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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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dear hearts,
is this thing,
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which I stress is nothing more
than the continuous analog of
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the summation of a power series.
This is the discrete version.
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This is by these perfectly
natural transformations the
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continuous version of the same
thing.
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It starts with a function
defined for positive values of
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t, and turns it into a function
of s.
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And, this is called the Laplace
transform.
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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
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of t, and you get out a function
of s.
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How could that be?
You know, for an operator,
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you put in 3x,
and you get out three if it's a
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differentiation operator.
In other words,
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when you have an operator,
the things we've been talking
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about the last two or three
weeks in one form or another,
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at least the variable doesn't
get changed.
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Well, but for a transform it
does, and that's why it's called
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a transform.
So, the difference between a
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transform and an operator is
that for a transform a function
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of t comes in,
but a function of s comes out.
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The variable gets changed,
whereas for an operator,
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f of t goes in and
what comes out is g of t,
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a function using the
same variable like
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differentiation is a typical
example of an operator,
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or the linear differential
operators we've been talking
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about.
Well, but this doesn't behave
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that way.
The variable does get changed.
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That's, in fact,
extremely important in the
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applications.
In the applications,
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t usually means the time,
and s very often,
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not always, but very often is a
variable measuring frequency,
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for instance.
But, so that's a peculiar thing
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that's hard to get used to.
But, a good thing is the fact
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that it's a linear transform.
In other words,
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it obeys the laws we'd love and
like that the Laplace
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transform-- oh,
I never gave you any notation
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for the laplace transform.
Hey, I'd better do that.
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Okay, so, some notation:
there are two notations that
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are used.
Your book mostly uses the
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notation that the laplace
transform of f of t is capital F
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of s,
uses the same letter but with
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the same capital.
Now, as you will see,
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there are some places you
absolutely cannot use that
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notation.
It may seem strange,
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looks perfectly natural.
There are certain laws you
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cannot express using that
notation.
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It's baffling.
But, if you can't do it this
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way, you can do it using this
notation instead.
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One or the other will almost
always work.
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So, I'll use my little squiggly
notation, but that's what I use.
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I think it's a little more
vivid, and the trouble is that
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this piles up too many
parentheses.
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And, that's always hard to
read.
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So, I like this better.
So, these are two alternate
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ways of saying the same thing.
The Laplace transform of this
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function is that one.
Okay, well, let's use,
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00:15:36 --> 00:15:42
for the linearity law,
it's definitely best.
262
00:15:39 --> 00:15:45
I really cannot express the
linearity law using the second
263
00:15:44 --> 00:15:50
notation, but using the first
notation, it's a breeze.
264
00:15:49 --> 00:15:55
The Laplace transform of the
sum of two functions is the sum
265
00:15:54 --> 00:16:00
of their Laplace transforms of
each of them separately.
266
00:16:00 --> 00:16:06
Or, better yet,
you could write it that way.
267
00:16:03 --> 00:16:09
Let's write it this way.
That way, it looks more like an
268
00:16:06 --> 00:16:12
operator, L of f plus L of g.
269
00:16:10 --> 00:16:16
And, of the same way,
if you take a function and
270
00:16:13 --> 00:16:19
multiply it by a constant and
take the laplace transform,
271
00:16:17 --> 00:16:23
you can pull the constant
outside.
272
00:16:19 --> 00:16:25
And, of course,
why are these true?
273
00:16:22 --> 00:16:28
These are true just because of
the form of the transform.
274
00:16:25 --> 00:16:31
If I add up f and g,
I simply add up the two
275
00:16:29 --> 00:16:35
corresponding integrals.
In other words,
276
00:16:33 --> 00:16:39
I'm using the fact that the
integral, this definite
277
00:16:37 --> 00:16:43
integral, is itself a linear
operator.
278
00:16:40 --> 00:16:46
Well, that's the general
setting.
279
00:16:43 --> 00:16:49
That's where it comes from,
and that's the notation for it.
280
00:16:47 --> 00:16:53
And, now we have to get to
work.
