Well, let's get started.
The topic for today is --
Sorry. Thank you.
For today and the next two lectures, we are going to be
studying Fourier series. Today will be an introduction
explaining what they are. And, I calculate them,
but I thought before we do that I ought to least give a couple
minutes oversight of why and where we're going with them,
and why they're coming into the course at this place at all.
So, the situation up to now is that we've been trying to solve
equations of the form y double prime plus a y prime,
constant coefficient second-order equations,
and the f of t was the input. So, we are considering
inhomogeneous equations. This is the input.
And so far, the response, then, is the solution equals
the corresponding solution, y of t,
maybe with some given initial conditions to pick out a special
one we call the response, the response to that particular
input. And now, over the last few
days, the inputs have been, however, extremely special.
For input, the basic input has been an exponential,
or sines and cosines. And, the trouble is that we
learn how to solve those. But the point is that those
seem extremely special. Now, the point of Fourier
series is to show you that they are not as special as they look.
The reason is that, let's put it this way,
that any reasonable f of t which is periodic,
it doesn't have to be even very reasonable.
It can be somewhat discontinuous,
although not terribly discontinuous,
which is periodic with period, maybe not the minimal period,
but some period two pi. Of course, sine t
and cosine t have the exact period two pi,
but if I change the frequency to an integer frequency like
sine 2t or sine 26 t,
two pie would still be a period, although would not be
the period. The period would be shorter.
The point is, such a thing can always be
represented as an infinite sum of sines and cosines.
So, it's going to look like this.
There's a constant term you have to put out front.
And then, the rest, instead of writing,
it's rather long to write unless you use summation
notation. So, I will.
So, it's a sum from n equal one to infinity integer values of n,
in other words, of a sine and a cosine.
It's customary to put the cosine first,
and with the frequency, the n indicates the frequency
of the thing. And, the bn is sine nt.
Now, why does that solve the
problem of general inputs for periodic functions,
at least if the period is two pi or some fraction of it?
Well, you could think of it this way.
I'll make a little table. I'll make a little table.
Let's look at, let's put over here the input,
and here, I'll put the response.
Okay, suppose the input is the function sine nt.
Well, in other words, if you just solve the problem,
you put a sine nt here, you know how to get the
answer, find a particular solution, in other words.
In fact, you do it by converting this to a complex
exponential, and then all the rigmarole we've been going
through. So, let's call the response
something. Let's call it y.
I'd better index it by n because it, of course,
is a response to this particular periodic function.
So, n of t, and if the input is cosine nt,
that also will have a response, yn.
Now, I really can't call them both by the same name.
So, why don't we put a little s up here to indicate that that's
the response to the sine. And here, I'll put a little c
to indicate what the answer to the cosine.
You're feeding cosine nt, what you get out is
this function. Now what?
Well, by the way, notice that if n is zero,
it's going to take care of a constant term,
too. In other words,
the reason there is a constant term out front is because that
corresponds to cosine of zero t, which is one.
Now, suppose I input instead an cosine nt.
All you do is multiply the answer by an.
Same here. Multiply the input by bn.
You multiply the response. That's because the equation is
a linear equation. And now, what am I going to do?
I'm going to add them up. If I add them up from the
different ends and take a count also, the n equals zero
corresponding to this first constant term,
the sum of all these according to my Fourier formula is going
to be f of t. What's the sum of this,
the corresponding responses? Well, that's going to be
summation a n y n c t plus b n y n,
the response to the sine.
That will be the sum from one to infinity, and there will be
some sort of constant term here. Let's just call it c1.
So, in other words, if this input produces that
response, and these are things which we can calculate,
we're led by this formula, Fourier's formula,
to the response to things which otherwise we would have not been
able to calculate, namely, any periodic function
of period two pi will have, the procedure will be,
you've got a periodic function of period two pi.
Find its Fourier series, and I'll show you how to do
that today. Find its Fourier series,
and then the response to that general f of t will be this
infinite series of functions, where these things are things
you already know how to calculate.
They are the responses to sines and cosines.
And, you just formed the sum with those coefficients.
Now, why does that work? It works by the superposition
principle. So, this is true.
The reason I can do the adding and multiplying by constant,
I'm using the superposition principle.
If this input produces that response, then the sum of a
bunch of inputs produces the sum of the corresponding responses.
