WEBVTT
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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
00:09:02.000 --> 00:09:08.000
numbers.
Don't forget,
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this is an improper integral
going all the way up to
00:09:08.000 --> 00:09:14.000
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,
00:10:05.000 --> 00:10:11.000
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
00:10:15.000 --> 00:10:21.000
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
00:10:35.000 --> 00:10:41.000
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.
00:11:01.000 --> 00:11:07.000
So, x has its name changed to e
00:11:06.000 --> 00:11:12.000
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
00:11:17.000 --> 00:11:23.000
power, but by the laws of
exponents, that means I simply
00:11:21.000 --> 00:11:27.000
multiply the exponent by t,
and dt.
00:11:24.000 --> 00:11:30.000
And now, since I'm calling the
function f of t,
00:11:28.000 --> 00:11:34.000
the output ought to be called
capital F.
00:11:32.000 --> 00:11:38.000
But it's now a function,
since I've changed the
00:11:35.000 --> 00:11:41.000
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,
00:11:42.000 --> 00:11:48.000
what's x?
x is e to the negative s,
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I guess.
I mean, no one would leave a
00:11:49.000 --> 00:11:55.000
function in that form.
It's simply a function of s.
00:11:53.000 --> 00:11:59.000
And, what is that?
So, what have we got,
00:11:57.000 --> 00:12:03.000
finally?
What we have,
00:11:58.000 --> 00:12:04.000
dear hearts,
is this thing,
00:12:00.000 --> 00:12:06.000
which I stress is nothing more
than the continuous analog of
00:12:05.000 --> 00:12:11.000
the summation of a power series.
This is the discrete version.
00:12:12.000 --> 00:12:18.000
This is by these perfectly
natural transformations the
00:12:16.000 --> 00:12:22.000
continuous version of the same
thing.
00:12:19.000 --> 00:12:25.000
It starts with a function
defined for positive values of
00:12:23.000 --> 00:12:29.000
t, and turns it into a function
of s.
00:12:26.000 --> 00:12:32.000
And, this is called the Laplace
transform.
00:12:29.000 --> 00:12:35.000
Now, if I've done my work
correctly, you should all be
00:12:34.000 --> 00:12:40.000
saying, oh, is that all?
But, I know you aren't.
00:12:37.000 --> 00:12:43.000
So, it's okay.
You'll get used to it.
00:12:42.000 --> 00:12:48.000
The first thing you have to get
used to is one thing some people
00:12:46.000 --> 00:12:52.000
never get used to,
which is you put in a function
00:12:50.000 --> 00:12:56.000
of t, and you get out a function
of s.
00:12:53.000 --> 00:12:59.000
How could that be?
You know, for an operator,
00:12:56.000 --> 00:13:02.000
you put in 3x,
and you get out three if it's a
00:13:00.000 --> 00:13:06.000
differentiation operator.
In other words,
00:13:04.000 --> 00:13:10.000
when you have an operator,
the things we've been talking
00:13:08.000 --> 00:13:14.000
about the last two or three
weeks in one form or another,
00:13:13.000 --> 00:13:19.000
at least the variable doesn't
get changed.
00:13:16.000 --> 00:13:22.000
Well, but for a transform it
does, and that's why it's called
00:13:21.000 --> 00:13:27.000
a transform.
So, the difference between a
00:13:24.000 --> 00:13:30.000
transform and an operator is
that for a transform a function
00:13:29.000 --> 00:13:35.000
of t comes in,
but a function of s comes out.
00:13:34.000 --> 00:13:40.000
The variable gets changed,
whereas for an operator,
00:13:37.000 --> 00:13:43.000
f of t goes in and
what comes out is g of t,
00:13:40.000 --> 00:13:46.000
a function using the
same variable like
00:13:43.000 --> 00:13:49.000
differentiation is a typical
example of an operator,
00:13:47.000 --> 00:13:53.000
or the linear differential
operators we've been talking
00:13:50.000 --> 00:13:56.000
about.
Well, but this doesn't behave
00:13:52.000 --> 00:13:58.000
that way.
The variable does get changed.
00:13:55.000 --> 00:14:01.000
That's, in fact,
extremely important in the
00:13:58.000 --> 00:14:04.000
applications.
In the applications,
00:14:01.000 --> 00:14:07.000
t usually means the time,
and s very often,
00:14:05.000 --> 00:14:11.000
not always, but very often is a
variable measuring frequency,
00:14:10.000 --> 00:14:16.000
for instance.
But, so that's a peculiar thing
00:14:14.000 --> 00:14:20.000
that's hard to get used to.
But, a good thing is the fact
00:14:18.000 --> 00:14:24.000
that it's a linear transform.
In other words,
00:14:22.000 --> 00:14:28.000
it obeys the laws we'd love and
like that the Laplace
00:14:26.000 --> 00:14:32.000
transform-- oh,
I never gave you any notation
00:14:30.000 --> 00:14:36.000
for the laplace transform.
Hey, I'd better do that.
00:14:35.000 --> 00:14:41.000
Okay, so, some notation:
there are two notations that
00:14:39.000 --> 00:14:45.000
are used.
Your book mostly uses the
00:14:41.000 --> 00:14:47.000
notation that the laplace
transform of f of t is capital F
00:14:45.000 --> 00:14:51.000
of s,
uses the same letter but with
00:14:48.000 --> 00:14:54.000
the same capital.
Now, as you will see,
00:14:51.000 --> 00:14:57.000
there are some places you
absolutely cannot use that
00:14:54.000 --> 00:15:00.000
notation.
It may seem strange,
00:14:56.000 --> 00:15:02.000
looks perfectly natural.
There are certain laws you
00:14:59.000 --> 00:15:05.000
cannot express using that
notation.
00:15:03.000 --> 00:15:09.000
It's baffling.
But, if you can't do it this
00:15:05.000 --> 00:15:11.000
way, you can do it using this
notation instead.
00:15:08.000 --> 00:15:14.000
One or the other will almost
always work.
00:15:11.000 --> 00:15:17.000
So, I'll use my little squiggly
notation, but that's what I use.
00:15:15.000 --> 00:15:21.000
I think it's a little more
vivid, and the trouble is that
00:15:19.000 --> 00:15:25.000
this piles up too many
parentheses.
00:15:21.000 --> 00:15:27.000
And, that's always hard to
read.
00:15:23.000 --> 00:15:29.000
So, I like this better.
So, these are two alternate
00:15:26.000 --> 00:15:32.000
ways of saying the same thing.