281
00:16:50 --> 00:16:56
The first thing to do to get
familiar with this is,
282
00:16:54 --> 00:17:00
obviously what we want to do is
say, okay, these were the
283
00:16:59 --> 00:17:05
transforms of some simple
discreet functions.
284
00:17:04 --> 00:17:10
Okay, suppose I put in some
familiar functions,
285
00:17:09 --> 00:17:15
f of t.
What do their Laplace
286
00:17:14 --> 00:17:20
transforms look like?
So, let's do that.
287
00:17:19 --> 00:17:25
So, one of the boards I should
keep stored.
288
00:17:24 --> 00:17:30
Why don't I store on this
board?
289
00:17:28 --> 00:17:34
I'll store on this board the
formulas as we get them.
290
00:17:37 --> 00:17:43
So, let's see,
what should we aim at,
291
00:17:39 --> 00:17:45
first?
Let's first find,
292
00:17:41 --> 00:17:47
and I'll do the calculations on
the sideboard,
293
00:17:44 --> 00:17:50
and we'll see how it works out.
I'm not very sure.
294
00:17:47 --> 00:17:53
In other words,
what's the Laplace transform of
295
00:17:51 --> 00:17:57
the function,
one?
296
00:17:52 --> 00:17:58
Well, there's an even easier
one.
297
00:17:54 --> 00:18:00
What's the Laplace transform of
the function zero?
298
00:17:57 --> 00:18:03
Answer: zero.
Very exciting.
299
00:18:00 --> 00:18:06
What's the Laplace transform of
one?
300
00:18:03 --> 00:18:09
Well, it doesn't turn out the
constant anymore than it turned
301
00:18:07 --> 00:18:13
out to be a constant up there.
Let's calculate it.
302
00:18:11 --> 00:18:17
Now, you can do these
calculations carefully,
303
00:18:14 --> 00:18:20
dotting all the i's,
or pretty carefully,
304
00:18:17 --> 00:18:23
or not carefully at all,
i.e.
305
00:18:19 --> 00:18:25
sloppily.
I'll let you be sloppy after,
306
00:18:21 --> 00:18:27
generally speaking,
you could be sloppy unless the
307
00:18:25 --> 00:18:31
directions tell you to be less
sloppy or to be careful,
308
00:18:29 --> 00:18:35
okay?
So, I'll do one carefully.
309
00:18:32 --> 00:18:38
Let's calculate the Laplace
transform of one carefully.
310
00:18:36 --> 00:18:42
Okay, in the beginning,
you've got nothing to use with
311
00:18:40 --> 00:18:46
the definition.
So, I have to calculate the
312
00:18:43 --> 00:18:49
integral from zero to infinity
of one, that's the f of t times
313
00:18:47 --> 00:18:53
e to the negative s t,
so I don't have to
314
00:18:51 --> 00:18:57
put in the one,
dt.
315
00:18:52 --> 00:18:58
All right, now,
let me remind you,
316
00:18:54 --> 00:19:00
this is an improper integral.
This is just about the first
317
00:18:58 --> 00:19:04
time in the course we've had an
improper integral.
318
00:19:01 --> 00:19:07
But, there are going to be a
lot of them over the next couple
319
00:19:06 --> 00:19:12
of weeks, nothing but.
All right, it's an improper
320
00:19:10 --> 00:19:16
integral.
That means we have to go back
321
00:19:12 --> 00:19:18
to the definition.
If you want to be careful,
322
00:19:15 --> 00:19:21
you have to go back to the
definition of improper integral.
323
00:19:19 --> 00:19:25
So, it's the limit,
as R goes to infinity,
324
00:19:21 --> 00:19:27
of what you get by integrating
only up as far as R.
325
00:19:24 --> 00:19:30
That's a definite integral.
That's a nice Riemann integral.
326
00:19:27 --> 00:19:33
So, this is what I have to
calculate.
327
00:19:31 --> 00:19:37
And, I have to take the limit
as R goes to infinity.