And, why is that? Why can I use the superposition
principle? Because the ODE is linear.
It's okay, since the ODE is linear.
That's what makes all this work.
Now, so what we're going to do today is I will show you how to
calculate those Fourier series. I will not be able to use it to
actually solve any differential equation.
It will take us pretty much all the period to show how to
calculate a Fourier series. And, okay, so I'm going to
solve differential equations on Monday.
Wrong. I probably won't even get to it
then because the calculation of a Fourier series is a sufficient
amount of work that you really want to know all the possible
tricks and shortcuts there are. Unfortunately,
they are not very clever tricks.
They are just obvious things. But, it will take me a period
to point out those obvious things, obvious in my sense if
not in yours. And, finally,
the third day, we'll solve differential
equations. I will actually carry out the
program. But the main thing we're going
to get out of it is another approach to resonance because
the things that we are going to be interested in are picking out
which of these terms may possibly produce resonance,
and therefore a very crazy response.
Some of the terms in the response suddenly get a much
bigger amplitude than this than you would normally have thought
they had because it's picking out resonant terms in the
Fourier series of the input. Okay, well, that's a big
mouthfu. Let's get started on
calculating. So, the program today is
calculate the Fourier series. Given f of t periodic,
having two pi as a period, find its Fourier series.
How, in other words, do I calculate those
coefficients, an and bn.
Now, the answer is not immediately apparent,
and it's really quite remarkable.
I think it's quite remarkable, anyway.
It's one of the basic things of higher mathematics.
And, what it depends upon are certain things called the
orthogonality relations. So, this is the place where
you've got to learn what such things are.
Well, I think it would be a good idea to have a general
definition, rather than immediately get into the
specifics. So, I'm going to call u of x,
u of t, I think I will use,
since Fourier analysis is most often applied when the variable
is time, I think I will stick to independent variable t all
period long, if I remember to, at any rate.
So, these are two continuous, or not very discontinuous
functions on minus pi. Let's make them periodic.
Let's say two pi is a period. So, functions,
for example like those guys, sine t, sine nt,
sine 22t,
and so on, say two pi is a period.
Well, I want them really on the whole real axis,
not there. Define for all real numbers.
Then, I say that they are orthogonal, perpendicular.
But nobody says perpendicular. Orthogonal is the word,
orthogonal on the interval minus pi to pi
if the integral, so, two are orthogonal.
Well, these two functions, if the integral from minus pi
to pi of u of t v of t, the product is zero,
that's called the orthogonality condition on minus pi to pi.
Now, well, it's just the
definition. I would love to go into a
little song and dance now on what the definition really
means, and what its application, why the word orthogonal is
used, because it really does have something to do with two
vectors being orthogonal in the sense in which you live it in
18.02. I'll have to put that on the
ice for the moment, and whether I get to it or not
depends on how fast I talk. But, you probably prefer I talk
slowly. So, let's compromise.
Anyway, that's the condition. And now, what I say is that
that Fourier, that blue Fourier series,
-- -- what finding the
coefficients an and bn depends upon is this theorem that the
collection of functions, as I look at this collection of
functions, sine nt for any value of the integer,
n, of course I can assume n is a positive integer because sine
of minus nt is the same as sine of nt.
And, cosine mt, let's give it a different,
so I don't want you to think they are exactly the same
integers. So, this is a big collection of
functions, as n runs from one to infinity-- Here,
I could let m be run from zero to infinity because cosine of
zero t means something.
It's a constant, one-- that any two distinct
ones, two distinct, you know, how can two things be
not different? Well, you know,
you talk about two coincident roots.
I'm just killing, doing a little overkill.
Any two distinct ones of these, two distinct members of the set
of this collection of, I don't know,
there's no way to say that, any two distinct ones are
orthogonal on this interval. Of course, they all have two pi
as a period for all of them. So, they form into this general
category that I'm talking about, but any two distinct ones are
orthogonal on the interval for minus pi to pi.
So, if I integrate from minus
pi to pi sine of three t times cosine of four t dt,
answer is zero.
If I integrate sine of 3t times
the sine of 60t, answer is zero.
The same thing with two cosines, or a sine and a cosine.
The only time you don't get zero is if you integrate,
if you make the two functions the same.
Now, how do you know that you could not possibly get the
answer is zero if the two functions are the same?
If the two functions are the same, then I'm integrating a
square. A square is always positive.