The Laplace transform of this
00:15:32.000 --> 00:15:38.000
function is that one.
Okay, well, let's use,
00:15:36.000 --> 00:15:42.000
for the linearity law,
it's definitely best.
00:15:39.000 --> 00:15:45.000
I really cannot express the
linearity law using the second
00:15:44.000 --> 00:15:50.000
notation, but using the first
notation, it's a breeze.
00:15:49.000 --> 00:15:55.000
The Laplace transform of the
sum of two functions is the sum
00:15:54.000 --> 00:16:00.000
of their Laplace transforms of
each of them separately.
00:16:00.000 --> 00:16:06.000
Or, better yet,
you could write it that way.
00:16:03.000 --> 00:16:09.000
Let's write it this way.
That way, it looks more like an
00:16:06.000 --> 00:16:12.000
operator, L of f plus L of g.
00:16:10.000 --> 00:16:16.000
And, of the same way,
if you take a function and
00:16:13.000 --> 00:16:19.000
multiply it by a constant and
take the laplace transform,
00:16:17.000 --> 00:16:23.000
you can pull the constant
outside.
00:16:19.000 --> 00:16:25.000
And, of course,
why are these true?
00:16:22.000 --> 00:16:28.000
These are true just because of
the form of the transform.
00:16:25.000 --> 00:16:31.000
If I add up f and g,
I simply add up the two
00:16:29.000 --> 00:16:35.000
corresponding integrals.
In other words,
00:16:33.000 --> 00:16:39.000
I'm using the fact that the
integral, this definite
00:16:37.000 --> 00:16:43.000
integral, is itself a linear
operator.
00:16:40.000 --> 00:16:46.000
Well, that's the general
setting.
00:16:43.000 --> 00:16:49.000
That's where it comes from,
and that's the notation for it.
00:16:47.000 --> 00:16:53.000
And, now we have to get to
work.
00:16:50.000 --> 00:16:56.000
The first thing to do to get
familiar with this is,
00:16:54.000 --> 00:17:00.000
obviously what we want to do is
say, okay, these were the
00:16:59.000 --> 00:17:05.000
transforms of some simple
discreet functions.
00:17:04.000 --> 00:17:10.000
Okay, suppose I put in some
familiar functions,
00:17:09.000 --> 00:17:15.000
f of t.
What do their Laplace
00:17:14.000 --> 00:17:20.000
transforms look like?
So, let's do that.
00:17:19.000 --> 00:17:25.000
So, one of the boards I should
keep stored.
00:17:24.000 --> 00:17:30.000
Why don't I store on this
board?
00:17:28.000 --> 00:17:34.000
I'll store on this board the
formulas as we get them.
00:17:37.000 --> 00:17:43.000
So, let's see,
what should we aim at,
00:17:39.000 --> 00:17:45.000
first?
Let's first find,
00:17:41.000 --> 00:17:47.000
and I'll do the calculations on
the sideboard,
00:17:44.000 --> 00:17:50.000
and we'll see how it works out.
I'm not very sure.
00:17:47.000 --> 00:17:53.000
In other words,
what's the Laplace transform of
00:17:51.000 --> 00:17:57.000
the function,
one?
00:17:52.000 --> 00:17:58.000
Well, there's an even easier
one.
00:17:54.000 --> 00:18:00.000
What's the Laplace transform of
the function zero?
00:17:57.000 --> 00:18:03.000
Answer: zero.
Very exciting.
00:18:00.000 --> 00:18:06.000
What's the Laplace transform of
one?
00:18:03.000 --> 00:18:09.000
Well, it doesn't turn out the
constant anymore than it turned
00:18:07.000 --> 00:18:13.000
out to be a constant up there.
Let's calculate it.
00:18:11.000 --> 00:18:17.000
Now, you can do these
calculations carefully,
00:18:14.000 --> 00:18:20.000
dotting all the i's,
or pretty carefully,
00:18:17.000 --> 00:18:23.000
or not carefully at all,
i.e.
00:18:19.000 --> 00:18:25.000
sloppily.
I'll let you be sloppy after,
00:18:21.000 --> 00:18:27.000
generally speaking,
you could be sloppy unless the
00:18:25.000 --> 00:18:31.000
directions tell you to be less
sloppy or to be careful,
00:18:29.000 --> 00:18:35.000
okay?
So, I'll do one carefully.
00:18:32.000 --> 00:18:38.000
Let's calculate the Laplace
transform of one carefully.
00:18:36.000 --> 00:18:42.000
Okay, in the beginning,
you've got nothing to use with
00:18:40.000 --> 00:18:46.000
the definition.
So, I have to calculate the
00:18:43.000 --> 00:18:49.000
integral from zero to infinity
of one, that's the f of t times
00:18:47.000 --> 00:18:53.000
e to the negative s t,
so I don't have to
00:18:51.000 --> 00:18:57.000
put in the one,
dt.
00:18:52.000 --> 00:18:58.000
All right, now,
let me remind you,
00:18:54.000 --> 00:19:00.000
this is an improper integral.
This is just about the first
00:18:58.000 --> 00:19:04.000
time in the course we've had an
improper integral.
00:19:01.000 --> 00:19:07.000
But, there are going to be a
lot of them over the next couple
00:19:06.000 --> 00:19:12.000
of weeks, nothing but.
All right, it's an improper
00:19:10.000 --> 00:19:16.000
integral.
That means we have to go back
00:19:12.000 --> 00:19:18.000
to the definition.
If you want to be careful,
00:19:15.000 --> 00:19:21.000
you have to go back to the
definition of improper integral.
00:19:19.000 --> 00:19:25.000
So, it's the limit,
as R goes to infinity,
00:19:21.000 --> 00:19:27.000
of what you get by integrating
only up as far as R.
00:19:24.000 --> 00:19:30.000
That's a definite integral.
That's a nice Riemann integral.
00:19:27.000 --> 00:19:33.000
So, this is what I have to
calculate.
00:19:31.000 --> 00:19:37.000
And, I have to take the limit
as R goes to infinity.
00:19:34.000 --> 00:19:40.000
Now, how do I calculate that?
Well, this integral is equal
00:19:37.000 --> 00:19:43.000
to, that's easy.
It's just integrating.
00:19:40.000 --> 00:19:46.000
Remember that you're
integrating with respect to t.
00:19:43.000 --> 00:19:49.000
So, s is a parameter.
It's like a constant,
00:19:45.000 --> 00:19:51.000
in other words.