328
00:19:34 --> 00:19:40
Now, how do I calculate that?
Well, this integral is equal
329
00:19:37 --> 00:19:43
to, that's easy.
It's just integrating.
330
00:19:40 --> 00:19:46
Remember that you're
integrating with respect to t.
331
00:19:43 --> 00:19:49
So, s is a parameter.
It's like a constant,
332
00:19:45 --> 00:19:51
in other words.
So, it's e to the minus s t,
333
00:19:48 --> 00:19:54
and when I differentiated,
334
00:19:50 --> 00:19:56
the derivative of this would
have negative s.
335
00:19:53 --> 00:19:59
So, to get rid of that negative
s, so the derivative is e to the
336
00:19:57 --> 00:20:03
minus s t.
You have to put minus s
337
00:20:00 --> 00:20:06
in the denominator.
And now, I'll want to evaluate
338
00:20:05 --> 00:20:11
that between zero and R.
And, what do I get?
339
00:20:09 --> 00:20:15
Well it is at the upper limit.
So, it's e to the minus s times
340
00:20:14 --> 00:20:20
R minus, at the lower limit,
it's t is equal to zero,
341
00:20:19 --> 00:20:25
so whatever s is,
it's one.
342
00:20:21 --> 00:20:27
And that's divided by this
constant up front,
343
00:20:25 --> 00:20:31
negative s. So,
344
00:20:28 --> 00:20:34
the answer is,
it is equal to the limit of,
345
00:20:32 --> 00:20:38
as R goes to infinity,
of e to the negative s R minus
346
00:20:37 --> 00:20:43
one divided by minus s.
347
00:20:43 --> 00:20:49
Now, what's that?
Well, as R goes to infinity,
348
00:20:47 --> 00:20:53
e to the minus 2R,
or minus 5R goes to zero,
349
00:20:52 --> 00:20:58
and the answer is minus one
over minus s.
350
00:20:57 --> 00:21:03
So, that's one over s.
And so, that's our answer.
351
00:21:02 --> 00:21:08
Let's put it up here.
It's one over s,
352
00:21:05 --> 00:21:11
except it isn't.
I made a mistake.
353
00:21:10 --> 00:21:16
Well, not mistake,
a little oversight.
354
00:21:16 --> 00:21:22
What's the oversight?
This is okay.
355
00:21:22 --> 00:21:28
This is okay.
This is okay.
356
00:21:26 --> 00:21:32
This is not okay.
This is okay.
357
00:21:31 --> 00:21:37
But that's not okay.
What's wrong?
358
00:21:38 --> 00:21:44
I did slight a verbal hand.
Maybe some of you have picked
359
00:21:41 --> 00:21:47
it up and were too embarrassed
to correct me,
360
00:21:44 --> 00:21:50
but I said like e to the minus
2R obviously goes to
361
00:21:48 --> 00:21:54
zero, and e to the minus 5R
goes to zero.
362
00:21:51 --> 00:21:57
How about e to the minus minus
3 R?
363
00:21:54 --> 00:22:00
Does that go to zero?
No, that's e to the 3R,
364
00:21:57 --> 00:22:03
which goes to infinity.
365
00:22:00 --> 00:22:06
The only time this goes to zero
is if s is a positive number.
366
00:22:05 --> 00:22:11
Minus s looks like a negative
number, but it's not,
367
00:22:10 --> 00:22:16
if s is equal to minus two.
So, this is only true if s is
368
00:22:16 --> 00:22:22
positive because only if s is
positive is this exponent really
369
00:22:22 --> 00:22:28
negative and large,
and therefore going to
370
00:22:26 --> 00:22:32
infinity, going to zero as R
goes to infinity.
371
00:22:30 --> 00:22:36
So, the answer is not one over
s.
372
00:22:34 --> 00:22:40
It is one over s,
s must positive.
