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Today, and for the next two
weeks, we are going to be

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studying what,
for many engineers and a few

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scientists is the most popular
method of solving any

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differential equation of the
kind that they happen to be,

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and that is to use the popular
machine called the Laplace

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transform.
Now, you will get proficient in

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using it by the end of the two
weeks.

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But, there is always a certain
amount of mystery that hangs

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around it.
People scratch their heads and

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can't figure out where it comes
from.

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And, that bothers them a lot.
In the past,

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I've usually promised to tell
you, the students at the end of

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the two weeks,
but I almost never have time.

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So, I'm going to break that
glorious tradition and tell you

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up front at the beginning,
where it comes from,

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and then talk very fast for the
rest of the period.

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Okay, a good way of thinking of
where the Laplace transform

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comes from, and a way which I
think dispels some of its

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mystery is by thinking of power
series.

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I think virtually all of you
have studied power series except

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possibly a few students who just
had 18.01 here last semester,

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and probably shouldn't be
taking 18.03 anyway,

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now.
But anyway, a power series

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looks like this:
summation (a)n x to the n.

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And, you sum that from,

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let's say, zero to infinity.
And, the typical thing you want

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to do with it is add it up to
find out what its sum is.

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Now, the only way I will depart
from tradition,

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instead of calling the sum some
generic name like f of x,

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in order to identify
the sum with the coefficients,

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a, I'll call it a of x.

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Now, I want to make just one
slight change in that.

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I want to use computer
notation, which doesn't use the

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subscript (a)n.
Instead, this,

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it thinks of as a function of
the discreet variable,

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n.
In other words,

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it's a function which assigns
to n equals zero,

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one, two, three real numbers.
That's what this sequence of

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coefficients really is.
So, the computer notation will

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look almost the same.
It's just that I will write

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this in functional notation as a
of n instead of (a)n.

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But, it still means the real
number associated with the

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positive integer,
n, and everything else is the

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same.
See, what I'm thinking of this

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as doing is taking this discreet
function, which gives me the

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sequence of coefficients of the
power series,

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and associating that with the
sum of the power series.

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Let me give you some very
simple examples,

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two very simple examples,
which I think you know.

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Suppose this is a function one.
Now, what do I mean by that?

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I mean it's the constant
function, one.

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To every positive integer,
it assigns the number one.

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Okay, what's a of x?
What I'm saying is,

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in other words,
in this fancy,

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mystifying form,
is all of these guys are one,

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what's a of x?
One plus x plus x squared plus

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x cubed.
Look, you are supposed to be

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born knowing what that adds up
to.

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It adds up to one over one
minus x,

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except that's the wrong answer.
What's wrong about it?

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It's not true for every value
of x.

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That's only true when x is such
that that series converges,

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and that is only true when x
lies between negative one and

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one.
So, it's not this function.

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It's this function with its
domain restricted to be less

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than one in absolute value.
What does that converge to?

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If x is bigger than one,
the answer is it doesn't

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converge.
There's nothing else you can

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put here.
Okay, let's take another

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function.
Suppose this is,

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let's see, one over n
you probably won't know.

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Let's take one you will know,
one over n factorial.

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Suppose a of n is the function

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one over n factorial,

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what's a of x?
So, what I'm asking is,

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what does this add up to when
the coefficient here is one over

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n factorial?

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What's summation x to the n
over n factorial?

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It is e to the x.

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And, this doesn't have to be
qualified because this is true

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for all values of x.
So, in other words,

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from this peculiar point of
view, I think of a power as

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summing the operation,
of summing a power series as

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taking a discreet function
defined for positive integers,

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or nonnegative integers,
and doing this funny process.

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And, out of it comes a
continuous function of some

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sort.
And, notice what goes in is the

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variable, n.
But, what comes out is the

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variable, x.
Well, that's perfectly natural.

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That's the way a power series
is set up.

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So, the question I ask is,
this is a discreet situation,

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a discreet summation.
Suppose I made the summation

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continuous instead of discreet.
So, I want the continuous

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analog of what I did over there.
Okay, what would a continuous

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analog be?
Well, instead of,

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I'll replace n zero,
one, two, that will be replaced

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by a continued,
that's a discreet variable.

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I'll replace it by a continuous
variable, t, which runs from

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zero to infinity,
and is allowed to take every

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real value in between instead of
being only allowed to take the

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values of the positive
nonnegative integers.

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Okay, well, if I want to use t
instead of n,

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I clearly cannot sum in the
usual way over all real numbers.

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But, the way the procedure
which replaces summation over

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all real numbers is integration.
So, what I'm going to do is

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replace that sum by the integral
from zero to infinity.

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That's like the sum from zero
to infinity of what?

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Well, of some function,
but now n is being replaced by

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the continuous variable,
t.

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So, this is going to be a
function of t.

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And, how about the rest of it?
The rest I will just copy,

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x to the n'th.
Well, instead of n I have to

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write t and dt.
And, what's the sum?

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Well, I'll call the sum,
what's the sum a function of?

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I integrate out the t.
So, that doesn't appear in the

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answer.
All that appears is this

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number, x, this parameter,
x.

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For each value of x,
like one, two,

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or 26.3, this integral has a
certain value,

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and I can calculate it.
So, this is going to end up as

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a function of x,
just as it did before.

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Now, I could leave it in that
form, but no mathematician would

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like to do that,
and very few engineers either.

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The reason is,
in general, when you do

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integration and differentiation,
you do not want to have as the

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base of an exponential something
like x.

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The only convenient thing to
have is e, and the reason is

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because it's only e that people
really like to differentiate,

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e to the something.
The only thing is that people

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really like to differentiate or
integrate.

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So, I'm going to make this look
a little better by converting x

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to the t to the base e.
I remember how to do that.

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You write x equals e to the log
x and so x to

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the t will be e to the
log x times t,

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if you want.
Now, the only problem is I want

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to make one more little change.
After all, I want to be able to

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calculate this integral.
And, it's clear that if t is

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going to infinity,
if I have a number here,

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for example,
like x equals two,

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that integral is really quite
unlikely to converge.

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For example,
if a of t were just

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the constant function,
one, the integral certainly

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wouldn't converge.
It would be horrible.

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That integral only has a chance
of converging if x is a number

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less than one,
so that when I take bigger and

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bigger powers of it,
I get smaller and smaller

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numbers.
Don't forget,

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this is an improper integral
going all the way up to

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infinity.
Those need treatment,

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delicate handling.
All right, so I really want x

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to be less than one.
Otherwise, that integral is

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very unlikely to converge.
I'd better have it positive,

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because if I allow it to be
negative I'm going to get into

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trouble with negative powers,
see what's minus one,

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for example,
to the one half when t is one

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half.
That's already imaginary.

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I don't want that.
If you've got an exponential,

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the base has got to be a
positive number.

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So, I want x to be a positive
number.

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All right, if x in my actual
practices going to lie between

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zero and one in order to make
the integral converge,

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how about log x?
Well, log x,

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if x is less than one,
so log x is going to be

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less than zero,
and it's going to go all the

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way down to negative infinity.
So, this means log x is

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negative.
In this interesting range of x,

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the log x is always going to be
negative.

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And now, I don't like that.
The first place I'd like to

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call this by a new variable
since no one uses log x as a

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variable.
And, it would make sense to

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make it a negative,
to make it negative,

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that is, to write log x is
equal to negative s.

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Let's put it on the other side,
in order that since log x is

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always going to be less than
zero, then s will always be

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positive.
And it's always more convenient

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to work with positive numbers
instead of negative numbers.

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So, if I make those changes,
what happens to the integral?