So, it's e to the minus s t,
00:19:48.000 --> 00:19:54.000
and when I differentiated,
00:19:50.000 --> 00:19:56.000
the derivative of this would
have negative s.
00:19:53.000 --> 00:19:59.000
So, to get rid of that negative
s, so the derivative is e to the
00:19:57.000 --> 00:20:03.000
minus s t.
You have to put minus s
00:20:00.000 --> 00:20:06.000
in the denominator.
And now, I'll want to evaluate
00:20:05.000 --> 00:20:11.000
that between zero and R.
And, what do I get?
00:20:09.000 --> 00:20:15.000
Well it is at the upper limit.
So, it's e to the minus s times
00:20:14.000 --> 00:20:20.000
R minus, at the lower limit,
it's t is equal to zero,
00:20:19.000 --> 00:20:25.000
so whatever s is,
it's one.
00:20:21.000 --> 00:20:27.000
And that's divided by this
constant up front,
00:20:25.000 --> 00:20:31.000
negative s. So,
00:20:28.000 --> 00:20:34.000
the answer is,
it is equal to the limit of,
00:20:32.000 --> 00:20:38.000
as R goes to infinity,
of e to the negative s R minus
00:20:37.000 --> 00:20:43.000
one divided by minus s.
00:20:43.000 --> 00:20:49.000
Now, what's that?
Well, as R goes to infinity,
00:20:47.000 --> 00:20:53.000
e to the minus 2R,
or minus 5R goes to zero,
00:20:52.000 --> 00:20:58.000
and the answer is minus one
over minus s.
00:20:57.000 --> 00:21:03.000
So, that's one over s.
And so, that's our answer.
00:21:02.000 --> 00:21:08.000
Let's put it up here.
It's one over s,
00:21:05.000 --> 00:21:11.000
except it isn't.
I made a mistake.
00:21:10.000 --> 00:21:16.000
Well, not mistake,
a little oversight.
00:21:16.000 --> 00:21:22.000
What's the oversight?
This is okay.
00:21:22.000 --> 00:21:28.000
This is okay.
This is okay.
00:21:26.000 --> 00:21:32.000
This is not okay.
This is okay.
00:21:31.000 --> 00:21:37.000
But that's not okay.
What's wrong?
00:21:38.000 --> 00:21:44.000
I did slight a verbal hand.
Maybe some of you have picked
00:21:41.000 --> 00:21:47.000
it up and were too embarrassed
to correct me,
00:21:44.000 --> 00:21:50.000
but I said like e to the minus
2R obviously goes to
00:21:48.000 --> 00:21:54.000
zero, and e to the minus 5R
goes to zero.
00:21:51.000 --> 00:21:57.000
How about e to the minus minus
3 R?
00:21:54.000 --> 00:22:00.000
Does that go to zero?
No, that's e to the 3R,
00:21:57.000 --> 00:22:03.000
which goes to infinity.
00:22:00.000 --> 00:22:06.000
The only time this goes to zero
is if s is a positive number.
00:22:05.000 --> 00:22:11.000
Minus s looks like a negative
number, but it's not,
00:22:10.000 --> 00:22:16.000
if s is equal to minus two.
So, this is only true if s is
00:22:16.000 --> 00:22:22.000
positive because only if s is
positive is this exponent really
00:22:22.000 --> 00:22:28.000
negative and large,
and therefore going to
00:22:26.000 --> 00:22:32.000
infinity, going to zero as R
goes to infinity.
00:22:30.000 --> 00:22:36.000
So, the answer is not one over
s.
00:22:34.000 --> 00:22:40.000
It is one over s,
s must positive.
00:22:39.000 --> 00:22:45.000
Now, once again,
here, people don't worry about
00:22:41.000 --> 00:22:47.000
this sort of thing with power
series because it seems very
00:22:45.000 --> 00:22:51.000
obvious, you know,
one over x,
00:22:48.000 --> 00:22:54.000
absolute value of x is less
than one,
00:22:51.000 --> 00:22:57.000
when it gets to be the Laplace
transform, just because the
00:22:54.000 --> 00:23:00.000
Laplace transform is mysterious,
the question is,
00:22:58.000 --> 00:23:04.000
okay, the Laplace transform is
one over s of one,
00:23:01.000 --> 00:23:07.000
well, Laplace transform of one
I understand is one over s if s
00:23:05.000 --> 00:23:11.000
is positive.
What is it if s is negative?
00:23:09.000 --> 00:23:15.000
Okay, right down in your little
books, this, but that down,
00:23:15.000 --> 00:23:21.000
what is it if s is negative,
and write underneath that,
00:23:21.000 --> 00:23:27.000
this question is meaningless.
It doesn't mean anything.
00:23:26.000 --> 00:23:32.000
I'll draw you a picture.
This is a picture of the
00:23:31.000 --> 00:23:37.000
Laplace transform of one.
It is that.
00:23:34.000 --> 00:23:40.000
It's one branch of this curve.
It does not include the branch
00:23:39.000 --> 00:23:45.000
on the left.
It doesn't because I showed you
00:23:43.000 --> 00:23:49.000
it doesn't.
That's all there is to it.
00:23:46.000 --> 00:23:52.000
Okay, so I did that carefully.
Now I'm going to get a little
00:23:50.000 --> 00:23:56.000
less careful.
What's the Laplace transform of
00:23:54.000 --> 00:24:00.000
e to the a t?
First of all,
00:23:57.000 --> 00:24:03.000
in general, the kind of
functions for which people like
00:24:01.000 --> 00:24:07.000
to calculate the Laplace
transform, and basically the
00:24:06.000 --> 00:24:12.000
only ones there will be in the
tables are exactly the sort of
00:24:10.000 --> 00:24:16.000
functions that you used in
solving linear equations with
00:24:15.000 --> 00:24:21.000
constant coefficients.
What kinds of functions entered
00:24:21.000 --> 00:24:27.000
in there?
Exponentials,
00:24:22.000 --> 00:24:28.000
sines and cosines,
but they were really complex
00:24:25.000 --> 00:24:31.000
exponentials,
right?
00:24:26.000 --> 00:24:32.000
e to the t sine t,
but that was really a
00:24:30.000 --> 00:24:36.000
complex exponential,
too, just a little more
00:24:33.000 --> 00:24:39.000
complicated one,
polynomials,
00:24:35.000 --> 00:24:41.000
and that's about it.
t times e to the t,
00:24:38.000 --> 00:24:44.000
that was okay,
too.
