Today, and for the next two weeks, we are going to be
studying what, for many engineers and a few
scientists is the most popular method of solving any
differential equation of the kind that they happen to be,
and that is to use the popular machine called the Laplace
transform. Now, you will get proficient in
using it by the end of the two weeks.
But, there is always a certain amount of mystery that hangs
around it. People scratch their heads and
can't figure out where it comes from.
And, that bothers them a lot. In the past,
I've usually promised to tell you, the students at the end of
the two weeks, but I almost never have time.
So, I'm going to break that glorious tradition and tell you
up front at the beginning, where it comes from,
and then talk very fast for the rest of the period.
Okay, a good way of thinking of where the Laplace transform
comes from, and a way which I think dispels some of its
mystery is by thinking of power series.
I think virtually all of you have studied power series except
possibly a few students who just had 18.01 here last semester,
and probably shouldn't be taking 18.03 anyway,
now. But anyway, a power series
looks like this: summation (a)n x to the n.
And, you sum that from,
let's say, zero to infinity. And, the typical thing you want
to do with it is add it up to find out what its sum is.
Now, the only way I will depart from tradition,
instead of calling the sum some generic name like f of x,
in order to identify the sum with the coefficients,
a, I'll call it a of x.
Now, I want to make just one slight change in that.
I want to use computer notation, which doesn't use the
subscript (a)n. Instead, this,
it thinks of as a function of the discreet variable,
n. In other words,
it's a function which assigns to n equals zero,
one, two, three real numbers. That's what this sequence of
coefficients really is. So, the computer notation will
look almost the same. It's just that I will write
this in functional notation as a of n instead of (a)n.
But, it still means the real number associated with the
positive integer, n, and everything else is the
same. See, what I'm thinking of this
as doing is taking this discreet function, which gives me the
sequence of coefficients of the power series,
and associating that with the sum of the power series.
Let me give you some very simple examples,
two very simple examples, which I think you know.
Suppose this is a function one. Now, what do I mean by that?
I mean it's the constant function, one.
To every positive integer, it assigns the number one.
Okay, what's a of x? What I'm saying is,
in other words, in this fancy,
mystifying form, is all of these guys are one,
what's a of x? One plus x plus x squared plus
x cubed. Look, you are supposed to be
born knowing what that adds up to.
It adds up to one over one minus x,
except that's the wrong answer. What's wrong about it?
It's not true for every value of x.
That's only true when x is such that that series converges,
and that is only true when x lies between negative one and
one. So, it's not this function.
It's this function with its domain restricted to be less
than one in absolute value. What does that converge to?
If x is bigger than one, the answer is it doesn't
converge. There's nothing else you can
put here. Okay, let's take another
function. Suppose this is,
let's see, one over n you probably won't know.
Let's take one you will know, one over n factorial.
Suppose a of n is the function
one over n factorial,
what's a of x? So, what I'm asking is,
what does this add up to when the coefficient here is one over
n factorial?
What's summation x to the n over n factorial?
It is e to the x.
And, this doesn't have to be qualified because this is true
for all values of x. So, in other words,
from this peculiar point of view, I think of a power as
summing the operation, of summing a power series as
taking a discreet function defined for positive integers,
or nonnegative integers, and doing this funny process.
And, out of it comes a continuous function of some
sort. And, notice what goes in is the
variable, n. But, what comes out is the
variable, x. Well, that's perfectly natural.
That's the way a power series is set up.
So, the question I ask is, this is a discreet situation,
a discreet summation. Suppose I made the summation
continuous instead of discreet. So, I want the continuous
analog of what I did over there. Okay, what would a continuous
analog be? Well, instead of,
I'll replace n zero, one, two, that will be replaced
by a continued, that's a discreet variable.
I'll replace it by a continuous variable, t, which runs from
zero to infinity, and is allowed to take every
real value in between instead of being only allowed to take the
values of the positive nonnegative integers.
Okay, well, if I want to use t instead of n,
I clearly cannot sum in the usual way over all real numbers.
But, the way the procedure which replaces summation over
all real numbers is integration. So, what I'm going to do is
replace that sum by the integral from zero to infinity.
That's like the sum from zero to infinity of what?
Well, of some function, but now n is being replaced by
the continuous variable, t.