373
00:22:39 --> 00:22:45
Now, once again,
here, people don't worry about
374
00:22:41 --> 00:22:47
this sort of thing with power
series because it seems very
375
00:22:45 --> 00:22:51
obvious, you know,
one over x,
376
00:22:48 --> 00:22:54
absolute value of x is less
than one,
377
00:22:51 --> 00:22:57
when it gets to be the Laplace
transform, just because the
378
00:22:54 --> 00:23:00
Laplace transform is mysterious,
the question is,
379
00:22:58 --> 00:23:04
okay, the Laplace transform is
one over s of one,
380
00:23:01 --> 00:23:07
well, Laplace transform of one
I understand is one over s if s
381
00:23:05 --> 00:23:11
is positive.
What is it if s is negative?
382
00:23:09 --> 00:23:15
Okay, right down in your little
books, this, but that down,
383
00:23:15 --> 00:23:21
what is it if s is negative,
and write underneath that,
384
00:23:21 --> 00:23:27
this question is meaningless.
It doesn't mean anything.
385
00:23:26 --> 00:23:32
I'll draw you a picture.
This is a picture of the
386
00:23:31 --> 00:23:37
Laplace transform of one.
It is that.
387
00:23:34 --> 00:23:40
It's one branch of this curve.
It does not include the branch
388
00:23:39 --> 00:23:45
on the left.
It doesn't because I showed you
389
00:23:43 --> 00:23:49
it doesn't.
That's all there is to it.
390
00:23:46 --> 00:23:52
Okay, so I did that carefully.
Now I'm going to get a little
391
00:23:50 --> 00:23:56
less careful.
What's the Laplace transform of
392
00:23:54 --> 00:24:00
e to the a t?
First of all,
393
00:23:57 --> 00:24:03
in general, the kind of
functions for which people like
394
00:24:01 --> 00:24:07
to calculate the Laplace
transform, and basically the
395
00:24:06 --> 00:24:12
only ones there will be in the
tables are exactly the sort of
396
00:24:10 --> 00:24:16
functions that you used in
solving linear equations with
397
00:24:15 --> 00:24:21
constant coefficients.
What kinds of functions entered
398
00:24:21 --> 00:24:27
in there?
Exponentials,
399
00:24:22 --> 00:24:28
sines and cosines,
but they were really complex
400
00:24:25 --> 00:24:31
exponentials,
right?
401
00:24:26 --> 00:24:32
e to the t sine t,
but that was really a
402
00:24:30 --> 00:24:36
complex exponential,
too, just a little more
403
00:24:33 --> 00:24:39
complicated one,
polynomials,
404
00:24:35 --> 00:24:41
and that's about it.
t times e to the t,
405
00:24:38 --> 00:24:44
that was okay,
too.
406
00:24:41 --> 00:24:47
These are the functions for
which people calculate the
407
00:24:44 --> 00:24:50
Laplace transform,
and all the other functions
408
00:24:46 --> 00:24:52
they don't calculate the Laplace
transforms.
409
00:24:49 --> 00:24:55
So, I don't mean to disappoint
you here.
410
00:24:52 --> 00:24:58
You're going to say,
oh, what, that same old stuff?
411
00:24:55 --> 00:25:01
For two more weeks,
we've got that same,
412
00:24:57 --> 00:25:03
well, the Laplace transform
does a lot of things much better
413
00:25:01 --> 00:25:07
than the methods we've been
using.
414
00:25:04 --> 00:25:10
And, I won't.
I'll sell it when I get a
415
00:25:06 --> 00:25:12
chance to, for now,
let's just get familiar with
416
00:25:09 --> 00:25:15
it.
All right, so while I'm not
417
00:25:11 --> 00:25:17
going to calculate e to the a t
for you,
418
00:25:14 --> 00:25:20
because I'd like instead to
just prove a simple formula
419
00:25:17 --> 00:25:23
which will just give that,
and will also give us e to the
420
00:25:21 --> 00:25:27
a t sine t.
It will give us a lot more,
421
00:25:24 --> 00:25:30
instead.
I'm going to calculate a
422
00:25:27 --> 00:25:33
formula for the Laplace
transform of this guy if you
423
00:25:30 --> 00:25:36
already know the Laplace
transform of it.