I'm integrating a square. A square is always positive,
and therefore I cannot get the answer, zero.
But, in the other cases, I might get the answer zero.
And the theorem is you always do.
Okay, so, why is this? Well, there are three ways to
prove this. It's like many fundamental
facts in mathematics. There are different ways of
going about it. By the way, along with the
theorem, I probably should have included, so,
I'm far away. But you might as well include,
because we're going to need it. What happens if you use the
same function? If I take U equal to V,
and in that case, as I've indicated,
you're not going to get the answer, zero.
But, what you will get is, so, in other words,
I'm just asking, what is the sine of
n t squared. That's a case where two of them
are the same. I use the same function.
What's that? Well, the answer is,
it's the same as what you will get if you integrate the cosine,
cosine squared n t dt.
And, the answer to either one
of these is pi. That's something you know how
to do from 18.01 or the equivalent thereof.
You can integrate sine squared. It's one of the things you had
to learn for whatever exam you took on methods of integration.
Anyway, so I'm not going to calculate this out.
The answer turns out to be pi. All right, now,
the ways to prove it are you can use trig identities.
And, I'm asking you in one of the early problems in the
problem set, identities, identities for the product of
sine and cosine, expressing it in a form in
which it's easy to integrate, and you can prove it that way.
Or, you can use, if you have forgotten the
trigonometric identities and want to get some more exercise
with complex-- you can use complex exponentials.
So, I'm asking you how to, in another part of the same
problem I'm asking you how to do it, do one of these,
at any rate, using complex exponentials.
And now, I'm going to use a mysterious third method another
way. I'm going to use the ODE.
I'm going to do that because this is the method.
It's not just sines and cosines which are orthogonal.
There are masses of orthogonal functions out there.
And, the way they are discovered, and the way you
prove they're orthogonal is not with trig identities and complex
exponentials because those only work with sines and cosines.
It is, instead, by going back to the
differential equation that they solve.
And that's, therefore, the method here that I'm going
to use here because this is the method which generalizes to many
other differential equations other than the simple ones
satisfied by sines and cosines. But anyway, that is the source.
So, the way the proof of these orthogonality conditions goes,
so I'm not going to do that. And, I'm going to assume that m
is different from n so that I'm not in either of these two
cases. What it depends on is,
what's the differential equation that all these
functions satisfy? Well, it's a different
differential equation depending upon the value of n,
-- -- but they look at essentially
the same. These satisfy the differential
equation, in other words, what they have in common.
The differential equation is, let's call it u.
It looks better. It's going to look better if
you let me call it u. u double prime plus,
well, n squared, so for the function sine n t
cosine n t, satisfy u double
prime plus n squared times u.
In other words, the frequency is n,
and therefore, this is a square of the
frequency is what you put here, equals zero.
In other words, what these functions have in
common is that they satisfy differential equations that look
like that. And the only thing that's
allowed to vary is the frequency, which is allowed to
change. The frequency is in this
coefficient of u. Now, the remarkable thing is
that's all you need to know. The fact that they satisfy the
differential equation, that's all you need to know to
prove the orthogonality relationship.
Okay, let's try to do it. Well, I need some notation.
So, I'm going to let un and vm be any two of the functions.
In other words, I'll assume m is different from
n. For example,
this one could be sine nt, and that could be
sine of mt, or this could be sine nt
and that could be cosine of mt.
You get the idea. Any two of those in the
subscript indicates whether what the n or the m is that are in
that. Any two, and I mean really two,
distinct, well, if I say that m is not n,
then they positively have to be different.
So, again, it's overkill with my two's-ness.
And, what I'm going to calculate, well,
first of all, from the equation,
I'm going to write the equation this way.
It says that u double prime is equal to minus n squared u.
That's true for any of these
guys. Of course, here,
it would be v double prime is equal to minus m squared
times v. You have to make those simple
adjustments. And now, what we're going to
calculate is the integral from minus pi to pi of un double
prime times vm dt.
Now, just bear with me.
Why am I going to do that? I can't explain what I'm going
to do that. But you won't ask me the
question in five minutes. But the point is,
this is highly un-symmetric. The u is differentiated twice.
The v isn't. So, those two functions-- but
there is a way of turning them into an expression which looks
extremely symmetric, where they are the same.