So, this is going to be a function of t.
And, how about the rest of it? The rest I will just copy,
x to the n'th. Well, instead of n I have to
write t and dt. And, what's the sum?
Well, I'll call the sum, what's the sum a function of?
I integrate out the t. So, that doesn't appear in the
answer. All that appears is this
number, x, this parameter, x.
For each value of x, like one, two,
or 26.3, this integral has a certain value,
and I can calculate it. So, this is going to end up as
a function of x, just as it did before.
Now, I could leave it in that form, but no mathematician would
like to do that, and very few engineers either.
The reason is, in general, when you do
integration and differentiation, you do not want to have as the
base of an exponential something like x.
The only convenient thing to have is e, and the reason is
because it's only e that people really like to differentiate,
e to the something. The only thing is that people
really like to differentiate or integrate.
So, I'm going to make this look a little better by converting x
to the t to the base e. I remember how to do that.
You write x equals e to the log x and so x to
the t will be e to the log x times t,
if you want. Now, the only problem is I want
to make one more little change. After all, I want to be able to
calculate this integral. And, it's clear that if t is
going to infinity, if I have a number here,
for example, like x equals two,
that integral is really quite unlikely to converge.
For example, if a of t were just
the constant function, one, the integral certainly
wouldn't converge. It would be horrible.
That integral only has a chance of converging if x is a number
less than one, so that when I take bigger and
bigger powers of it, I get smaller and smaller
numbers. Don't forget,
this is an improper integral going all the way up to
infinity. Those need treatment,
delicate handling. All right, so I really want x
to be less than one. Otherwise, that integral is
very unlikely to converge. I'd better have it positive,
because if I allow it to be negative I'm going to get into
trouble with negative powers, see what's minus one,
for example, to the one half when t is one
half. That's already imaginary.
I don't want that. If you've got an exponential,
the base has got to be a positive number.
So, I want x to be a positive number.
All right, if x in my actual practices going to lie between
zero and one in order to make the integral converge,
how about log x? Well, log x,
if x is less than one, so log x is going to be
less than zero, and it's going to go all the
way down to negative infinity. So, this means log x is
negative. In this interesting range of x,
the log x is always going to be negative.
And now, I don't like that. The first place I'd like to
call this by a new variable since no one uses log x as a
variable. And, it would make sense to
make it a negative, to make it negative,
that is, to write log x is equal to negative s.
Let's put it on the other side, in order that since log x is
always going to be less than zero, then s will always be
positive. And it's always more convenient
to work with positive numbers instead of negative numbers.
So, if I make those changes, what happens to the integral?
Well, I stress, all these changes are just
cosmetic to make things a little easier to work with in terms of
symbols. First of all,
the a I'm going to change. I don't want to call it a of t
because most people don't call functions a of t.
They call them f of t. So, I'll call it f of t.
x is e to the log x, which is e to the minus s.
So, x has its name changed to e
to the minus s. In other words,
I'm using as the new variable not x any longer but s in order
that the base be e. t, I now raise this to the t'th
power, but by the laws of exponents, that means I simply
multiply the exponent by t, and dt.
And now, since I'm calling the function f of t,
the output ought to be called capital F.
But it's now a function, since I've changed the
variable, of s. It's no longer a function of x.
If you like, you may think of this as a of,
what's x? x is e to the negative s,
I guess. I mean, no one would leave a
function in that form. It's simply a function of s.
And, what is that? So, what have we got,
finally? What we have,
dear hearts, is this thing,
which I stress is nothing more than the continuous analog of
the summation of a power series. This is the discrete version.
This is by these perfectly natural transformations the
continuous version of the same thing.
It starts with a function defined for positive values of
t, and turns it into a function of s.
And, this is called the Laplace transform.
Now, if I've done my work correctly, you should all be
saying, oh, is that all? But, I know you aren't.
So, it's okay. You'll get used to it.
The first thing you have to get used to is one thing some people
never get used to, which is you put in a function
of t, and you get out a function of s.
How could that be? You know, for an operator,
you put in 3x, and you get out three if it's a
differentiation operator. In other words,
when you have an operator, the things we've been talking
about the last two or three weeks in one form or another,
at least the variable doesn't get changed.