424
00:25:34 --> 00:25:40
Now, see, this falls in that
category because this is really
425
00:25:38 --> 00:25:44
e to the a t times one.
426
00:25:41 --> 00:25:47
But, I already know the Laplace
transform of one.
427
00:25:44 --> 00:25:50
So that's, if I can get a
general formula for this,
428
00:25:48 --> 00:25:54
I'll be able to get the formula
for e to the a t as a
429
00:25:53 --> 00:25:59
consequence.
So, let's look for this Laplace
430
00:25:56 --> 00:26:02
transform.
Now, it's really easy.
431
00:25:59 --> 00:26:05
Let's see, where am I doing
calculations?
432
00:26:02 --> 00:26:08
Over here.
Okay, so we've got e.
433
00:26:05 --> 00:26:11
So, I want to calculate the
Laplace transform e to the a t f
434
00:26:09 --> 00:26:15
of t.
So I'm going to say that's the
435
00:26:13 --> 00:26:19
integral from zero to infinity
of e to the a t times f of t.
436
00:26:16 --> 00:26:22
And now, the rest I copy.
437
00:26:19 --> 00:26:25
That's the function part of it
that goes to the input,
438
00:26:23 --> 00:26:29
and then there's the other
part.
439
00:26:25 --> 00:26:31
This part is called the kernel,
by the way, but don't worry
440
00:26:29 --> 00:26:35
about that.
However, if you drop it in
441
00:26:33 --> 00:26:39
conversation,
people will look at you and
442
00:26:36 --> 00:26:42
say, gee, they know something I
don't.
443
00:26:39 --> 00:26:45
And you will.
You know that it's the kernel.
444
00:26:43 --> 00:26:49
Okay, well, now,
what kind of formula can I be
445
00:26:47 --> 00:26:53
looking for?
Clearly, I can only be looking
446
00:26:51 --> 00:26:57
for a formula which expresses it
in terms of the Laplace
447
00:26:56 --> 00:27:02
transform of f of t.
Let's calculate and see what we
448
00:27:02 --> 00:27:08
get.
Now, what would you do to that
449
00:27:04 --> 00:27:10
thing to make?
Well, obviously,
450
00:27:06 --> 00:27:12
the thing to do is to combine
the two exponentials.
451
00:27:09 --> 00:27:15
So, that's going to be the
integral from zero to infinity
452
00:27:13 --> 00:27:19
of f of t.
e, now, I'd like to put it,
453
00:27:16 --> 00:27:22
to combine the exponentials in
such a way that it has,
454
00:27:19 --> 00:27:25
still, that same form,
so, I'm going to begin with
455
00:27:23 --> 00:27:29
that negative sign,
and then see what the rest of
456
00:27:26 --> 00:27:32
it has to be.
What is it going to be?
457
00:27:30 --> 00:27:36
Well, minus s t and
plus a t,
458
00:27:34 --> 00:27:40
but I can make that minus a
here, and it will come out
459
00:27:39 --> 00:27:45
right.
So, it's minus s t plus a t,
460
00:27:42 --> 00:27:48
and there are the two parts,
461
00:27:46 --> 00:27:52
those two factors,
dt.
462
00:27:48 --> 00:27:54
So, what's that?
That's the Laplace transform.
463
00:27:52 --> 00:27:58
If the a weren't there,
this would be the Laplace
464
00:27:56 --> 00:28:02
transform of f of t.
What is it with the a there?
465
00:28:03 --> 00:28:09
It's the Laplace transform of f
of t,
466
00:28:08 --> 00:28:14
except that instead of the
variable, s has been replaced by
467
00:28:15 --> 00:28:21
the variable s minus a.
468
00:28:19 --> 00:28:25
I'll give you a second to
digest that.
469
00:28:24 --> 00:28:30
Well, you digest it while I'm
writing it because that's the
470
00:28:30 --> 00:28:36
answer.
And, the way this is most often
471
00:28:35 --> 00:28:41
used, I have to qualify it for
the value.