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Well, I stress,
all these changes are just

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cosmetic to make things a little
easier to work with in terms of

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symbols.
First of all,

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the a I'm going to change.
I don't want to call it a of t

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because most people
don't call functions a of t.

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They call them f of t.
So, I'll call it f of t.

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x is e to the log x,
which is e to the minus s.

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So, x has its name changed to e

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to the minus s.
In other words,

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I'm using as the new variable
not x any longer but s in order

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that the base be e.
t, I now raise this to the t'th

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power, but by the laws of
exponents, that means I simply

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multiply the exponent by t,
and dt.

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And now, since I'm calling the
function f of t,

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the output ought to be called
capital F.

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But it's now a function,
since I've changed the

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variable, of s.
It's no longer a function of x.

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If you like,
you may think of this as a of,

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what's x?
x is e to the negative s,

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I guess.
I mean, no one would leave a

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function in that form.
It's simply a function of s.

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And, what is that?
So, what have we got,

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finally?
What we have,

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00:11:58,000 --> 00:12:04,000
dear hearts,
is this thing,

202
00:12:00,000 --> 00:12:06,000
which I stress is nothing more
than the continuous analog of

203
00:12:05,000 --> 00:12:11,000
the summation of a power series.
This is the discrete version.

204
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This is by these perfectly
natural transformations the

205
00:12:16,000 --> 00:12:22,000
continuous version of the same
thing.

206
00:12:19,000 --> 00:12:25,000
It starts with a function
defined for positive values of

207
00:12:23,000 --> 00:12:29,000
t, and turns it into a function
of s.

208
00:12:26,000 --> 00:12:32,000
And, this is called the Laplace
transform.

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00:12:29,000 --> 00:12:35,000
Now, if I've done my work
correctly, you should all be

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saying, oh, is that all?
But, I know you aren't.

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So, it's okay.
You'll get used to it.

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The first thing you have to get
used to is one thing some people

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never get used to,
which is you put in a function

214
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of t, and you get out a function
of s.

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00:12:53,000 --> 00:12:59,000
How could that be?
You know, for an operator,

216
00:12:56,000 --> 00:13:02,000
you put in 3x,
and you get out three if it's a

217
00:13:00,000 --> 00:13:06,000
differentiation operator.
In other words,

218
00:13:04,000 --> 00:13:10,000
when you have an operator,
the things we've been talking

219
00:13:08,000 --> 00:13:14,000
about the last two or three
weeks in one form or another,

220
00:13:13,000 --> 00:13:19,000
at least the variable doesn't
get changed.

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00:13:16,000 --> 00:13:22,000
Well, but for a transform it
does, and that's why it's called

222
00:13:21,000 --> 00:13:27,000
a transform.
So, the difference between a

223
00:13:24,000 --> 00:13:30,000
transform and an operator is
that for a transform a function

224
00:13:29,000 --> 00:13:35,000
of t comes in,
but a function of s comes out.

225
00:13:34,000 --> 00:13:40,000
The variable gets changed,
whereas for an operator,

226
00:13:37,000 --> 00:13:43,000
f of t goes in and
what comes out is g of t,

227
00:13:40,000 --> 00:13:46,000
a function using the
same variable like

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00:13:43,000 --> 00:13:49,000
differentiation is a typical
example of an operator,

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00:13:47,000 --> 00:13:53,000
or the linear differential
operators we've been talking

230
00:13:50,000 --> 00:13:56,000
about.
Well, but this doesn't behave

231
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that way.
The variable does get changed.

232
00:13:55,000 --> 00:14:01,000
That's, in fact,
extremely important in the

233
00:13:58,000 --> 00:14:04,000
applications.
In the applications,

234
00:14:01,000 --> 00:14:07,000
t usually means the time,
and s very often,

235
00:14:05,000 --> 00:14:11,000
not always, but very often is a
variable measuring frequency,

236
00:14:10,000 --> 00:14:16,000
for instance.
But, so that's a peculiar thing

237
00:14:14,000 --> 00:14:20,000
that's hard to get used to.
But, a good thing is the fact

238
00:14:18,000 --> 00:14:24,000
that it's a linear transform.
In other words,

239
00:14:22,000 --> 00:14:28,000
it obeys the laws we'd love and
like that the Laplace

240
00:14:26,000 --> 00:14:32,000
transform-- oh,
I never gave you any notation

241
00:14:30,000 --> 00:14:36,000
for the laplace transform.
Hey, I'd better do that.

242
00:14:35,000 --> 00:14:41,000
Okay, so, some notation:
there are two notations that

243
00:14:39,000 --> 00:14:45,000
are used.
Your book mostly uses the

244
00:14:41,000 --> 00:14:47,000
notation that the laplace
transform of f of t is capital F

245
00:14:45,000 --> 00:14:51,000
of s,
uses the same letter but with

246
00:14:48,000 --> 00:14:54,000
the same capital.
Now, as you will see,

247
00:14:51,000 --> 00:14:57,000
there are some places you
absolutely cannot use that

248
00:14:54,000 --> 00:15:00,000
notation.
It may seem strange,

249
00:14:56,000 --> 00:15:02,000
looks perfectly natural.
There are certain laws you

250
00:14:59,000 --> 00:15:05,000
cannot express using that
notation.

251
00:15:03,000 --> 00:15:09,000
It's baffling.
But, if you can't do it this

252
00:15:05,000 --> 00:15:11,000
way, you can do it using this
notation instead.

253
00:15:08,000 --> 00:15:14,000
One or the other will almost
always work.

254
00:15:11,000 --> 00:15:17,000
So, I'll use my little squiggly
notation, but that's what I use.

255
00:15:15,000 --> 00:15:21,000
I think it's a little more
vivid, and the trouble is that

256
00:15:19,000 --> 00:15:25,000
this piles up too many
parentheses.

257
00:15:21,000 --> 00:15:27,000
And, that's always hard to
read.

258
00:15:23,000 --> 00:15:29,000
So, I like this better.
So, these are two alternate

259
00:15:26,000 --> 00:15:32,000
ways of saying the same thing.
The Laplace transform of this

260
00:15:32,000 --> 00:15:38,000
function is that one.
Okay, well, let's use,

261
00:15:36,000 --> 00:15:42,000
for the linearity law,
it's definitely best.

262
00:15:39,000 --> 00:15:45,000
I really cannot express the
linearity law using the second

263
00:15:44,000 --> 00:15:50,000
notation, but using the first
notation, it's a breeze.

264
00:15:49,000 --> 00:15:55,000
The Laplace transform of the
sum of two functions is the sum

265
00:15:54,000 --> 00:16:00,000
of their Laplace transforms of
each of them separately.

266
00:16:00,000 --> 00:16:06,000
Or, better yet,
you could write it that way.

267
00:16:03,000 --> 00:16:09,000
Let's write it this way.
That way, it looks more like an

268
00:16:06,000 --> 00:16:12,000
operator, L of f plus L of g.

269
00:16:10,000 --> 00:16:16,000
And, of the same way,
if you take a function and

270
00:16:13,000 --> 00:16:19,000
multiply it by a constant and
take the laplace transform,

271
00:16:17,000 --> 00:16:23,000
you can pull the constant
outside.

272
00:16:19,000 --> 00:16:25,000
And, of course,
why are these true?

273
00:16:22,000 --> 00:16:28,000
These are true just because of
the form of the transform.

274
00:16:25,000 --> 00:16:31,000
If I add up f and g,
I simply add up the two

275
00:16:29,000 --> 00:16:35,000
corresponding integrals.
In other words,

276
00:16:33,000 --> 00:16:39,000
I'm using the fact that the
integral, this definite

277
00:16:37,000 --> 00:16:43,000
integral, is itself a linear
operator.