And the way to do that is I want to get rid of one of these
primes here and put one on here. The way to do that is if you
want to integrate one of these guys, and differentiate this one
to make them look the same, that's called integration by
parts, the most important theoretical method you learned
in 18.01 even though you didn't know that it was the most
important theoretical method. Okay, we're going to use it now
as a basis for Fourier series. Okay, so I'm going to integrate
by parts. Now, the first thing you do,
of course, when you integrate by parts is you just do the
integration. You don't do differentiation.
So, the first thing looks like this.
And, that's to be evaluated between negative pi and pi.
In doing integration by parts between limits,
minus what you get by doing both.
You do both, the integration and the
differentiation. And, again, evaluate that
between limits. Now, I'm just going to BS my
way through this. This is zero.
I don't care what the un's, which un you picked and which
vm you picked. The answer here is always going
to be zero. Instead of wasting six boards
trying to write out the argument, let me wave my hands.
Okay, it's clear, for example,
that a v is a sine, sine mt.
Of course it's zero because the sine vanishes at both pi and
minus pi. If the un were a cosine,
after I differentiate it, it became a sine.
And so, now it's this side guy that's zero at both ends.
So, the only case in which we might have a little doubt is if
this is a cosine, and after differentiation,
this is also a cosine. In other words,
it might look like cosine, after, this cosine nt times
cosine mt. But, I claim that that's zero,
too. Why?
Because the cosines are even functions, and therefore,
they have the same value at both ends.
So, if I subtract the value evaluated at pi,
and subtract the value of minus pi, again zero because I have
the same value at both ends. So, by this entirely convincing
argument, no matter what combination of sines and cosines
I have here, the answer to that part will always be zero.
So, by calculation, but thought calculation;
it's just a waste of time to write anything out.
You stare at it until you agree that it's so.
And now, I've taken, by this integration by parts,
I've taken this highly un-symmetric expression and
turned it into something in which the u and the v are
treated exactly alike. Well, good, that's nice,
but why? Why did I go to this trouble?
Okay, now we're going to use the fact that this satisfies the
differential equation, in other words,
that u double prime is equal to minus n,
I'm sorry, I should have subscripted this.
If that's the solution, then this is equal to,
times. You have to put in a subscript
otherwise. The n wouldn't matter.
All right, I'm now going to take that expression,
and evaluate it differently. un double prime vm dt
is equal to, well, un double prime,
because it satisfies the differential equation is equal
to that. So, what is this?
This is minus n squared times the integral from
negative pi to pi, and I'm replacing un double
prime by minus n squared un.
I pulled the minus n squared out.
So, it's un here, and the other factor is vm dt.
Now, that's the proof. Huh?
What do you mean that's the proof?
Okay, well, I'll first state it, why intuitively that's the
end of the argument. And then, I'll spell it out a
little more detail, but the more detail you make
for this, the more obscure it gets instead of,
look, I just showed you that this is symmetric in u and v,
after you massage it a little bit.
Here, I'm calculating it a different way.
Is this symmetric in u and v? Well, the answer is yes or no.
Is this symmetric at u and v? No.
Why? Because of the n.
The n favors u. We have what is called a
paradox. This thing is symmetric in u
and v because I can show it is. And, it's not symmetric in u
and v because I can show it is. I can show it's not symmetric
because it favors the n. Now, there's only one possible
resolution of that paradox. Both would be symmetric if what
were true? Pardon?
Negative pi. All right, let me write it this
way. Okay, never mind.
You see, the only way this can happen is if this expression is
zero. In other words,
the only way something can be both symmetric and not symmetric
is if it's zero all the time. And, that's what we're trying
to prove, that this is zero. But, instead of doing it that
way, let me show you. This is equal to that,
and therefore, two things according to Euclid,
two things equal to the same thing are equal to each other.
So, this equals that, which, in turn,
therefore, equals what I would have gotten.
I'm just saying the symmetry of different way,
what I would have gotten if I had done this calculation.
And, that turns out to be minus m squared times the integral
from minus pi to pi of un vm dt.
So, these two are equal because
they are both equal to this. This is equal to that.
This equals that. Therefore, how can this equal
that unless the integral is zero?
How's that? Remember, m is different from
n. So, what this proves is,
therefore, the integral from negative pi to pi of un vm dt is
equal to zero,
at least if m is different from n.
Now, there is one case I didn't include.