Well, but for a transform it does, and that's why it's called
a transform. So, the difference between a
transform and an operator is that for a transform a function
of t comes in, but a function of s comes out.
The variable gets changed, whereas for an operator,
f of t goes in and what comes out is g of t,
a function using the same variable like
differentiation is a typical example of an operator,
or the linear differential operators we've been talking
about. Well, but this doesn't behave
that way. The variable does get changed.
That's, in fact, extremely important in the
applications. In the applications,
t usually means the time, and s very often,
not always, but very often is a variable measuring frequency,
for instance. But, so that's a peculiar thing
that's hard to get used to. But, a good thing is the fact
that it's a linear transform. In other words,
it obeys the laws we'd love and like that the Laplace
transform-- oh, I never gave you any notation
for the laplace transform. Hey, I'd better do that.
Okay, so, some notation: there are two notations that
are used. Your book mostly uses the
notation that the laplace transform of f of t is capital F
of s, uses the same letter but with
the same capital. Now, as you will see,
there are some places you absolutely cannot use that
notation. It may seem strange,
looks perfectly natural. There are certain laws you
cannot express using that notation.
It's baffling. But, if you can't do it this
way, you can do it using this notation instead.
One or the other will almost always work.
So, I'll use my little squiggly notation, but that's what I use.
I think it's a little more vivid, and the trouble is that
this piles up too many parentheses.
And, that's always hard to read.
So, I like this better. So, these are two alternate
ways of saying the same thing. The Laplace transform of this
function is that one. Okay, well, let's use,
for the linearity law, it's definitely best.
I really cannot express the linearity law using the second
notation, but using the first notation, it's a breeze.
The Laplace transform of the sum of two functions is the sum
of their Laplace transforms of each of them separately.
Or, better yet, you could write it that way.
Let's write it this way. That way, it looks more like an
operator, L of f plus L of g.
And, of the same way, if you take a function and
multiply it by a constant and take the laplace transform,
you can pull the constant outside.
And, of course, why are these true?
These are true just because of the form of the transform.
If I add up f and g, I simply add up the two
corresponding integrals. In other words,
I'm using the fact that the integral, this definite
integral, is itself a linear operator.
Well, that's the general setting.
That's where it comes from, and that's the notation for it.
And, now we have to get to work.
The first thing to do to get familiar with this is,
obviously what we want to do is say, okay, these were the
transforms of some simple discreet functions.
Okay, suppose I put in some familiar functions,
f of t. What do their Laplace
transforms look like? So, let's do that.
So, one of the boards I should keep stored.
Why don't I store on this board?
I'll store on this board the formulas as we get them.
So, let's see, what should we aim at,
first? Let's first find,
and I'll do the calculations on the sideboard,
and we'll see how it works out. I'm not very sure.
In other words, what's the Laplace transform of
the function, one?
Well, there's an even easier one.
What's the Laplace transform of the function zero?
Answer: zero. Very exciting.
What's the Laplace transform of one?
Well, it doesn't turn out the constant anymore than it turned
out to be a constant up there. Let's calculate it.
Now, you can do these calculations carefully,
dotting all the i's, or pretty carefully,
or not carefully at all, i.e.
sloppily. I'll let you be sloppy after,
generally speaking, you could be sloppy unless the
directions tell you to be less sloppy or to be careful,
okay? So, I'll do one carefully.
Let's calculate the Laplace transform of one carefully.
Okay, in the beginning, you've got nothing to use with
the definition. So, I have to calculate the
integral from zero to infinity of one, that's the f of t times
e to the negative s t, so I don't have to
put in the one, dt.
All right, now, let me remind you,
this is an improper integral. This is just about the first
time in the course we've had an improper integral.
But, there are going to be a lot of them over the next couple
of weeks, nothing but. All right, it's an improper
integral. That means we have to go back
to the definition. If you want to be careful,
you have to go back to the definition of improper integral.
So, it's the limit, as R goes to infinity,
of what you get by integrating only up as far as R.
That's a definite integral. That's a nice Riemann integral.
So, this is what I have to calculate.
And, I have to take the limit as R goes to infinity.
Now, how do I calculate that? Well, this integral is equal
to, that's easy. It's just integrating.
Remember that you're integrating with respect to t.
So, s is a parameter. It's like a constant,
in other words. So, it's e to the minus s t,
and when I differentiated,
the derivative of this would have negative s.