278
00:16:40,000 --> 00:16:46,000
Well, that's the general
setting.

279
00:16:43,000 --> 00:16:49,000
That's where it comes from,
and that's the notation for it.

280
00:16:47,000 --> 00:16:53,000
And, now we have to get to
work.

281
00:16:50,000 --> 00:16:56,000
The first thing to do to get
familiar with this is,

282
00:16:54,000 --> 00:17:00,000
obviously what we want to do is
say, okay, these were the

283
00:16:59,000 --> 00:17:05,000
transforms of some simple
discreet functions.

284
00:17:04,000 --> 00:17:10,000
Okay, suppose I put in some
familiar functions,

285
00:17:09,000 --> 00:17:15,000
f of t.
What do their Laplace

286
00:17:14,000 --> 00:17:20,000
transforms look like?
So, let's do that.

287
00:17:19,000 --> 00:17:25,000
So, one of the boards I should
keep stored.

288
00:17:24,000 --> 00:17:30,000
Why don't I store on this
board?

289
00:17:28,000 --> 00:17:34,000
I'll store on this board the
formulas as we get them.

290
00:17:37,000 --> 00:17:43,000
So, let's see,
what should we aim at,

291
00:17:39,000 --> 00:17:45,000
first?
Let's first find,

292
00:17:41,000 --> 00:17:47,000
and I'll do the calculations on
the sideboard,

293
00:17:44,000 --> 00:17:50,000
and we'll see how it works out.
I'm not very sure.

294
00:17:47,000 --> 00:17:53,000
In other words,
what's the Laplace transform of

295
00:17:51,000 --> 00:17:57,000
the function,
one?

296
00:17:52,000 --> 00:17:58,000
Well, there's an even easier
one.

297
00:17:54,000 --> 00:18:00,000
What's the Laplace transform of
the function zero?

298
00:17:57,000 --> 00:18:03,000
Answer: zero.
Very exciting.

299
00:18:00,000 --> 00:18:06,000
What's the Laplace transform of
one?

300
00:18:03,000 --> 00:18:09,000
Well, it doesn't turn out the
constant anymore than it turned

301
00:18:07,000 --> 00:18:13,000
out to be a constant up there.
Let's calculate it.

302
00:18:11,000 --> 00:18:17,000
Now, you can do these
calculations carefully,

303
00:18:14,000 --> 00:18:20,000
dotting all the i's,
or pretty carefully,

304
00:18:17,000 --> 00:18:23,000
or not carefully at all,
i.e.

305
00:18:19,000 --> 00:18:25,000
sloppily.
I'll let you be sloppy after,

306
00:18:21,000 --> 00:18:27,000
generally speaking,
you could be sloppy unless the

307
00:18:25,000 --> 00:18:31,000
directions tell you to be less
sloppy or to be careful,

308
00:18:29,000 --> 00:18:35,000
okay?
So, I'll do one carefully.

309
00:18:32,000 --> 00:18:38,000
Let's calculate the Laplace
transform of one carefully.

310
00:18:36,000 --> 00:18:42,000
Okay, in the beginning,
you've got nothing to use with

311
00:18:40,000 --> 00:18:46,000
the definition.
So, I have to calculate the

312
00:18:43,000 --> 00:18:49,000
integral from zero to infinity
of one, that's the f of t times

313
00:18:47,000 --> 00:18:53,000
e to the negative s t,
so I don't have to

314
00:18:51,000 --> 00:18:57,000
put in the one,
dt.

315
00:18:52,000 --> 00:18:58,000
All right, now,
let me remind you,

316
00:18:54,000 --> 00:19:00,000
this is an improper integral.
This is just about the first

317
00:18:58,000 --> 00:19:04,000
time in the course we've had an
improper integral.

318
00:19:01,000 --> 00:19:07,000
But, there are going to be a
lot of them over the next couple

319
00:19:06,000 --> 00:19:12,000
of weeks, nothing but.
All right, it's an improper

320
00:19:10,000 --> 00:19:16,000
integral.
That means we have to go back

321
00:19:12,000 --> 00:19:18,000
to the definition.
If you want to be careful,

322
00:19:15,000 --> 00:19:21,000
you have to go back to the
definition of improper integral.

323
00:19:19,000 --> 00:19:25,000
So, it's the limit,
as R goes to infinity,

324
00:19:21,000 --> 00:19:27,000
of what you get by integrating
only up as far as R.

325
00:19:24,000 --> 00:19:30,000
That's a definite integral.
That's a nice Riemann integral.

326
00:19:27,000 --> 00:19:33,000
So, this is what I have to
calculate.

327
00:19:31,000 --> 00:19:37,000
And, I have to take the limit
as R goes to infinity.

328
00:19:34,000 --> 00:19:40,000
Now, how do I calculate that?
Well, this integral is equal

329
00:19:37,000 --> 00:19:43,000
to, that's easy.
It's just integrating.

330
00:19:40,000 --> 00:19:46,000
Remember that you're
integrating with respect to t.

331
00:19:43,000 --> 00:19:49,000
So, s is a parameter.
It's like a constant,

332
00:19:45,000 --> 00:19:51,000
in other words.
So, it's e to the minus s t,

333
00:19:48,000 --> 00:19:54,000
and when I differentiated,

334
00:19:50,000 --> 00:19:56,000
the derivative of this would
have negative s.

335
00:19:53,000 --> 00:19:59,000
So, to get rid of that negative
s, so the derivative is e to the

336
00:19:57,000 --> 00:20:03,000
minus s t.
You have to put minus s

337
00:20:00,000 --> 00:20:06,000
in the denominator.
And now, I'll want to evaluate

338
00:20:05,000 --> 00:20:11,000
that between zero and R.
And, what do I get?

339
00:20:09,000 --> 00:20:15,000
Well it is at the upper limit.
So, it's e to the minus s times

340
00:20:14,000 --> 00:20:20,000
R minus, at the lower limit,
it's t is equal to zero,

341
00:20:19,000 --> 00:20:25,000
so whatever s is,
it's one.

342
00:20:21,000 --> 00:20:27,000
And that's divided by this
constant up front,

343
00:20:25,000 --> 00:20:31,000
negative s. So,

344
00:20:28,000 --> 00:20:34,000
the answer is,
it is equal to the limit of,

345
00:20:32,000 --> 00:20:38,000
as R goes to infinity,
of e to the negative s R minus

346
00:20:37,000 --> 00:20:43,000
one divided by minus s.

347
00:20:43,000 --> 00:20:49,000
Now, what's that?
Well, as R goes to infinity,

348
00:20:47,000 --> 00:20:53,000
e to the minus 2R,
or minus 5R goes to zero,

349
00:20:52,000 --> 00:20:58,000
and the answer is minus one
over minus s.

350
00:20:57,000 --> 00:21:03,000
So, that's one over s.
And so, that's our answer.

351
00:21:02,000 --> 00:21:08,000
Let's put it up here.
It's one over s,

352
00:21:05,000 --> 00:21:11,000
except it isn't.
I made a mistake.

353
00:21:10,000 --> 00:21:16,000
Well, not mistake,
a little oversight.

354
00:21:16,000 --> 00:21:22,000
What's the oversight?
This is okay.

355
00:21:22,000 --> 00:21:28,000
This is okay.
This is okay.

356
00:21:26,000 --> 00:21:32,000
This is not okay.
This is okay.

357
00:21:31,000 --> 00:21:37,000
But that's not okay.
What's wrong?

358
00:21:38,000 --> 00:21:44,000
I did slight a verbal hand.
Maybe some of you have picked

359
00:21:41,000 --> 00:21:47,000
it up and were too embarrassed
to correct me,

360
00:21:44,000 --> 00:21:50,000
but I said like e to the minus
2R obviously goes to

361
00:21:48,000 --> 00:21:54,000
zero, and e to the minus 5R
goes to zero.