Which case didn't I include? un times un is not supposed to
be zero. So, in that case,
I don't have to worry about, but there is a case that I
didn't. For example,
something like the cosine of nt times the sine of nt.
Here, the m is the same as the
n. Nonetheless,
I am claiming that this is zero because these aren't the same
function. One is a cosine.
Why is that zero? Can you see mentally that
that's zero? Mentally?
Well, this is trying to be in another life,
it's trying to be one half the sine of two nt, right?
And obviously the integral of
sine of two nt is zero between minus pi and pi
because you integrate it,
and it turns out to be zero. You integrate it to a cosine,
which has the same value of both ends.
Well, that was a lot of talking.
If this proof is too abstract for you, I won't ask you to
reproduce it on an exam. You can go with the proofs
using trigonometric identities, and/or complex exponentials.
But, you ought to know at least one of those,
and for the problem set I'm asking you to fool around a
little with at least two of them.
Okay, now, what has this got to do with the problem we started
with originally? The problem is to explain this
blue series. So, our problem is,
how, from this, am I going to get the terms of
this blue series? So, given f of t,
two pi s a period. Find the an and the bn.
Okay, let's focus on the an. The bn is the same.
Once you know how to do one, you know how to do the other.
So, here's the idea. Again, it goes back to the
something you learned at the very beginning of 18.02,
but I don't think it took. But maybe some of you will
recognize it. So, what I'm going to do is
write it. Here's the term we're looking
for here, this one. Okay, and there are others.
It's an infinite series that goes on forever.
And now, to make the argument, I've got to put it one more
term here. So, I'm going to put in ak
cosine kt. I don't mean to imply that that
k could be more than n, in which case I should have
written it here. I could have also used equally
well bk sine kt here, and I could have put it
there. This is just some other term.
This is the an, and this is the one we want.
And, this is some other term. Okay, all right,
now, what you do is, to get the an,
what you do is you multiply everything through by,
you focus on the one you want, so it's dot,
dot, dot, dot, dot, and you multiply by cosine
nt. So, it's ak cosine kt times
cosine nt.
Of course, that gets multiplied, too.
But, the one we want also gets multiplied, an.
And, it becomes, when I multiply by cosine nt,
cosine squared nt,
and now, I hope you can see what's going to happen.
Now, oops, I didn't multiply the f of t,
sorry. It's the oldest trick in the
book. I now integrate everything from
minus, so I don't endlessly recopy.
I'll integrate by putting it up in yellow chalk,
and you are left to your own devices.
This is definitely a colored pen type of course.
Okay, so, you want to integrate from minus pi to pi?
Good. Just integrate everything on
the right hand side, also, from minus pi to pi.
Plus, these are the guys just to indicate that I haven't,
they are out there, too.
And now, what happens? What's this?
Zero. Every term is zero because of
the orthogonality relations. They are all of the form,
a constant times cosine nt times something different from
cosine nt, sine kt,
cosine kt, or even that constant term.
All of the other terms are zero, and the only one which
survives is this one. And, what's its value?
The integral from minus pi to pi of cosine squared,
I put that up somewhere. It's right here,
down there? It is pi.
So, this term turns into an pi, an, dragged along,
but this, the integral of the square of the cosine turns out
to be pi. And so, the end result is that
we get a formula for an. What is an?
an is, well, an times pi,
all these terms of zero, and nothing is left but this
left-hand side. And therefore,
an times pi is the integral from negative pi to pi of f of t
times cosine nt dt.
But, that's an times pi.
Therefore, if I want just an, I have to divide it by pi.
And, that's the formula for the coefficient an.
The argument is exactly the same if you want bn,
but I will write it down for the sake of completeness,
as they say, and to give you a chance to
digest what I've done, you know, 30 seconds to digest
it. Sine nt dt.
And, that's because the argument is the same.
And, the integral of sine squared nt is also
pi. So, there's no difference
there. Now, there's only one little
caution. It have to be a little careful.
This is n one, two, and so on,
and this is also n one, two, and unfortunately,
the constant term is a slight exception.
We better look at that specifically because if you
forget it, you can get them to gross, gross,
gross errors. How about the constant term?
Suppose I repeat the argument for that in miniature.
There is a constant term plus other stuff, a typical other
stuff, an cosine, let's say.
How am I going to get that constant term?