So, to get rid of that negative s, so the derivative is e to the
minus s t. You have to put minus s
in the denominator. And now, I'll want to evaluate
that between zero and R. And, what do I get?
Well it is at the upper limit. So, it's e to the minus s times
R minus, at the lower limit, it's t is equal to zero,
so whatever s is, it's one.
And that's divided by this constant up front,
negative s. So,
the answer is, it is equal to the limit of,
as R goes to infinity, of e to the negative s R minus
one divided by minus s.
Now, what's that? Well, as R goes to infinity,
e to the minus 2R, or minus 5R goes to zero,
and the answer is minus one over minus s.
So, that's one over s. And so, that's our answer.
Let's put it up here. It's one over s,
except it isn't. I made a mistake.
Well, not mistake, a little oversight.
What's the oversight? This is okay.
This is okay.
This is not okay. This is okay.
But that's not okay. What's wrong?
I did slight a verbal hand. Maybe some of you have picked
it up and were too embarrassed to correct me,
but I said like e to the minus 2R obviously goes to
zero, and e to the minus 5R goes to zero.
How about e to the minus minus 3 R?
Does that go to zero? No, that's e to the 3R,
which goes to infinity.
The only time this goes to zero is if s is a positive number.
Minus s looks like a negative number, but it's not,
if s is equal to minus two. So, this is only true if s is
positive because only if s is positive is this exponent really
negative and large, and therefore going to
infinity, going to zero as R goes to infinity.
So, the answer is not one over s.
It is one over s, s must positive.
Now, once again, here, people don't worry about
this sort of thing with power series because it seems very
obvious, you know, one over x,
absolute value of x is less than one,
when it gets to be the Laplace transform, just because the
Laplace transform is mysterious, the question is,
okay, the Laplace transform is one over s of one,
well, Laplace transform of one I understand is one over s if s
is positive. What is it if s is negative?
Okay, right down in your little books, this, but that down,
what is it if s is negative, and write underneath that,
this question is meaningless. It doesn't mean anything.
I'll draw you a picture. This is a picture of the
Laplace transform of one. It is that.
It's one branch of this curve. It does not include the branch
on the left. It doesn't because I showed you
it doesn't. That's all there is to it.
Okay, so I did that carefully. Now I'm going to get a little
less careful. What's the Laplace transform of
e to the a t? First of all,
in general, the kind of functions for which people like
to calculate the Laplace transform, and basically the
only ones there will be in the tables are exactly the sort of
functions that you used in solving linear equations with
constant coefficients. What kinds of functions entered
in there? Exponentials,
sines and cosines, but they were really complex
exponentials, right?
e to the t sine t, but that was really a
complex exponential, too, just a little more
complicated one, polynomials,
and that's about it. t times e to the t,
that was okay, too.
These are the functions for which people calculate the
Laplace transform, and all the other functions
they don't calculate the Laplace transforms.
So, I don't mean to disappoint you here.
You're going to say, oh, what, that same old stuff?
For two more weeks, we've got that same,
well, the Laplace transform does a lot of things much better
than the methods we've been using.
And, I won't. I'll sell it when I get a
chance to, for now, let's just get familiar with
it. All right, so while I'm not
going to calculate e to the a t for you,
because I'd like instead to just prove a simple formula
which will just give that, and will also give us e to the
a t sine t. It will give us a lot more,
instead. I'm going to calculate a
formula for the Laplace transform of this guy if you
already know the Laplace transform of it.
Now, see, this falls in that category because this is really
e to the a t times one.
But, I already know the Laplace transform of one.
So that's, if I can get a general formula for this,
I'll be able to get the formula for e to the a t as a
consequence. So, let's look for this Laplace
transform. Now, it's really easy.
Let's see, where am I doing calculations?
Over here. Okay, so we've got e.
So, I want to calculate the Laplace transform e to the a t f
of t. So I'm going to say that's the
integral from zero to infinity of e to the a t times f of t.
And now, the rest I copy.
That's the function part of it that goes to the input,
and then there's the other part.
This part is called the kernel, by the way, but don't worry
about that. However, if you drop it in
conversation, people will look at you and
say, gee, they know something I don't.
And you will. You know that it's the kernel.