362
00:21:51,000 --> 00:21:57,000
How about e to the minus minus
3 R?

363
00:21:54,000 --> 00:22:00,000
Does that go to zero?
No, that's e to the 3R,

364
00:21:57,000 --> 00:22:03,000
which goes to infinity.

365
00:22:00,000 --> 00:22:06,000
The only time this goes to zero
is if s is a positive number.

366
00:22:05,000 --> 00:22:11,000
Minus s looks like a negative
number, but it's not,

367
00:22:10,000 --> 00:22:16,000
if s is equal to minus two.
So, this is only true if s is

368
00:22:16,000 --> 00:22:22,000
positive because only if s is
positive is this exponent really

369
00:22:22,000 --> 00:22:28,000
negative and large,
and therefore going to

370
00:22:26,000 --> 00:22:32,000
infinity, going to zero as R
goes to infinity.

371
00:22:30,000 --> 00:22:36,000
So, the answer is not one over
s.

372
00:22:34,000 --> 00:22:40,000
It is one over s,
s must positive.

373
00:22:39,000 --> 00:22:45,000
Now, once again,
here, people don't worry about

374
00:22:41,000 --> 00:22:47,000
this sort of thing with power
series because it seems very

375
00:22:45,000 --> 00:22:51,000
obvious, you know,
one over x,

376
00:22:48,000 --> 00:22:54,000
absolute value of x is less
than one,

377
00:22:51,000 --> 00:22:57,000
when it gets to be the Laplace
transform, just because the

378
00:22:54,000 --> 00:23:00,000
Laplace transform is mysterious,
the question is,

379
00:22:58,000 --> 00:23:04,000
okay, the Laplace transform is
one over s of one,

380
00:23:01,000 --> 00:23:07,000
well, Laplace transform of one
I understand is one over s if s

381
00:23:05,000 --> 00:23:11,000
is positive.
What is it if s is negative?

382
00:23:09,000 --> 00:23:15,000
Okay, right down in your little
books, this, but that down,

383
00:23:15,000 --> 00:23:21,000
what is it if s is negative,
and write underneath that,

384
00:23:21,000 --> 00:23:27,000
this question is meaningless.
It doesn't mean anything.

385
00:23:26,000 --> 00:23:32,000
I'll draw you a picture.
This is a picture of the

386
00:23:31,000 --> 00:23:37,000
Laplace transform of one.
It is that.

387
00:23:34,000 --> 00:23:40,000
It's one branch of this curve.
It does not include the branch

388
00:23:39,000 --> 00:23:45,000
on the left.
It doesn't because I showed you

389
00:23:43,000 --> 00:23:49,000
it doesn't.
That's all there is to it.

390
00:23:46,000 --> 00:23:52,000
Okay, so I did that carefully.
Now I'm going to get a little

391
00:23:50,000 --> 00:23:56,000
less careful.
What's the Laplace transform of

392
00:23:54,000 --> 00:24:00,000
e to the a t?
First of all,

393
00:23:57,000 --> 00:24:03,000
in general, the kind of
functions for which people like

394
00:24:01,000 --> 00:24:07,000
to calculate the Laplace
transform, and basically the

395
00:24:06,000 --> 00:24:12,000
only ones there will be in the
tables are exactly the sort of

396
00:24:10,000 --> 00:24:16,000
functions that you used in
solving linear equations with

397
00:24:15,000 --> 00:24:21,000
constant coefficients.
What kinds of functions entered

398
00:24:21,000 --> 00:24:27,000
in there?
Exponentials,

399
00:24:22,000 --> 00:24:28,000
sines and cosines,
but they were really complex

400
00:24:25,000 --> 00:24:31,000
exponentials,
right?

401
00:24:26,000 --> 00:24:32,000
e to the t sine t,
but that was really a

402
00:24:30,000 --> 00:24:36,000
complex exponential,
too, just a little more

403
00:24:33,000 --> 00:24:39,000
complicated one,
polynomials,

404
00:24:35,000 --> 00:24:41,000
and that's about it.
t times e to the t,

405
00:24:38,000 --> 00:24:44,000
that was okay,
too.

406
00:24:41,000 --> 00:24:47,000
These are the functions for
which people calculate the

407
00:24:44,000 --> 00:24:50,000
Laplace transform,
and all the other functions

408
00:24:46,000 --> 00:24:52,000
they don't calculate the Laplace
transforms.

409
00:24:49,000 --> 00:24:55,000
So, I don't mean to disappoint
you here.

410
00:24:52,000 --> 00:24:58,000
You're going to say,
oh, what, that same old stuff?

411
00:24:55,000 --> 00:25:01,000
For two more weeks,
we've got that same,

412
00:24:57,000 --> 00:25:03,000
well, the Laplace transform
does a lot of things much better

413
00:25:01,000 --> 00:25:07,000
than the methods we've been
using.

414
00:25:04,000 --> 00:25:10,000
And, I won't.
I'll sell it when I get a

415
00:25:06,000 --> 00:25:12,000
chance to, for now,
let's just get familiar with

416
00:25:09,000 --> 00:25:15,000
it.
All right, so while I'm not

417
00:25:11,000 --> 00:25:17,000
going to calculate e to the a t
for you,

418
00:25:14,000 --> 00:25:20,000
because I'd like instead to
just prove a simple formula

419
00:25:17,000 --> 00:25:23,000
which will just give that,
and will also give us e to the

420
00:25:21,000 --> 00:25:27,000
a t sine t.
It will give us a lot more,

421
00:25:24,000 --> 00:25:30,000
instead.
I'm going to calculate a

422
00:25:27,000 --> 00:25:33,000
formula for the Laplace
transform of this guy if you

423
00:25:30,000 --> 00:25:36,000
already know the Laplace
transform of it.

424
00:25:34,000 --> 00:25:40,000
Now, see, this falls in that
category because this is really

425
00:25:38,000 --> 00:25:44,000
e to the a t times one.

426
00:25:41,000 --> 00:25:47,000
But, I already know the Laplace
transform of one.

427
00:25:44,000 --> 00:25:50,000
So that's, if I can get a
general formula for this,

428
00:25:48,000 --> 00:25:54,000
I'll be able to get the formula
for e to the a t as a

429
00:25:53,000 --> 00:25:59,000
consequence.
So, let's look for this Laplace

430
00:25:56,000 --> 00:26:02,000
transform.
Now, it's really easy.

431
00:25:59,000 --> 00:26:05,000
Let's see, where am I doing
calculations?

432
00:26:02,000 --> 00:26:08,000
Over here.
Okay, so we've got e.

433
00:26:05,000 --> 00:26:11,000
So, I want to calculate the
Laplace transform e to the a t f

434
00:26:09,000 --> 00:26:15,000
of t.
So I'm going to say that's the

435
00:26:13,000 --> 00:26:19,000
integral from zero to infinity
of e to the a t times f of t.

436
00:26:16,000 --> 00:26:22,000
And now, the rest I copy.

437
00:26:19,000 --> 00:26:25,000
That's the function part of it
that goes to the input,

438
00:26:23,000 --> 00:26:29,000
and then there's the other
part.

439
00:26:25,000 --> 00:26:31,000
This part is called the kernel,
by the way, but don't worry

440
00:26:29,000 --> 00:26:35,000
about that.
However, if you drop it in

441
00:26:33,000 --> 00:26:39,000
conversation,
people will look at you and

442
00:26:36,000 --> 00:26:42,000
say, gee, they know something I
don't.

443
00:26:39,000 --> 00:26:45,000
And you will.
You know that it's the kernel.