Well, if you think of this as sort of like a constant times,
the reason is the constant is because it's being multiplied by
cosine zero t. So, that suggests I should
multiply by one. In other words,
what I should do is simply integrate this from negative pi
to pi, f of t dt.
What's the answer? Well, this integrated from
minus pi to pi is how much? It's c zero times two pi,
right? And, the other terms all give
me zero. Every other term is zero
because if you integrate cosine nt or sine nt
over a complete period, you always get zero.
There is as much area above the axis or below.
Or, you can look at two special cases.
Anyway, you always get zero. It's the same thing with sine
here. So, the answer is that c zero
is equal to, is a little special.
You don't just put n equals zero here because then
you would lose a factor of two. So, c zero should be one
over two pi times this integral.
Now, there are two kinds of people in the world,
the ones who learn two separate formulas, and the ones who just
learn two separate notations. So, what most people do is they
say, look, I want this to be always the formula for a zero.
That means, even when n is zero, I want this to be the
formula. Well, then you are not going to
get the right leading term. Instead of getting c zero,
you're going to get twice it, and therefore,
the formula is, the Fourier series,
therefore, isn't written this way.
It's written-- If you want an a zero there,
calculate it by this formula. Then, you've got to write not c
zero, but a zero over two.
I think you will be happiest if I have to give you advice.
I think you'll be happiest remembering a single formula for
the an's and bn's, in which case you have to
remember that the constant leading term is a zero over two
if you insist on using that formula.
Otherwise, you have to learn a special formula for the leading
coefficient, namely one over two pi instead of one
over pi. Well, am I really going to
calculate a Fourier series in four minutes?
Not very likely, but I'll give it a brave
college try. Anyway, you will be doing a
great deal of it, and your book has lots and lots
of examples, too many, in fact.
It ruined all the good examples by calculating them for you.
But, I will at least outline. Do you want me to spend three
minutes outlining a calculation just so you have something to
work on in the next boring class you are in?
Let's see, so I'll just put a few key things on the board.
I would advise you to sit still for this.
Otherwise you're going to hack it, and take twice as long as
you should, even though I knew you've been up to 3:00 in the
morning doing your problem set. Cheer up.
I got up at 6:00 to make up the new one.
So, we're even. This should be zero here.
So, here's minus pi. Here's pi.
Here's one, negative one. The function starts out like
that, and now to be periodic, it then has to continue on in
the same way. So, I think that's enough of
its path through life to indicate how it runs.
This is a typical square-away function, sometimes it's called.
It's an odd function. It goes equally above and below
the axis. Now, the integrals,
when you calculate them, the an is going to be,
okay, look, the an is going to turn out to be zero.
Let me, instead, and you will get that with a
little hacking. I'm much more worried about
what you'll do with the bn's. Also, next Monday you'll see
intuitively that the an is zero, in which case you won't even
bother trying to calculate it. How about the bn,
though? Well, you see,
because the function is discontinuous,
so, this is my input. My f of t is that
orange discontinuous function. The bn is going to be,
I have to break it into two parts.
In the first part, the function is negative one.
And there, I will be integrating from minus pi to pi
of the function, which is minus one times the
sine of nt dt.
And then, there's another part,
sorry, minus pi to zero. The other part I integrate from
zero to pi of what? Well, f of t is now plus one.
And so, I simply integrate sine
nt dt. Now, each of these is a
perfectly simple integral. The only question is how you
combine them. So, this is,
after you calculate it, it will be (one minus cosine n
pi) all over n.
And, this part will turn out to be (one minus cosine n pi) over
n also. And therefore,
the answer will be two minus two cosine, two over n times,
right, two minus, two times (one minus cosine n
pi) over n.
No, okay, now, what's this?
This is minus one if n is odd. It's plus one if n is even.
Now, either you can work with it this way, or you can combine
the two of them into a single expression.
Its minus one to the nth power takes care of both of
them. But, the way the answer is
normally expressed, it would be minus two over n,
two over n times, if n is even,
I get zero. If n is odd,
I get two. So, times two,
if n is odd, and zero if n is even.
So, it's four over n, or it's zero,
and the final series is a sum of those coefficients times the
appropriate-- cosine or sine? Sine terms because the cosine
terms were all coefficients, all turned out to be zero.
I'm sorry I didn't have the chance to do that calculation in
detail. But, I think that's enough
sketch for you to be able to do the rest of it yourself.