00:24:41.000 --> 00:24:47.000
These are the functions for
which people calculate the
00:24:44.000 --> 00:24:50.000
Laplace transform,
and all the other functions
00:24:46.000 --> 00:24:52.000
they don't calculate the Laplace
transforms.
00:24:49.000 --> 00:24:55.000
So, I don't mean to disappoint
you here.
00:24:52.000 --> 00:24:58.000
You're going to say,
oh, what, that same old stuff?
00:24:55.000 --> 00:25:01.000
For two more weeks,
we've got that same,
00:24:57.000 --> 00:25:03.000
well, the Laplace transform
does a lot of things much better
00:25:01.000 --> 00:25:07.000
than the methods we've been
using.
00:25:04.000 --> 00:25:10.000
And, I won't.
I'll sell it when I get a
00:25:06.000 --> 00:25:12.000
chance to, for now,
let's just get familiar with
00:25:09.000 --> 00:25:15.000
it.
All right, so while I'm not
00:25:11.000 --> 00:25:17.000
going to calculate e to the a t
for you,
00:25:14.000 --> 00:25:20.000
because I'd like instead to
just prove a simple formula
00:25:17.000 --> 00:25:23.000
which will just give that,
and will also give us e to the
00:25:21.000 --> 00:25:27.000
a t sine t.
It will give us a lot more,
00:25:24.000 --> 00:25:30.000
instead.
I'm going to calculate a
00:25:27.000 --> 00:25:33.000
formula for the Laplace
transform of this guy if you
00:25:30.000 --> 00:25:36.000
already know the Laplace
transform of it.
00:25:34.000 --> 00:25:40.000
Now, see, this falls in that
category because this is really
00:25:38.000 --> 00:25:44.000
e to the a t times one.
00:25:41.000 --> 00:25:47.000
But, I already know the Laplace
transform of one.
00:25:44.000 --> 00:25:50.000
So that's, if I can get a
general formula for this,
00:25:48.000 --> 00:25:54.000
I'll be able to get the formula
for e to the a t as a
00:25:53.000 --> 00:25:59.000
consequence.
So, let's look for this Laplace
00:25:56.000 --> 00:26:02.000
transform.
Now, it's really easy.
00:25:59.000 --> 00:26:05.000
Let's see, where am I doing
calculations?
00:26:02.000 --> 00:26:08.000
Over here.
Okay, so we've got e.
00:26:05.000 --> 00:26:11.000
So, I want to calculate the
Laplace transform e to the a t f
00:26:09.000 --> 00:26:15.000
of t.
So I'm going to say that's the
00:26:13.000 --> 00:26:19.000
integral from zero to infinity
of e to the a t times f of t.
00:26:16.000 --> 00:26:22.000
And now, the rest I copy.
00:26:19.000 --> 00:26:25.000
That's the function part of it
that goes to the input,
00:26:23.000 --> 00:26:29.000
and then there's the other
part.
00:26:25.000 --> 00:26:31.000
This part is called the kernel,
by the way, but don't worry
00:26:29.000 --> 00:26:35.000
about that.
However, if you drop it in
00:26:33.000 --> 00:26:39.000
conversation,
people will look at you and
00:26:36.000 --> 00:26:42.000
say, gee, they know something I
don't.
00:26:39.000 --> 00:26:45.000
And you will.
You know that it's the kernel.
00:26:43.000 --> 00:26:49.000
Okay, well, now,
what kind of formula can I be
00:26:47.000 --> 00:26:53.000
looking for?
Clearly, I can only be looking
00:26:51.000 --> 00:26:57.000
for a formula which expresses it
in terms of the Laplace
00:26:56.000 --> 00:27:02.000
transform of f of t.
Let's calculate and see what we
00:27:02.000 --> 00:27:08.000
get.
Now, what would you do to that
00:27:04.000 --> 00:27:10.000
thing to make?
Well, obviously,
00:27:06.000 --> 00:27:12.000
the thing to do is to combine
the two exponentials.
00:27:09.000 --> 00:27:15.000
So, that's going to be the
integral from zero to infinity
00:27:13.000 --> 00:27:19.000
of f of t.
e, now, I'd like to put it,
00:27:16.000 --> 00:27:22.000
to combine the exponentials in
such a way that it has,
00:27:19.000 --> 00:27:25.000
still, that same form,
so, I'm going to begin with
00:27:23.000 --> 00:27:29.000
that negative sign,
and then see what the rest of
00:27:26.000 --> 00:27:32.000
it has to be.
What is it going to be?
00:27:30.000 --> 00:27:36.000
Well, minus s t and
plus a t,
00:27:34.000 --> 00:27:40.000
but I can make that minus a
here, and it will come out
00:27:39.000 --> 00:27:45.000
right.
So, it's minus s t plus a t,
00:27:42.000 --> 00:27:48.000
and there are the two parts,
00:27:46.000 --> 00:27:52.000
those two factors,
dt.
00:27:48.000 --> 00:27:54.000
So, what's that?
That's the Laplace transform.
00:27:52.000 --> 00:27:58.000
If the a weren't there,
this would be the Laplace
00:27:56.000 --> 00:28:02.000
transform of f of t.
What is it with the a there?
00:28:03.000 --> 00:28:09.000
It's the Laplace transform of f
of t,
00:28:08.000 --> 00:28:14.000
except that instead of the
variable, s has been replaced by
00:28:15.000 --> 00:28:21.000
the variable s minus a.
00:28:19.000 --> 00:28:25.000
I'll give you a second to
digest that.
00:28:24.000 --> 00:28:30.000
Well, you digest it while I'm
writing it because that's the
00:28:30.000 --> 00:28:36.000
answer.
And, the way this is most often
00:28:35.000 --> 00:28:41.000
used, I have to qualify it for
the value.
00:28:38.000 --> 00:28:44.000
So, if F of s is good
for s positive,
00:28:42.000 --> 00:28:48.000
the way it would be,
for example,
00:28:45.000 --> 00:28:51.000
if I used the function one
here, then to finish that off,
00:28:50.000 --> 00:28:56.000
then, F of s minus a will be,
00:28:53.000 --> 00:28:59.000
this will be good when s is
bigger than a.
00:28:57.000 --> 00:29:03.000
Why is that?
Well, because this is true.
00:29:02.000 --> 00:29:08.000
This is true.
If s minus a is
00:29:06.000 --> 00:29:12.000
positive, that's the condition.