444
00:26:43,000 --> 00:26:49,000
Okay, well, now,
what kind of formula can I be

445
00:26:47,000 --> 00:26:53,000
looking for?
Clearly, I can only be looking

446
00:26:51,000 --> 00:26:57,000
for a formula which expresses it
in terms of the Laplace

447
00:26:56,000 --> 00:27:02,000
transform of f of t.
Let's calculate and see what we

448
00:27:02,000 --> 00:27:08,000
get.
Now, what would you do to that

449
00:27:04,000 --> 00:27:10,000
thing to make?
Well, obviously,

450
00:27:06,000 --> 00:27:12,000
the thing to do is to combine
the two exponentials.

451
00:27:09,000 --> 00:27:15,000
So, that's going to be the
integral from zero to infinity

452
00:27:13,000 --> 00:27:19,000
of f of t.
e, now, I'd like to put it,

453
00:27:16,000 --> 00:27:22,000
to combine the exponentials in
such a way that it has,

454
00:27:19,000 --> 00:27:25,000
still, that same form,
so, I'm going to begin with

455
00:27:23,000 --> 00:27:29,000
that negative sign,
and then see what the rest of

456
00:27:26,000 --> 00:27:32,000
it has to be.
What is it going to be?

457
00:27:30,000 --> 00:27:36,000
Well, minus s t and
plus a t,

458
00:27:34,000 --> 00:27:40,000
but I can make that minus a
here, and it will come out

459
00:27:39,000 --> 00:27:45,000
right.
So, it's minus s t plus a t,

460
00:27:42,000 --> 00:27:48,000
and there are the two parts,

461
00:27:46,000 --> 00:27:52,000
those two factors,
dt.

462
00:27:48,000 --> 00:27:54,000
So, what's that?
That's the Laplace transform.

463
00:27:52,000 --> 00:27:58,000
If the a weren't there,
this would be the Laplace

464
00:27:56,000 --> 00:28:02,000
transform of f of t.
What is it with the a there?

465
00:28:03,000 --> 00:28:09,000
It's the Laplace transform of f
of t,

466
00:28:08,000 --> 00:28:14,000
except that instead of the
variable, s has been replaced by

467
00:28:15,000 --> 00:28:21,000
the variable s minus a.

468
00:28:19,000 --> 00:28:25,000
I'll give you a second to
digest that.

469
00:28:24,000 --> 00:28:30,000
Well, you digest it while I'm
writing it because that's the

470
00:28:30,000 --> 00:28:36,000
answer.
And, the way this is most often

471
00:28:35,000 --> 00:28:41,000
used, I have to qualify it for
the value.

472
00:28:38,000 --> 00:28:44,000
So, if F of s is good
for s positive,

473
00:28:42,000 --> 00:28:48,000
the way it would be,
for example,

474
00:28:45,000 --> 00:28:51,000
if I used the function one
here, then to finish that off,

475
00:28:50,000 --> 00:28:56,000
then, F of s minus a will be,

476
00:28:53,000 --> 00:28:59,000
this will be good when s is
bigger than a.

477
00:28:57,000 --> 00:29:03,000
Why is that?
Well, because this is true.

478
00:29:02,000 --> 00:29:08,000
This is true.
If s minus a is

479
00:29:06,000 --> 00:29:12,000
positive, that's the condition.
That's what this Laplace

480
00:29:11,000 --> 00:29:17,000
transform is good.
But that simply says that s

481
00:29:15,000 --> 00:29:21,000
should be bigger than a.

482
00:29:18,000 --> 00:29:24,000
And, since this doesn't look
pretty, let me try to make it

483
00:29:23,000 --> 00:29:29,000
look a little bit prettier.
So, let's write it.

484
00:29:29,000 --> 00:29:35,000
So, this is assuming F of s is
for s greater than zero.

485
00:29:34,000 --> 00:29:40,000
Now, this is called something.

486
00:29:38,000 --> 00:29:44,000
This is called,
well, what would you call it?

487
00:29:43,000 --> 00:29:49,000
On the left side,
you multiply by an exponential.

488
00:29:47,000 --> 00:29:53,000
On the right,
you translate.

489
00:29:50,000 --> 00:29:56,000
You shift the argument over by
a.

490
00:29:53,000 --> 00:29:59,000
So, this is called,
gulp, the exponential shift.

491
00:29:58,000 --> 00:30:04,000
What?
Well, I'll call it the formula.

492
00:30:02,000 --> 00:30:08,000
The thing before,
when we talked about operators,

493
00:30:06,000 --> 00:30:12,000
we called it the exponential
shift rule or the exponential

494
00:30:10,000 --> 00:30:16,000
shift law.
But, in fact,

495
00:30:12,000 --> 00:30:18,000
this is, in a way,
a disguised form of the same

496
00:30:16,000 --> 00:30:22,000
law.
And, engineers who typically do

497
00:30:19,000 --> 00:30:25,000
all their work using the Laplace
transform and don't use

498
00:30:23,000 --> 00:30:29,000
operators, this is the form of
the exponential shift law that

499
00:30:28,000 --> 00:30:34,000
they would know.
What you can do with one,

500
00:30:33,000 --> 00:30:39,000
you can do with the other.
You can now use both.

501
00:30:37,000 --> 00:30:43,000
So, what's the answer to e to
the a t?

502
00:30:41,000 --> 00:30:47,000
Well, the answer is,
I'm supposed to,

503
00:30:44,000 --> 00:30:50,000
e to the a t times one,
the Laplace transform of one is

504
00:30:48,000 --> 00:30:54,000
one over s.
And, therefore,

505
00:30:51,000 --> 00:30:57,000
what I do is to multiply by e
to the a t, I change s to s

506
00:30:56,000 --> 00:31:02,000
minus a .
And so, that's the answer.

507
00:31:00,000 --> 00:31:06,000
Let's see, what else don't we
know?

508
00:31:05,000 --> 00:31:11,000
Well, how about sines and
cosines?

509
00:31:08,000 --> 00:31:14,000
Well, the way to do sines and
cosines is by making the

510
00:31:15,000 --> 00:31:21,000
observation that this formula
also works when a is a complex

511
00:31:22,000 --> 00:31:28,000
number.
So, can use also for a a

512
00:31:26,000 --> 00:31:32,000
complex number, for e to
the a plus b i times t.

513
00:31:31,000 --> 00:31:37,000
The Laplace transform of e to

514
00:31:37,000 --> 00:31:43,000
the a plus b i times t is one
over s minus a plus b i.

515
00:31:43,000 --> 00:31:49,000
And again, it will be for s
bigger than a.

516
00:31:47,000 --> 00:31:53,000
So, let's calculate the Laplace
transform of,

517
00:31:50,000 --> 00:31:56,000
let's say, well,
I've got to cover up something.

518
00:31:54,000 --> 00:32:00,000
Okay, so, that's the Laplace
transform.

519
00:31:57,000 --> 00:32:03,000
I've got to remember that.
So, let's calculate the Laplace

520
00:32:02,000 --> 00:32:08,000
transform of,
let's say, sine of a t

521
00:32:05,000 --> 00:32:11,000
and cosine a t.

522
00:32:08,000 --> 00:32:14,000
What do you get for that?
Well, just for a little

523
00:32:12,000 --> 00:32:18,000
variety, we could do it by using
that formula,

524
00:32:15,000 --> 00:32:21,000
and taking its real and
imaginary parts.

525
00:32:18,000 --> 00:32:24,000
Since some of you had so much
difficulty with the backwards

526
00:32:23,000 --> 00:32:29,000
Euler formula,
he is a good case where you

527
00:32:26,000 --> 00:32:32,000
could use it.
Suppose you want to calculate

528
00:32:29,000 --> 00:32:35,000
the Laplace transform of cosine
a t.