Okay, well, now, what kind of formula can I be
looking for? Clearly, I can only be looking
for a formula which expresses it in terms of the Laplace
transform of f of t. Let's calculate and see what we
get. Now, what would you do to that
thing to make? Well, obviously,
the thing to do is to combine the two exponentials.
So, that's going to be the integral from zero to infinity
of f of t. e, now, I'd like to put it,
to combine the exponentials in such a way that it has,
still, that same form, so, I'm going to begin with
that negative sign, and then see what the rest of
it has to be. What is it going to be?
Well, minus s t and plus a t,
but I can make that minus a here, and it will come out
right. So, it's minus s t plus a t,
and there are the two parts,
those two factors, dt.
So, what's that? That's the Laplace transform.
If the a weren't there, this would be the Laplace
transform of f of t. What is it with the a there?
It's the Laplace transform of f of t,
except that instead of the variable, s has been replaced by
the variable s minus a.
I'll give you a second to digest that.
Well, you digest it while I'm writing it because that's the
answer. And, the way this is most often
used, I have to qualify it for the value.
So, if F of s is good for s positive,
the way it would be, for example,
if I used the function one here, then to finish that off,
then, F of s minus a will be,
this will be good when s is bigger than a.
Why is that? Well, because this is true.
This is true. If s minus a is
positive, that's the condition. That's what this Laplace
transform is good. But that simply says that s
should be bigger than a.
And, since this doesn't look pretty, let me try to make it
look a little bit prettier. So, let's write it.
So, this is assuming F of s is for s greater than zero.
Now, this is called something.
This is called, well, what would you call it?
On the left side, you multiply by an exponential.
On the right, you translate.
You shift the argument over by a.
So, this is called, gulp, the exponential shift.
What? Well, I'll call it the formula.
The thing before, when we talked about operators,
we called it the exponential shift rule or the exponential
shift law. But, in fact,
this is, in a way, a disguised form of the same
law. And, engineers who typically do
all their work using the Laplace transform and don't use
operators, this is the form of the exponential shift law that
they would know. What you can do with one,
you can do with the other. You can now use both.
So, what's the answer to e to the a t?
Well, the answer is, I'm supposed to,
e to the a t times one, the Laplace transform of one is
one over s. And, therefore,
what I do is to multiply by e to the a t, I change s to s
minus a . And so, that's the answer.
Let's see, what else don't we know?
Well, how about sines and cosines?
Well, the way to do sines and cosines is by making the
observation that this formula also works when a is a complex
number. So, can use also for a a
complex number, for e to the a plus b i times t.
The Laplace transform of e to
the a plus b i times t is one over s minus a plus b i.
And again, it will be for s bigger than a.
So, let's calculate the Laplace transform of,
let's say, well, I've got to cover up something.
Okay, so, that's the Laplace transform.
I've got to remember that. So, let's calculate the Laplace
transform of, let's say, sine of a t
and cosine a t.
What do you get for that? Well, just for a little
variety, we could do it by using that formula,
and taking its real and imaginary parts.
Since some of you had so much difficulty with the backwards
Euler formula, he is a good case where you
could use it. Suppose you want to calculate
the Laplace transform of cosine a t.
Well, I'm going to write that using, I want to calculate using
complex exponentials. The way I will do it is by
using the backwards Euler formula.
So, this is e to the i a t plus e to the minus i a t divided by
two.
Remember, the foreword Euler formula would say e to the i a t
equals cosine a t plus i sine a t.
That expresses the complex
exponential in terms of sines and cosines.
This is the backward formula, which just read it backwards,
expressing cosines and sines in terms of complex exponentials
instead. Both formulas are useful,
almost equally useful, in fact.
And anyway, just remind you of it, let's use this one.
Okay, what's the Laplace transform, then,
of cosine a t? Well, by linearity,
it's equal to one half the Laplace transform of this guy
plus the Laplace transform of that guy.
And, what are those? Well, the Laplace transform of
e to the i a t is one over s minus i a,
and the Laplace transform of the other guy is
one divided by s plus i a.
Now, of course, this has become out to be a
real function. This is real.
Every integral is real. This must come out to be real.
This looks kind of complex, but it isn't.
I know automatically that this is going to be a real function.
How I know that? Well, mentally,
you can combine the terms and calculate.