That's what this Laplace
00:29:11.000 --> 00:29:17.000
transform is good.
But that simply says that s
00:29:15.000 --> 00:29:21.000
should be bigger than a.
00:29:18.000 --> 00:29:24.000
And, since this doesn't look
pretty, let me try to make it
00:29:23.000 --> 00:29:29.000
look a little bit prettier.
So, let's write it.
00:29:29.000 --> 00:29:35.000
So, this is assuming F of s is
for s greater than zero.
00:29:34.000 --> 00:29:40.000
Now, this is called something.
00:29:38.000 --> 00:29:44.000
This is called,
well, what would you call it?
00:29:43.000 --> 00:29:49.000
On the left side,
you multiply by an exponential.
00:29:47.000 --> 00:29:53.000
On the right,
you translate.
00:29:50.000 --> 00:29:56.000
You shift the argument over by
a.
00:29:53.000 --> 00:29:59.000
So, this is called,
gulp, the exponential shift.
00:29:58.000 --> 00:30:04.000
What?
Well, I'll call it the formula.
00:30:02.000 --> 00:30:08.000
The thing before,
when we talked about operators,
00:30:06.000 --> 00:30:12.000
we called it the exponential
shift rule or the exponential
00:30:10.000 --> 00:30:16.000
shift law.
But, in fact,
00:30:12.000 --> 00:30:18.000
this is, in a way,
a disguised form of the same
00:30:16.000 --> 00:30:22.000
law.
And, engineers who typically do
00:30:19.000 --> 00:30:25.000
all their work using the Laplace
transform and don't use
00:30:23.000 --> 00:30:29.000
operators, this is the form of
the exponential shift law that
00:30:28.000 --> 00:30:34.000
they would know.
What you can do with one,
00:30:33.000 --> 00:30:39.000
you can do with the other.
You can now use both.
00:30:37.000 --> 00:30:43.000
So, what's the answer to e to
the a t?
00:30:41.000 --> 00:30:47.000
Well, the answer is,
I'm supposed to,
00:30:44.000 --> 00:30:50.000
e to the a t times one,
the Laplace transform of one is
00:30:48.000 --> 00:30:54.000
one over s.
And, therefore,
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
00:30:56.000 --> 00:31:02.000
minus a .
And so, that's the answer.
00:31:00.000 --> 00:31:06.000
Let's see, what else don't we
know?
00:31:05.000 --> 00:31:11.000
Well, how about sines and
cosines?
00:31:08.000 --> 00:31:14.000
Well, the way to do sines and
cosines is by making the
00:31:15.000 --> 00:31:21.000
observation that this formula
also works when a is a complex
00:31:22.000 --> 00:31:28.000
number.
So, can use also for a a
00:31:26.000 --> 00:31:32.000
complex number, for e to
the a plus b i times t.
00:31:31.000 --> 00:31:37.000
The Laplace transform of e to
00:31:37.000 --> 00:31:43.000
the a plus b i times t is one
over s minus a plus b i.
00:31:43.000 --> 00:31:49.000
And again, it will be for s
bigger than a.
00:31:47.000 --> 00:31:53.000
So, let's calculate the Laplace
transform of,
00:31:50.000 --> 00:31:56.000
let's say, well,
I've got to cover up something.
00:31:54.000 --> 00:32:00.000
Okay, so, that's the Laplace
transform.
00:31:57.000 --> 00:32:03.000
I've got to remember that.
So, let's calculate the Laplace
00:32:02.000 --> 00:32:08.000
transform of,
let's say, sine of a t
00:32:05.000 --> 00:32:11.000
and cosine a t.
00:32:08.000 --> 00:32:14.000
What do you get for that?
Well, just for a little
00:32:12.000 --> 00:32:18.000
variety, we could do it by using
that formula,
00:32:15.000 --> 00:32:21.000
and taking its real and
imaginary parts.
00:32:18.000 --> 00:32:24.000
Since some of you had so much
difficulty with the backwards
00:32:23.000 --> 00:32:29.000
Euler formula,
he is a good case where you
00:32:26.000 --> 00:32:32.000
could use it.
Suppose you want to calculate
00:32:29.000 --> 00:32:35.000
the Laplace transform of cosine
a t.
00:32:35.000 --> 00:32:41.000
Well, I'm going to write that
using, I want to calculate using
00:32:39.000 --> 00:32:45.000
complex exponentials.
The way I will do it is by
00:32:43.000 --> 00:32:49.000
using the backwards Euler
formula.
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
00:32:50.000 --> 00:32:56.000
two.
00:32:52.000 --> 00:32:58.000
Remember, the foreword Euler
formula would say e to the i a t
00:32:57.000 --> 00:33:03.000
equals cosine a t plus i sine a
t.
00:33:01.000 --> 00:33:07.000
That expresses the complex
00:33:04.000 --> 00:33:10.000
exponential in terms of sines
and cosines.
00:33:07.000 --> 00:33:13.000
This is the backward formula,
which just read it backwards,
00:33:11.000 --> 00:33:17.000
expressing cosines and sines in
terms of complex exponentials
00:33:15.000 --> 00:33:21.000
instead.
Both formulas are useful,
00:33:17.000 --> 00:33:23.000
almost equally useful,
in fact.
00:33:19.000 --> 00:33:25.000
And anyway, just remind you of
it, let's use this one.
00:33:23.000 --> 00:33:29.000
Okay, what's the Laplace
transform, then,
00:33:26.000 --> 00:33:32.000
of cosine a t?
Well, by linearity,
00:33:30.000 --> 00:33:36.000
it's equal to one half the
Laplace transform of this guy
00:33:35.000 --> 00:33:41.000
plus the Laplace transform of
that guy.
00:33:38.000 --> 00:33:44.000
And, what are those?
Well, the Laplace transform of
00:33:43.000 --> 00:33:49.000
e to the i a t is one over s
minus i a,
00:33:48.000 --> 00:33:54.000
and the Laplace
transform of the other guy is
00:33:53.000 --> 00:33:59.000
one divided by s plus i a.
00:33:56.000 --> 00:34:02.000
Now, of course,
this has become out to be a
00:34:00.000 --> 00:34:06.000
real function.
This is real.
00:34:03.000 --> 00:34:09.000
Every integral is real.
This must come out to be real.
00:34:07.000 --> 00:34:13.000
This looks kind of complex,
but it isn't.
00:34:10.000 --> 00:34:16.000
I know automatically that this
is going to be a real function.