529
00:32:35,000 --> 00:32:41,000
Well, I'm going to write that
using, I want to calculate using

530
00:32:39,000 --> 00:32:45,000
complex exponentials.
The way I will do it is by

531
00:32:43,000 --> 00:32:49,000
using the backwards Euler
formula.

532
00:32:45,000 --> 00:32:51,000
So, this is e to the i a t plus
e to the minus i a t divided by

533
00:32:50,000 --> 00:32:56,000
two.

534
00:32:52,000 --> 00:32:58,000
Remember, the foreword Euler
formula would say e to the i a t

535
00:32:57,000 --> 00:33:03,000
equals cosine a t plus i sine a
t.

536
00:33:01,000 --> 00:33:07,000
That expresses the complex

537
00:33:04,000 --> 00:33:10,000
exponential in terms of sines
and cosines.

538
00:33:07,000 --> 00:33:13,000
This is the backward formula,
which just read it backwards,

539
00:33:11,000 --> 00:33:17,000
expressing cosines and sines in
terms of complex exponentials

540
00:33:15,000 --> 00:33:21,000
instead.
Both formulas are useful,

541
00:33:17,000 --> 00:33:23,000
almost equally useful,
in fact.

542
00:33:19,000 --> 00:33:25,000
And anyway, just remind you of
it, let's use this one.

543
00:33:23,000 --> 00:33:29,000
Okay, what's the Laplace
transform, then,

544
00:33:26,000 --> 00:33:32,000
of cosine a t?
Well, by linearity,

545
00:33:30,000 --> 00:33:36,000
it's equal to one half the
Laplace transform of this guy

546
00:33:35,000 --> 00:33:41,000
plus the Laplace transform of
that guy.

547
00:33:38,000 --> 00:33:44,000
And, what are those?
Well, the Laplace transform of

548
00:33:43,000 --> 00:33:49,000
e to the i a t is one over s
minus i a,

549
00:33:48,000 --> 00:33:54,000
and the Laplace
transform of the other guy is

550
00:33:53,000 --> 00:33:59,000
one divided by s plus i a.

551
00:33:56,000 --> 00:34:02,000
Now, of course,
this has become out to be a

552
00:34:00,000 --> 00:34:06,000
real function.
This is real.

553
00:34:03,000 --> 00:34:09,000
Every integral is real.
This must come out to be real.

554
00:34:07,000 --> 00:34:13,000
This looks kind of complex,
but it isn't.

555
00:34:10,000 --> 00:34:16,000
I know automatically that this
is going to be a real function.

556
00:34:15,000 --> 00:34:21,000
How I know that?
Well, mentally,

557
00:34:17,000 --> 00:34:23,000
you can combine the terms and
calculate.

558
00:34:20,000 --> 00:34:26,000
But, I know even before that.
Remember, there are two ways to

559
00:34:24,000 --> 00:34:30,000
see that something is real.
You can calculate it and see

560
00:34:28,000 --> 00:34:34,000
that its imaginary part is zero,
hack, or without any

561
00:34:32,000 --> 00:34:38,000
calculation, if you change i to
minus i,

562
00:34:36,000 --> 00:34:42,000
and you get the same thing,
it must be real.

563
00:34:41,000 --> 00:34:47,000
Now, if I change i to minus i
in this expression,

564
00:34:45,000 --> 00:34:51,000
what happens?
If I change i to minus i,

565
00:34:48,000 --> 00:34:54,000
this term turns into that one,
and this one turns into that

566
00:34:54,000 --> 00:35:00,000
one.
Conclusion: the sum of the two

567
00:34:57,000 --> 00:35:03,000
is unchanged.
And therefore,

568
00:34:59,000 --> 00:35:05,000
this is real.
Well, of course,

569
00:35:03,000 --> 00:35:09,000
in the time I took to make that
argument, I could have actually

570
00:35:07,000 --> 00:35:13,000
calculated it.
So, what the heck,

571
00:35:09,000 --> 00:35:15,000
let's calculate it?
So, you do the high school

572
00:35:12,000 --> 00:35:18,000
thing, and it's this guy plus
that guy on top,

573
00:35:15,000 --> 00:35:21,000
which makes 2s.
I on the bottom is the product

574
00:35:18,000 --> 00:35:24,000
of those, which by now you
should know the product of two

575
00:35:22,000 --> 00:35:28,000
complex numbers.
A product of a number and its

576
00:35:25,000 --> 00:35:31,000
complex conjugate is the sum of
the squares.

577
00:35:29,000 --> 00:35:35,000
So, what's the answer?
The twos cancel,

578
00:35:31,000 --> 00:35:37,000
and the answer is that the
Laplace transform of cosine a t

579
00:35:35,000 --> 00:35:41,000
is s over s squared plus a
squared.

580
00:35:39,000 --> 00:35:45,000
And, that will be true as,

581
00:35:41,000 --> 00:35:47,000
in general, it's true up there
for positive values of s only.

582
00:35:46,000 --> 00:35:52,000
And, the sine a t,
you can calculate that in

583
00:35:49,000 --> 00:35:55,000
recitation tomorrow.
The answer to that is a divided

584
00:35:53,000 --> 00:35:59,000
by s squared plus a squared.

585
00:35:56,000 --> 00:36:02,000
You would get the same answers
if you took the real and

586
00:36:00,000 --> 00:36:06,000
imaginary parts of that
expression.

587
00:36:04,000 --> 00:36:10,000
It's another way of getting at
the recitations tomorrow;

588
00:36:08,000 --> 00:36:14,000
we'll get practice in
calculating other functions

589
00:36:11,000 --> 00:36:17,000
related to these by using these
formulas, and also from scratch

590
00:36:16,000 --> 00:36:22,000
directly from the definition of
the Laplace transform.

591
00:36:20,000 --> 00:36:26,000
Well, there are two things
which we still should do.

592
00:36:24,000 --> 00:36:30,000
The first is I want to get you
started with calculating inverse

593
00:36:29,000 --> 00:36:35,000
Laplace transforms.
And, the reason for doing that

594
00:36:33,000 --> 00:36:39,000
is, in other words,
I've started with f of t,

595
00:36:36,000 --> 00:36:42,000
and we've been focusing
on what is capital F of s?

596
00:36:40,000 --> 00:36:46,000
But, you will find that when

597
00:36:42,000 --> 00:36:48,000
you go to solve differential
equations, by far,

598
00:36:45,000 --> 00:36:51,000
the hardest part of the
procedure is you get F of s.

599
00:36:49,000 --> 00:36:55,000
The Laplace transform of the
answer, and you have to convert

600
00:36:52,000 --> 00:36:58,000
that back into the answer in
terms of t that you were looking

601
00:36:56,000 --> 00:37:02,000
for.
In other words,

602
00:36:58,000 --> 00:37:04,000
the main step in the procedure
that you are going to be using

603
00:37:01,000 --> 00:37:07,000
for solving differential
equations is,

604
00:37:04,000 --> 00:37:10,000
and the hardest part of the
step will be to calculate

605
00:37:07,000 --> 00:37:13,000
inverse laplace transforms.
Now, you think that could be

606
00:37:13,000 --> 00:37:19,000
done by tables,
but, in fact,

607
00:37:15,000 --> 00:37:21,000
it can't unless the tables are
too long to be useful.

608
00:37:19,000 --> 00:37:25,000
You have to do a certain amount
of work yourself.

609
00:37:23,000 --> 00:37:29,000
And, the certain amount of work
that you have to do yourself

610
00:37:28,000 --> 00:37:34,000
involves partial fractions
decompositions.