But, I know even before that. Remember, there are two ways to
see that something is real. You can calculate it and see
that its imaginary part is zero, hack, or without any
calculation, if you change i to minus i,
and you get the same thing, it must be real.
Now, if I change i to minus i in this expression,
what happens? If I change i to minus i,
this term turns into that one, and this one turns into that
one. Conclusion: the sum of the two
is unchanged. And therefore,
this is real. Well, of course,
in the time I took to make that argument, I could have actually
calculated it. So, what the heck,
let's calculate it? So, you do the high school
thing, and it's this guy plus that guy on top,
which makes 2s. I on the bottom is the product
of those, which by now you should know the product of two
complex numbers. A product of a number and its
complex conjugate is the sum of the squares.
So, what's the answer? The twos cancel,
and the answer is that the Laplace transform of cosine a t
is s over s squared plus a squared.
And, that will be true as,
in general, it's true up there for positive values of s only.
And, the sine a t, you can calculate that in
recitation tomorrow. The answer to that is a divided
by s squared plus a squared.
You would get the same answers if you took the real and
imaginary parts of that expression.
It's another way of getting at the recitations tomorrow;
we'll get practice in calculating other functions
related to these by using these formulas, and also from scratch
directly from the definition of the Laplace transform.
Well, there are two things which we still should do.
The first is I want to get you started with calculating inverse
Laplace transforms. And, the reason for doing that
is, in other words, I've started with f of t,
and we've been focusing on what is capital F of s?
But, you will find that when
you go to solve differential equations, by far,
the hardest part of the procedure is you get F of s.
The Laplace transform of the answer, and you have to convert
that back into the answer in terms of t that you were looking
for. In other words,
the main step in the procedure that you are going to be using
for solving differential equations is,
and the hardest part of the step will be to calculate
inverse laplace transforms. Now, you think that could be
done by tables, but, in fact,
it can't unless the tables are too long to be useful.
You have to do a certain amount of work yourself.
And, the certain amount of work that you have to do yourself
involves partial fractions decompositions.
And, in case you were wondering which you are not,
the reason you learned partial fractions in 18.01 was not to
learn those silly integrals, but he learned it so that when
you got to 18.03 you would be able to calculate,
solve differential equations by using Laplace transforms.
Sorry. That's life.
Now, so a certain amount of the recitation time tomorrow will be
devoted to reminding you how to do partial fractions since you
haven't done it in a while, and I assume,
yeah, we had that, I think.
Okay, now, they also remind you of the most efficient method,
which about half of you have had, and the rest think you
might have had, but really aren't sure.
So, here's the answer. We want to find out what it's
inverse Laplace transform is. What you have to do,
it normally won't be in the tables like this.
You have to put it in a form in which it will be in the tables.
As you do that, you have to make partial
fractions decompositions, which, to do it quickly,
so if you don't know what I'm doing now, or you think you once
knew but don't quite remember, go to recitation tomorrow.
To get the coefficient here, I cover up s,
and I put s equals zero because that's the law.
To get this coefficient, I cover up s plus three and I
put s equals a negative three because that's what you're
supposed to do. Put s equal negative three,
you get minus one third. This is equal to that.
In this form, I don't know what the inverse
Laplace form is, but in this form,
I certainly do know with the inverse Laplace transform
because the inverse Laplace transform is linear,
and because each of these guys especially occurs in those
tables. Well, what's this?
Well, it's whatever the Laplace transform of,
inverse Laplace transform of one over s is multiplied by one
third. Well, the inverse Laplace
transform of one over s is one. So, it's one third times one.
How about the other guy? Minus one third,
the inverse Laplace transform of one over s plus three,
that's this formula. a is negative three,
and that makes e to the minus 3t.
So, if this was the Laplace transform of the solution to the
differential equation, then the solution in terms of t
was this function. Now, you'll get lots of
practice in that. All I'm doing now is signaling
that that's the most important and difficult step of the
procedure, and that, please, start getting practice.
Get up to snuff doing that procedure.
Okay, in the time remaining, I want to add one formula to
this list, and that is going to be the Laplace transform of,
we still haven't done polynomials.
And now, to polynomials, because the Laplace transform
is linear, all I have to do is know what the Laplace transform
of, the individual term of a polynomial.