00:34:15.000 --> 00:34:21.000
How I know that?
Well, mentally,
00:34:17.000 --> 00:34:23.000
you can combine the terms and
calculate.
00:34:20.000 --> 00:34:26.000
But, I know even before that.
Remember, there are two ways to
00:34:24.000 --> 00:34:30.000
see that something is real.
You can calculate it and see
00:34:28.000 --> 00:34:34.000
that its imaginary part is zero,
hack, or without any
00:34:32.000 --> 00:34:38.000
calculation, if you change i to
minus i,
00:34:36.000 --> 00:34:42.000
and you get the same thing,
it must be real.
00:34:41.000 --> 00:34:47.000
Now, if I change i to minus i
in this expression,
00:34:45.000 --> 00:34:51.000
what happens?
If I change i to minus i,
00:34:48.000 --> 00:34:54.000
this term turns into that one,
and this one turns into that
00:34:54.000 --> 00:35:00.000
one.
Conclusion: the sum of the two
00:34:57.000 --> 00:35:03.000
is unchanged.
And therefore,
00:34:59.000 --> 00:35:05.000
this is real.
Well, of course,
00:35:03.000 --> 00:35:09.000
in the time I took to make that
argument, I could have actually
00:35:07.000 --> 00:35:13.000
calculated it.
So, what the heck,
00:35:09.000 --> 00:35:15.000
let's calculate it?
So, you do the high school
00:35:12.000 --> 00:35:18.000
thing, and it's this guy plus
that guy on top,
00:35:15.000 --> 00:35:21.000
which makes 2s.
I on the bottom is the product
00:35:18.000 --> 00:35:24.000
of those, which by now you
should know the product of two
00:35:22.000 --> 00:35:28.000
complex numbers.
A product of a number and its
00:35:25.000 --> 00:35:31.000
complex conjugate is the sum of
the squares.
00:35:29.000 --> 00:35:35.000
So, what's the answer?
The twos cancel,
00:35:31.000 --> 00:35:37.000
and the answer is that the
Laplace transform of cosine a t
00:35:35.000 --> 00:35:41.000
is s over s squared plus a
squared.
00:35:39.000 --> 00:35:45.000
And, that will be true as,
00:35:41.000 --> 00:35:47.000
in general, it's true up there
for positive values of s only.
00:35:46.000 --> 00:35:52.000
And, the sine a t,
you can calculate that in
00:35:49.000 --> 00:35:55.000
recitation tomorrow.
The answer to that is a divided
00:35:53.000 --> 00:35:59.000
by s squared plus a squared.
00:35:56.000 --> 00:36:02.000
You would get the same answers
if you took the real and
00:36:00.000 --> 00:36:06.000
imaginary parts of that
expression.
00:36:04.000 --> 00:36:10.000
It's another way of getting at
the recitations tomorrow;
00:36:08.000 --> 00:36:14.000
we'll get practice in
calculating other functions
00:36:11.000 --> 00:36:17.000
related to these by using these
formulas, and also from scratch
00:36:16.000 --> 00:36:22.000
directly from the definition of
the Laplace transform.
00:36:20.000 --> 00:36:26.000
Well, there are two things
which we still should do.
00:36:24.000 --> 00:36:30.000
The first is I want to get you
started with calculating inverse
00:36:29.000 --> 00:36:35.000
Laplace transforms.
And, the reason for doing that
00:36:33.000 --> 00:36:39.000
is, in other words,
I've started with f of t,
00:36:36.000 --> 00:36:42.000
and we've been focusing
on what is capital F of s?
00:36:40.000 --> 00:36:46.000
But, you will find that when
00:36:42.000 --> 00:36:48.000
you go to solve differential
equations, by far,
00:36:45.000 --> 00:36:51.000
the hardest part of the
procedure is you get F of s.
00:36:49.000 --> 00:36:55.000
The Laplace transform of the
answer, and you have to convert
00:36:52.000 --> 00:36:58.000
that back into the answer in
terms of t that you were looking
00:36:56.000 --> 00:37:02.000
for.
In other words,
00:36:58.000 --> 00:37:04.000
the main step in the procedure
that you are going to be using
00:37:01.000 --> 00:37:07.000
for solving differential
equations is,
00:37:04.000 --> 00:37:10.000
and the hardest part of the
step will be to calculate
00:37:07.000 --> 00:37:13.000
inverse laplace transforms.
Now, you think that could be
00:37:13.000 --> 00:37:19.000
done by tables,
but, in fact,
00:37:15.000 --> 00:37:21.000
it can't unless the tables are
too long to be useful.
00:37:19.000 --> 00:37:25.000
You have to do a certain amount
of work yourself.
00:37:23.000 --> 00:37:29.000
And, the certain amount of work
that you have to do yourself
00:37:28.000 --> 00:37:34.000
involves partial fractions
decompositions.
00:37:33.000 --> 00:37:39.000
And, in case you were wondering
which you are not,
00:37:36.000 --> 00:37:42.000
the reason you learned partial
fractions in 18.01 was not to
00:37:41.000 --> 00:37:47.000
learn those silly integrals,
but he learned it so that when
00:37:45.000 --> 00:37:51.000
you got to 18.03 you would be
able to calculate,
00:37:48.000 --> 00:37:54.000
solve differential equations by
using Laplace transforms.
00:37:53.000 --> 00:37:59.000
Sorry.
That's life.
00:37:54.000 --> 00:38:00.000
Now, so a certain amount of the
recitation time tomorrow will be
00:37:59.000 --> 00:38:05.000
devoted to reminding you how to
do partial fractions since you
00:38:03.000 --> 00:38:09.000
haven't done it in a while,
and I assume,
00:38:06.000 --> 00:38:12.000
yeah, we had that,
I think.
00:38:10.000 --> 00:38:16.000
Okay, now, they also remind you
of the most efficient method,
00:38:16.000 --> 00:38:22.000
which about half of you have
had, and the rest think you
00:38:23.000 --> 00:38:29.000
might have had,
but really aren't sure.
00:38:27.000 --> 00:38:33.000
So, here's the answer.
We want to find out what it's
00:38:33.000 --> 00:38:39.000
inverse Laplace transform is.
What you have to do,
00:38:39.000 --> 00:38:45.000
it normally won't be in the
tables like this.
00:38:42.000 --> 00:38:48.000
You have to put it in a form in
which it will be in the tables.