611
00:37:33,000 --> 00:37:39,000
And, in case you were wondering
which you are not,

612
00:37:36,000 --> 00:37:42,000
the reason you learned partial
fractions in 18.01 was not to

613
00:37:41,000 --> 00:37:47,000
learn those silly integrals,
but he learned it so that when

614
00:37:45,000 --> 00:37:51,000
you got to 18.03 you would be
able to calculate,

615
00:37:48,000 --> 00:37:54,000
solve differential equations by
using Laplace transforms.

616
00:37:53,000 --> 00:37:59,000
Sorry.
That's life.

617
00:37:54,000 --> 00:38:00,000
Now, so a certain amount of the
recitation time tomorrow will be

618
00:37:59,000 --> 00:38:05,000
devoted to reminding you how to
do partial fractions since you

619
00:38:03,000 --> 00:38:09,000
haven't done it in a while,
and I assume,

620
00:38:06,000 --> 00:38:12,000
yeah, we had that,
I think.

621
00:38:10,000 --> 00:38:16,000
Okay, now, they also remind you
of the most efficient method,

622
00:38:16,000 --> 00:38:22,000
which about half of you have
had, and the rest think you

623
00:38:23,000 --> 00:38:29,000
might have had,
but really aren't sure.

624
00:38:27,000 --> 00:38:33,000
So, here's the answer.
We want to find out what it's

625
00:38:33,000 --> 00:38:39,000
inverse Laplace transform is.
What you have to do,

626
00:38:39,000 --> 00:38:45,000
it normally won't be in the
tables like this.

627
00:38:42,000 --> 00:38:48,000
You have to put it in a form in
which it will be in the tables.

628
00:38:46,000 --> 00:38:52,000
As you do that,
you have to make partial

629
00:38:48,000 --> 00:38:54,000
fractions decompositions,
which, to do it quickly,

630
00:38:51,000 --> 00:38:57,000
so if you don't know what I'm
doing now, or you think you once

631
00:38:55,000 --> 00:39:01,000
knew but don't quite remember,
go to recitation tomorrow.

632
00:38:59,000 --> 00:39:05,000
To get the coefficient here,
I cover up s,

633
00:39:02,000 --> 00:39:08,000
and I put s equals zero
because that's the law.

634
00:39:07,000 --> 00:39:13,000
To get this coefficient,
I cover up s plus three and I

635
00:39:10,000 --> 00:39:16,000
put s equals a negative three
because that's what you're

636
00:39:14,000 --> 00:39:20,000
supposed to do.
Put s equal negative three,

637
00:39:17,000 --> 00:39:23,000
you get minus one third.
This is equal to that.

638
00:39:20,000 --> 00:39:26,000
In this form,
I don't know what the inverse

639
00:39:23,000 --> 00:39:29,000
Laplace form is,
but in this form,

640
00:39:25,000 --> 00:39:31,000
I certainly do know with the
inverse Laplace transform

641
00:39:29,000 --> 00:39:35,000
because the inverse Laplace
transform is linear,

642
00:39:32,000 --> 00:39:38,000
and because each of these guys
especially occurs in those

643
00:39:36,000 --> 00:39:42,000
tables.
Well, what's this?

644
00:39:40,000 --> 00:39:46,000
Well, it's whatever the Laplace
transform of,

645
00:39:43,000 --> 00:39:49,000
inverse Laplace transform of
one over s is multiplied by one

646
00:39:49,000 --> 00:39:55,000
third.
Well, the inverse Laplace

647
00:39:51,000 --> 00:39:57,000
transform of one over s is one.
So, it's one third times one.

648
00:39:56,000 --> 00:40:02,000
How about the other guy?
Minus one third,

649
00:40:00,000 --> 00:40:06,000
the inverse Laplace transform
of one over s plus three,

650
00:40:05,000 --> 00:40:11,000
that's this formula.
a is negative three,

651
00:40:09,000 --> 00:40:15,000
and that makes e to the minus
3t.

652
00:40:13,000 --> 00:40:19,000
So, if this was the Laplace
transform of the solution to the

653
00:40:18,000 --> 00:40:24,000
differential equation,
then the solution in terms of t

654
00:40:22,000 --> 00:40:28,000
was this function.
Now, you'll get lots of

655
00:40:26,000 --> 00:40:32,000
practice in that.
All I'm doing now is signaling

656
00:40:30,000 --> 00:40:36,000
that that's the most important
and difficult step of the

657
00:40:34,000 --> 00:40:40,000
procedure, and that,
please, start getting practice.

658
00:40:40,000 --> 00:40:46,000
Get up to snuff doing that
procedure.

659
00:40:42,000 --> 00:40:48,000
Okay, in the time remaining,
I want to add one formula to

660
00:40:46,000 --> 00:40:52,000
this list, and that is going to
be the Laplace transform of,

661
00:40:50,000 --> 00:40:56,000
we still haven't done
polynomials.

662
00:40:52,000 --> 00:40:58,000
And now, to polynomials,
because the Laplace transform

663
00:40:56,000 --> 00:41:02,000
is linear, all I have to do is
know what the Laplace transform

664
00:41:00,000 --> 00:41:06,000
of, the individual term of a
polynomial.

665
00:41:04,000 --> 00:41:10,000
In other words,
what the Laplace transform of t

666
00:41:07,000 --> 00:41:13,000
to the n,
where n is some positive

667
00:41:10,000 --> 00:41:16,000
integer?
Well, let's bravely start

668
00:41:13,000 --> 00:41:19,000
trying to calculate it.
Integral from zero to infinity

669
00:41:17,000 --> 00:41:23,000
t to the n e to the negative st
dt.

670
00:41:22,000 --> 00:41:28,000
Now, I think you can see that

671
00:41:26,000 --> 00:41:32,000
the method you should use is
integration by part because this

672
00:41:31,000 --> 00:41:37,000
is a product of two things,
one of which you would like to

673
00:41:35,000 --> 00:41:41,000
differentiate a lot of times,
in fact, and the other won't

674
00:41:40,000 --> 00:41:46,000
hurt to integrate it because
it's very easy to integrate.

675
00:41:44,000 --> 00:41:50,000
So, this factor is going to be

676
00:41:48,000 --> 00:41:54,000
the one that's to be
differentiated,

677
00:41:50,000 --> 00:41:56,000
and this is the factor that
will be pleased to integrate it.

678
00:41:54,000 --> 00:42:00,000
Let's get started and see what
we can get out of it.

679
00:41:57,000 --> 00:42:03,000
Well, this time I'm going to
be, well, I'd better be a little

680
00:42:01,000 --> 00:42:07,000
careful because there's a point
here that's tricky.

681
00:42:06,000 --> 00:42:12,000
Okay, the first step of
integration by parts is you only

682
00:42:10,000 --> 00:42:16,000
do the integration.
You don't do the

683
00:42:12,000 --> 00:42:18,000
differentiation.
Remember, the variable is t.

684
00:42:15,000 --> 00:42:21,000
The s is just a parameter.
It's just a constant.

685
00:42:19,000 --> 00:42:25,000
It's hanging around,
not knowing what to do.

686
00:42:22,000 --> 00:42:28,000
Okay, so the first step is you
don't do the differentiation.

687
00:42:26,000 --> 00:42:32,000
You only do the integration.
Evaluate it between limits,

688
00:42:30,000 --> 00:42:36,000
and then you put a minus sign
before you forget to do it.

689
00:42:36,000 --> 00:42:42,000
And then, integral zero to
infinity.