In other words, what the Laplace transform of t
to the n, where n is some positive
integer? Well, let's bravely start
trying to calculate it. Integral from zero to infinity
t to the n e to the negative st dt.
Now, I think you can see that
the method you should use is integration by part because this
is a product of two things, one of which you would like to
differentiate a lot of times, in fact, and the other won't
hurt to integrate it because it's very easy to integrate.
So, this factor is going to be
the one that's to be differentiated,
and this is the factor that will be pleased to integrate it.
Let's get started and see what we can get out of it.
Well, this time I'm going to be, well, I'd better be a little
careful because there's a point here that's tricky.
Okay, the first step of integration by parts is you only
do the integration. You don't do the
differentiation. Remember, the variable is t.
The s is just a parameter. It's just a constant.
It's hanging around, not knowing what to do.
Okay, so the first step is you don't do the differentiation.
You only do the integration. Evaluate it between limits,
and then you put a minus sign before you forget to do it.
And then, integral zero to infinity.
Now you do both operations. So, it's n t to the n
minus one, and you also do the
integration. Okay, let's consider each of
these pieces in turn. Now, this piece,
well, there's no problem with the lower limit,
zero, because when t is equal to zero, this factor is zero,
and the thing disappears as long as n is one or higher.
So, it's minus zero here at the lower limit.
The question is, what is at the upper limit?
So, what I have to do is find out, what is the limit?
The limit, as t goes to infinity, that's what's
happening up there, of t to the n times e to the
negative s t divided by minus s.
Well, as t goes to infinity, this goes to infinity,
of course. This had better go to zero
unless I want an answer, infinity, which won't do me any
good. If this goes to zero,
s had better be positive. So, I'd better be restricting
myself to that case. Okay, so let's assume that s is
positive so that this minus s really is a negative number.
Okay, then I have a chance. So, this is going to be the
limit. Let's write it in a more
familiar form with that down below.
So, it's t to the n. That's going to infinity.
But, the bottom is e to the minus s t.
But now, it's plus s t. And, that's going to infinity,
too, because s is positive. So, the two guys are racing,
and the question is, oh, I lost a minus s here.
So, oh... equals minus one over s.
How's that? So, the question is only,
which guy wins? In the race to infinity,
which one wins, and how do you decide?
And, the answer, of course, is that's the bottom
that wins. The exponential always wins,
and it's because of L'Hopital's rule.
You differentiate top and bottom.
Nothing much happens to the bottom.
It gets another factor of s, but the top goes down to t to
the n minus one. L'Hopital it again,
and again, and again, and again, and again until
finally you've reduced the top to t to the zero where it's
defenseless and just sitting there, and nothing's happened to
the bottom. It's still got e to the s t.
and that goes to infinity.
So, the answer is, this is zero by n applications
of L'Hopital's rule. Or, if you're very clever,
you can do it in one, but I won't tell you how.
So, the answer is that this is zero.
At the upper limit, it's also zero at least if s is
positive, which is the case we're considering.
That leaves the rest of this. All right, let's pull the
constants out front. That's plus.
Two negatives make a plus. n over s,
now, what's left? The integral from zero to
infinity of t to the n minus one, e to the minus s t dt.
But, what on Earth is that? That is n over s times
the Laplace transform of t to the n minus one.
We got a reduction for it. We don't get the answer in one
step. But, we get a reduction
formula. And, it says that the Laplace
transform, let me write it this way for once.
The first way is now better, is equal to n over s times the
Laplace transform of n minus t to the n minus one.
Okay, the next step,
this would be n over s times n minus one over s times the
Laplace transform of t to the n minus two.
If I can continue,
I finally get in the top n times n minus one times all the
way down to one divided by the same number of s's,
n of them, times the Laplace transform of t to the zero,
finally. See, one, zero,
n minus one. And so, what's the final
answer? It is n factorial over s to the
what power? Well, the Laplace transform of
this is one over s. So, the answer is it's s to the
n plus one, n of them here plus an extra
one coming from the one over s here.
And, that's the answer. The Laplace transform of t to
the n, oddly enough,
is more complicated, and looks a little different
from these. It's n factorial over s to the
n plus one.
And, with that, you can now calculate the
Laplace transform of anything in sight, and tomorrow you will.