00:38:46.000 --> 00:38:52.000
As you do that,
you have to make partial
00:38:48.000 --> 00:38:54.000
fractions decompositions,
which, to do it quickly,
00:38:51.000 --> 00:38:57.000
so if you don't know what I'm
doing now, or you think you once
00:38:55.000 --> 00:39:01.000
knew but don't quite remember,
go to recitation tomorrow.
00:38:59.000 --> 00:39:05.000
To get the coefficient here,
I cover up s,
00:39:02.000 --> 00:39:08.000
and I put s equals zero
because that's the law.
00:39:07.000 --> 00:39:13.000
To get this coefficient,
I cover up s plus three and I
00:39:10.000 --> 00:39:16.000
put s equals a negative three
because that's what you're
00:39:14.000 --> 00:39:20.000
supposed to do.
Put s equal negative three,
00:39:17.000 --> 00:39:23.000
you get minus one third.
This is equal to that.
00:39:20.000 --> 00:39:26.000
In this form,
I don't know what the inverse
00:39:23.000 --> 00:39:29.000
Laplace form is,
but in this form,
00:39:25.000 --> 00:39:31.000
I certainly do know with the
inverse Laplace transform
00:39:29.000 --> 00:39:35.000
because the inverse Laplace
transform is linear,
00:39:32.000 --> 00:39:38.000
and because each of these guys
especially occurs in those
00:39:36.000 --> 00:39:42.000
tables.
Well, what's this?
00:39:40.000 --> 00:39:46.000
Well, it's whatever the Laplace
transform of,
00:39:43.000 --> 00:39:49.000
inverse Laplace transform of
one over s is multiplied by one
00:39:49.000 --> 00:39:55.000
third.
Well, the inverse Laplace
00:39:51.000 --> 00:39:57.000
transform of one over s is one.
So, it's one third times one.
00:39:56.000 --> 00:40:02.000
How about the other guy?
Minus one third,
00:40:00.000 --> 00:40:06.000
the inverse Laplace transform
of one over s plus three,
00:40:05.000 --> 00:40:11.000
that's this formula.
a is negative three,
00:40:09.000 --> 00:40:15.000
and that makes e to the minus
3t.
00:40:13.000 --> 00:40:19.000
So, if this was the Laplace
transform of the solution to the
00:40:18.000 --> 00:40:24.000
differential equation,
then the solution in terms of t
00:40:22.000 --> 00:40:28.000
was this function.
Now, you'll get lots of
00:40:26.000 --> 00:40:32.000
practice in that.
All I'm doing now is signaling
00:40:30.000 --> 00:40:36.000
that that's the most important
and difficult step of the
00:40:34.000 --> 00:40:40.000
procedure, and that,
please, start getting practice.
00:40:40.000 --> 00:40:46.000
Get up to snuff doing that
procedure.
00:40:42.000 --> 00:40:48.000
Okay, in the time remaining,
I want to add one formula to
00:40:46.000 --> 00:40:52.000
this list, and that is going to
be the Laplace transform of,
00:40:50.000 --> 00:40:56.000
we still haven't done
polynomials.
00:40:52.000 --> 00:40:58.000
And now, to polynomials,
because the Laplace transform
00:40:56.000 --> 00:41:02.000
is linear, all I have to do is
know what the Laplace transform
00:41:00.000 --> 00:41:06.000
of, the individual term of a
polynomial.
00:41:04.000 --> 00:41:10.000
In other words,
what the Laplace transform of t
00:41:07.000 --> 00:41:13.000
to the n,
where n is some positive
00:41:10.000 --> 00:41:16.000
integer?
Well, let's bravely start
00:41:13.000 --> 00:41:19.000
trying to calculate it.
Integral from zero to infinity
00:41:17.000 --> 00:41:23.000
t to the n e to the negative st
dt.
00:41:22.000 --> 00:41:28.000
Now, I think you can see that
00:41:26.000 --> 00:41:32.000
the method you should use is
integration by part because this
00:41:31.000 --> 00:41:37.000
is a product of two things,
one of which you would like to
00:41:35.000 --> 00:41:41.000
differentiate a lot of times,
in fact, and the other won't
00:41:40.000 --> 00:41:46.000
hurt to integrate it because
it's very easy to integrate.
00:41:44.000 --> 00:41:50.000
So, this factor is going to be
00:41:48.000 --> 00:41:54.000
the one that's to be
differentiated,
00:41:50.000 --> 00:41:56.000
and this is the factor that
will be pleased to integrate it.
00:41:54.000 --> 00:42:00.000
Let's get started and see what
we can get out of it.
00:41:57.000 --> 00:42:03.000
Well, this time I'm going to
be, well, I'd better be a little
00:42:01.000 --> 00:42:07.000
careful because there's a point
here that's tricky.
00:42:06.000 --> 00:42:12.000
Okay, the first step of
integration by parts is you only
00:42:10.000 --> 00:42:16.000
do the integration.
You don't do the
00:42:12.000 --> 00:42:18.000
differentiation.
Remember, the variable is t.
00:42:15.000 --> 00:42:21.000
The s is just a parameter.
It's just a constant.
00:42:19.000 --> 00:42:25.000
It's hanging around,
not knowing what to do.
00:42:22.000 --> 00:42:28.000
Okay, so the first step is you
don't do the differentiation.
00:42:26.000 --> 00:42:32.000
You only do the integration.
Evaluate it between limits,
00:42:30.000 --> 00:42:36.000
and then you put a minus sign
before you forget to do it.
00:42:36.000 --> 00:42:42.000
And then, integral zero to
infinity.
00:42:38.000 --> 00:42:44.000
Now you do both operations.
So, it's n t to the n
00:42:43.000 --> 00:42:49.000
minus one,
and you also do the
00:42:46.000 --> 00:42:52.000
integration.
Okay, let's consider each of
00:42:49.000 --> 00:42:55.000
these pieces in turn.
Now, this piece,
00:42:52.000 --> 00:42:58.000
well, there's no problem with
the lower limit,
00:42:55.000 --> 00:43:01.000
zero, because when t is equal
to zero, this factor is zero,
00:43:00.000 --> 00:43:06.000
and the thing disappears as
long as n is one or higher.
00:43:06.000 --> 00:43:12.000
So, it's minus zero here at the
lower limit.
00:43:10.000 --> 00:43:16.000
The question is,
what is at the upper limit?
00:43:15.000 --> 00:43:21.000
So, what I have to do is find
out, what is the limit?