690
00:42:38,000 --> 00:42:44,000
Now you do both operations.
So, it's n t to the n

691
00:42:43,000 --> 00:42:49,000
minus one,
and you also do the

692
00:42:46,000 --> 00:42:52,000
integration.
Okay, let's consider each of

693
00:42:49,000 --> 00:42:55,000
these pieces in turn.
Now, this piece,

694
00:42:52,000 --> 00:42:58,000
well, there's no problem with
the lower limit,

695
00:42:55,000 --> 00:43:01,000
zero, because when t is equal
to zero, this factor is zero,

696
00:43:00,000 --> 00:43:06,000
and the thing disappears as
long as n is one or higher.

697
00:43:06,000 --> 00:43:12,000
So, it's minus zero here at the
lower limit.

698
00:43:10,000 --> 00:43:16,000
The question is,
what is at the upper limit?

699
00:43:15,000 --> 00:43:21,000
So, what I have to do is find
out, what is the limit?

700
00:43:20,000 --> 00:43:26,000
The limit, as t goes to
infinity, that's what's

701
00:43:25,000 --> 00:43:31,000
happening up there,
of t to the n times e to the

702
00:43:30,000 --> 00:43:36,000
negative s t divided
by minus s.

703
00:43:37,000 --> 00:43:43,000
Well, as t goes to infinity,
this goes to infinity,

704
00:43:40,000 --> 00:43:46,000
of course.
This had better go to zero

705
00:43:43,000 --> 00:43:49,000
unless I want an answer,
infinity, which won't do me any

706
00:43:46,000 --> 00:43:52,000
good.
If this goes to zero,

707
00:43:48,000 --> 00:43:54,000
s had better be positive.
So, I'd better be restricting

708
00:43:52,000 --> 00:43:58,000
myself to that case.
Okay, so let's assume that s is

709
00:43:56,000 --> 00:44:02,000
positive so that this minus s
really is a negative number.

710
00:44:01,000 --> 00:44:07,000
Okay, then I have a chance.
So, this is going to be the

711
00:44:04,000 --> 00:44:10,000
limit.
Let's write it in a more

712
00:44:06,000 --> 00:44:12,000
familiar form with that down
below.

713
00:44:08,000 --> 00:44:14,000
So, it's t to the n.
That's going to infinity.

714
00:44:12,000 --> 00:44:18,000
But, the bottom is e to the
minus s t.

715
00:44:15,000 --> 00:44:21,000
But now, it's plus s t.
And, that's going to infinity,

716
00:44:19,000 --> 00:44:25,000
too, because s is positive.
So, the two guys are racing,

717
00:44:23,000 --> 00:44:29,000
and the question is,
oh, I lost a minus s here.

718
00:44:26,000 --> 00:44:32,000
So, oh... equals minus
one over s.

719
00:44:30,000 --> 00:44:36,000
How's that?
So, the question is only,

720
00:44:33,000 --> 00:44:39,000
which guy wins?
In the race to infinity,

721
00:44:35,000 --> 00:44:41,000
which one wins,
and how do you decide?

722
00:44:38,000 --> 00:44:44,000
And, the answer,
of course, is that's the bottom

723
00:44:41,000 --> 00:44:47,000
that wins.
The exponential always wins,

724
00:44:43,000 --> 00:44:49,000
and it's because of L'Hopital's
rule.

725
00:44:45,000 --> 00:44:51,000
You differentiate top and
bottom.

726
00:44:47,000 --> 00:44:53,000
Nothing much happens to the
bottom.

727
00:44:49,000 --> 00:44:55,000
It gets another factor of s,
but the top goes down to t to

728
00:44:53,000 --> 00:44:59,000
the n minus one.
L'Hopital it again,

729
00:44:56,000 --> 00:45:02,000
and again, and again,
and again, and again until

730
00:44:59,000 --> 00:45:05,000
finally you've reduced the top
to t to the zero where it's

731
00:45:03,000 --> 00:45:09,000
defenseless and just sitting
there, and nothing's happened to

732
00:45:06,000 --> 00:45:12,000
the bottom.
It's still got e to the s t.

733
00:45:11,000 --> 00:45:17,000
and that goes to infinity.

734
00:45:14,000 --> 00:45:20,000
So, the answer is,
this is zero by n applications

735
00:45:19,000 --> 00:45:25,000
of L'Hopital's rule.
Or, if you're very clever,

736
00:45:23,000 --> 00:45:29,000
you can do it in one,
but I won't tell you how.

737
00:45:27,000 --> 00:45:33,000
So, the answer is that this is
zero.

738
00:45:32,000 --> 00:45:38,000
At the upper limit,
it's also zero at least if s is

739
00:45:35,000 --> 00:45:41,000
positive, which is the case
we're considering.

740
00:45:38,000 --> 00:45:44,000
That leaves the rest of this.
All right, let's pull the

741
00:45:41,000 --> 00:45:47,000
constants out front.
That's plus.

742
00:45:43,000 --> 00:45:49,000
Two negatives make a plus.
n over s,

743
00:45:46,000 --> 00:45:52,000
now, what's left?
The integral from zero to

744
00:45:49,000 --> 00:45:55,000
infinity of t to the n minus
one, e to the minus s t dt.

745
00:45:57,000 --> 00:46:03,000
But, what on Earth is that?
That is n over s times

746
00:46:00,000 --> 00:46:06,000
the Laplace transform of t to
the n minus one.

747
00:46:04,000 --> 00:46:10,000
We got a reduction for it.
We don't get the answer in one

748
00:46:08,000 --> 00:46:14,000
step.
But, we get a reduction

749
00:46:11,000 --> 00:46:17,000
formula.
And, it says that the Laplace

750
00:46:13,000 --> 00:46:19,000
transform, let me write it this
way for once.

751
00:46:16,000 --> 00:46:22,000
The first way is now better,
is equal to n over s times the

752
00:46:20,000 --> 00:46:26,000
Laplace transform of n minus t
to the n minus one.

753
00:46:24,000 --> 00:46:30,000
Okay, the next step,

754
00:46:27,000 --> 00:46:33,000
this would be n over s times n
minus one over s times the

755
00:46:30,000 --> 00:46:36,000
Laplace transform of t to the n
minus two.

756
00:46:34,000 --> 00:46:40,000
If I can continue,

757
00:46:38,000 --> 00:46:44,000
I finally get in the top n
times n minus one times all the

758
00:46:44,000 --> 00:46:50,000
way down to one divided by the
same number of s's,

759
00:46:48,000 --> 00:46:54,000
n of them, times the Laplace
transform of t to the zero,

760
00:46:53,000 --> 00:46:59,000
finally.
See, one, zero,

761
00:46:55,000 --> 00:47:01,000
n minus one.
And so, what's the final

762
00:46:59,000 --> 00:47:05,000
answer?
It is n factorial over s to the

763
00:47:02,000 --> 00:47:08,000
what power?
Well, the Laplace transform of

764
00:47:07,000 --> 00:47:13,000
this is one over s.
So, the answer is it's s to the

765
00:47:13,000 --> 00:47:19,000
n plus one,
n of them here plus an extra

766
00:47:18,000 --> 00:47:24,000
one coming from the one over s
here.

767
00:47:21,000 --> 00:47:27,000
And, that's the answer.
The Laplace transform of t to

768
00:47:26,000 --> 00:47:32,000
the n, oddly enough,

769
00:47:29,000 --> 00:47:35,000
is more complicated,
and looks a little different

770
00:47:33,000 --> 00:47:39,000
from these.
It's n factorial over s to the

771
00:47:37,000 --> 00:47:43,000
n plus one.

772
00:47:40,000 --> 00:47:46,000
And, with that,
you can now calculate the

773
00:47:43,000 --> 00:47:49,000
Laplace transform of anything in
sight, and tomorrow you will.