:57.000
And now, what do I do?
Well, now I've got,
00:47:54.000 --> 00:48:00.000
from minus L to L.
So, all I'm allowed to do is
00:47:59.000 --> 00:48:05.000
keep repeating the values.
In other words,
00:48:03.000 --> 00:48:09.000
apply the theory of Fourier
series to this guy,
00:48:06.000 --> 00:48:12.000
use a cosine series because
it's an even function,
00:48:10.000 --> 00:48:16.000
and then everything you want to
do, you say, okay,
00:48:14.000 --> 00:48:20.000
all the rest of this is
garbage.
00:48:16.000 --> 00:48:22.000
I only really care about it
from here to here.
00:48:20.000 --> 00:48:26.000
And, that's what you will plug
into your differential equation
00:48:24.000 --> 00:48:30.000
on the right-hand side,
just that part of it,
00:48:28.000 --> 00:48:34.000
just this part of it.
How about the odd extension?
00:48:33.000 --> 00:48:39.000
What would that look like?
Okay, the odd extension,
00:48:37.000 --> 00:48:43.000
here I start like this.
And now, to extend it to be an
00:48:41.000 --> 00:48:47.000
odd function,
I have to make it go down in
00:48:44.000 --> 00:48:50.000
exactly the same way it went up.
And, what do I do here?
00:48:49.000 --> 00:48:55.000
I have to make it start
repeating its values so it will
00:48:53.000 --> 00:48:59.000
look like this.
So, the odd extension is going
00:48:57.000 --> 00:49:03.000
to be discontinuous in this
case.
00:49:01.000 --> 00:49:07.000
And, what's the Fourier series
going to converge to?
00:49:05.000 --> 00:49:11.000
Well, in each case,
to the average,
00:49:07.000 --> 00:49:13.000
to the midpoint of the jump,
and the odd extension looks
00:49:12.000 --> 00:49:18.000
like this, and this will give me
assigned series.
00:49:16.000 --> 00:49:22.000
Okay, you've got lots of
problems to do.