00:43:20.000 --> 00:43:26.000
The limit, as t goes to
infinity, that's what's
00:43:25.000 --> 00:43:31.000
happening up there,
of t to the n times e to the
00:43:30.000 --> 00:43:36.000
negative s t divided
by minus s.
00:43:37.000 --> 00:43:43.000
Well, as t goes to infinity,
this goes to infinity,
00:43:40.000 --> 00:43:46.000
of course.
This had better go to zero
00:43:43.000 --> 00:43:49.000
unless I want an answer,
infinity, which won't do me any
00:43:46.000 --> 00:43:52.000
good.
If this goes to zero,
00:43:48.000 --> 00:43:54.000
s had better be positive.
So, I'd better be restricting
00:43:52.000 --> 00:43:58.000
myself to that case.
Okay, so let's assume that s is
00:43:56.000 --> 00:44:02.000
positive so that this minus s
really is a negative number.
00:44:01.000 --> 00:44:07.000
Okay, then I have a chance.
So, this is going to be the
00:44:04.000 --> 00:44:10.000
limit.
Let's write it in a more
00:44:06.000 --> 00:44:12.000
familiar form with that down
below.
00:44:08.000 --> 00:44:14.000
So, it's t to the n.
That's going to infinity.
00:44:12.000 --> 00:44:18.000
But, the bottom is e to the
minus s t.
00:44:15.000 --> 00:44:21.000
But now, it's plus s t.
And, that's going to infinity,
00:44:19.000 --> 00:44:25.000
too, because s is positive.
So, the two guys are racing,
00:44:23.000 --> 00:44:29.000
and the question is,
oh, I lost a minus s here.
00:44:26.000 --> 00:44:32.000
So, oh... equals minus
one over s.
00:44:30.000 --> 00:44:36.000
How's that?
So, the question is only,
00:44:33.000 --> 00:44:39.000
which guy wins?
In the race to infinity,
00:44:35.000 --> 00:44:41.000
which one wins,
and how do you decide?
00:44:38.000 --> 00:44:44.000
And, the answer,
of course, is that's the bottom
00:44:41.000 --> 00:44:47.000
that wins.
The exponential always wins,
00:44:43.000 --> 00:44:49.000
and it's because of L'Hopital's
rule.
00:44:45.000 --> 00:44:51.000
You differentiate top and
bottom.
00:44:47.000 --> 00:44:53.000
Nothing much happens to the
bottom.
00:44:49.000 --> 00:44:55.000
It gets another factor of s,
but the top goes down to t to
00:44:53.000 --> 00:44:59.000
the n minus one.
L'Hopital it again,
00:44:56.000 --> 00:45:02.000
and again, and again,
and again, and again until
00:44:59.000 --> 00:45:05.000
finally you've reduced the top
to t to the zero where it's
00:45:03.000 --> 00:45:09.000
defenseless and just sitting
there, and nothing's happened to
00:45:06.000 --> 00:45:12.000
the bottom.
It's still got e to the s t.
00:45:11.000 --> 00:45:17.000
and that goes to infinity.
00:45:14.000 --> 00:45:20.000
So, the answer is,
this is zero by n applications
00:45:19.000 --> 00:45:25.000
of L'Hopital's rule.
Or, if you're very clever,
00:45:23.000 --> 00:45:29.000
you can do it in one,
but I won't tell you how.
00:45:27.000 --> 00:45:33.000
So, the answer is that this is
zero.
00:45:32.000 --> 00:45:38.000
At the upper limit,
it's also zero at least if s is
00:45:35.000 --> 00:45:41.000
positive, which is the case
we're considering.
00:45:38.000 --> 00:45:44.000
That leaves the rest of this.
All right, let's pull the
00:45:41.000 --> 00:45:47.000
constants out front.
That's plus.
00:45:43.000 --> 00:45:49.000
Two negatives make a plus.
n over s,
00:45:46.000 --> 00:45:52.000
now, what's left?
The integral from zero to
00:45:49.000 --> 00:45:55.000
infinity of t to the n minus
one, e to the minus s t dt.
00:45:57.000 --> 00:46:03.000
But, what on Earth is that?
That is n over s times
00:46:00.000 --> 00:46:06.000
the Laplace transform of t to
the n minus one.
00:46:04.000 --> 00:46:10.000
We got a reduction for it.
We don't get the answer in one
00:46:08.000 --> 00:46:14.000
step.
But, we get a reduction
00:46:11.000 --> 00:46:17.000
formula.
And, it says that the Laplace
00:46:13.000 --> 00:46:19.000
transform, let me write it this
way for once.
00:46:16.000 --> 00:46:22.000
The first way is now better,
is equal to n over s times the
00:46:20.000 --> 00:46:26.000
Laplace transform of n minus t
to the n minus one.
00:46:24.000 --> 00:46:30.000
Okay, the next step,
00:46:27.000 --> 00:46:33.000
this would be n over s times n
minus one over s times the
00:46:30.000 --> 00:46:36.000
Laplace transform of t to the n
minus two.
00:46:34.000 --> 00:46:40.000
If I can continue,
00:46:38.000 --> 00:46:44.000
I finally get in the top n
times n minus one times all the
00:46:44.000 --> 00:46:50.000
way down to one divided by the
same number of s's,
00:46:48.000 --> 00:46:54.000
n of them, times the Laplace
transform of t to the zero,
00:46:53.000 --> 00:46:59.000
finally.
See, one, zero,
00:46:55.000 --> 00:47:01.000
n minus one.
And so, what's the final
00:46:59.000 --> 00:47:05.000
answer?
It is n factorial over s to the
00:47:02.000 --> 00:47:08.000
what power?
Well, the Laplace transform of
00:47:07.000 --> 00:47:13.000
this is one over s.
So, the answer is it's s to the
00:47:13.000 --> 00:47:19.000
n plus one,
n of them here plus an extra
00:47:18.000 --> 00:47:24.000
one coming from the one over s
here.
00:47:21.000 --> 00:47:27.000
And, that's the answer.
The Laplace transform of t to
00:47:26.000 --> 00:47:32.000
the n, oddly enough,
00:47:29.000 --> 00:47:35.000
is more complicated,
and looks a little different
00:47:33.000 --> 00:47:39.000
from these.
It's n factorial over s to the
00:47:37.000 --> 00:47:43.000
n plus one.
00:47:40.000 --> 00:47:46.000
And, with that,
you can now calculate the
00:47:43.000 --> 00:47:49.000
Laplace transform of anything in
sight, and tomorrow you will.