472
00:28:38 --> 00:28:44
So, if F of s is good
for s positive,
473
00:28:42 --> 00:28:48
the way it would be,
for example,
474
00:28:45 --> 00:28:51
if I used the function one
here, then to finish that off,
475
00:28:50 --> 00:28:56
then, F of s minus a will be,
476
00:28:53 --> 00:28:59
this will be good when s is
bigger than a.
477
00:28:57 --> 00:29:03
Why is that?
Well, because this is true.
478
00:29:02 --> 00:29:08
This is true.
If s minus a is
479
00:29:06 --> 00:29:12
positive, that's the condition.
That's what this Laplace
480
00:29:11 --> 00:29:17
transform is good.
But that simply says that s
481
00:29:15 --> 00:29:21
should be bigger than a.
482
00:29:18 --> 00:29:24
And, since this doesn't look
pretty, let me try to make it
483
00:29:23 --> 00:29:29
look a little bit prettier.
So, let's write it.
484
00:29:29 --> 00:29:35
So, this is assuming F of s is
for s greater than zero.
485
00:29:34 --> 00:29:40
Now, this is called something.
486
00:29:38 --> 00:29:44
This is called,
well, what would you call it?
487
00:29:43 --> 00:29:49
On the left side,
you multiply by an exponential.
488
00:29:47 --> 00:29:53
On the right,
you translate.
489
00:29:50 --> 00:29:56
You shift the argument over by
a.
490
00:29:53 --> 00:29:59
So, this is called,
gulp, the exponential shift.
491
00:29:58 --> 00:30:04
What?
Well, I'll call it the formula.
492
00:30:02 --> 00:30:08
The thing before,
when we talked about operators,
493
00:30:06 --> 00:30:12
we called it the exponential
shift rule or the exponential
494
00:30:10 --> 00:30:16
shift law.
But, in fact,
495
00:30:12 --> 00:30:18
this is, in a way,
a disguised form of the same
496
00:30:16 --> 00:30:22
law.
And, engineers who typically do
497
00:30:19 --> 00:30:25
all their work using the Laplace
transform and don't use
498
00:30:23 --> 00:30:29
operators, this is the form of
the exponential shift law that
499
00:30:28 --> 00:30:34
they would know.
What you can do with one,
500
00:30:33 --> 00:30:39
you can do with the other.
You can now use both.
501
00:30:37 --> 00:30:43
So, what's the answer to e to
the a t?
502
00:30:41 --> 00:30:47
Well, the answer is,
I'm supposed to,
503
00:30:44 --> 00:30:50
e to the a t times one,
the Laplace transform of one is
504
00:30:48 --> 00:30:54
one over s.
And, therefore,
505
00:30:51 --> 00:30:57
what I do is to multiply by e
to the a t, I change s to s
506
00:30:56 --> 00:31:02
minus a .
And so, that's the answer.
507
00:31:00 --> 00:31:06
Let's see, what else don't we
know?
508
00:31:05 --> 00:31:11
Well, how about sines and
cosines?
509
00:31:08 --> 00:31:14
Well, the way to do sines and
cosines is by making the
510
00:31:15 --> 00:31:21
observation that this formula
also works when a is a complex
511
00:31:22 --> 00:31:28
number.
So, can use also for a a
512
00:31:26 --> 00:31:32
complex number, for e to
the a plus b i times t.
513
00:31:31 --> 00:31:37
The Laplace transform of e to
514
00:31:37 --> 00:31:43
the a plus b i times t is one
over s minus a plus b i.
515
00:31:40 --> 00:31:46
516
00:31:43 --> 00:31:49
And again, it will be for s
bigger than a.
517
00:31:47 --> 00:31:53
So, let's calculate the Laplace
transform of,
518
00:31:50 --> 00:31:56
let's say, well,
I've got to cover up something.
519
00:31:54 --> 00:32:00
Okay, so, that's the Laplace
transform.
520
00:31:57 --> 00:32:03
I've got to remember that.
So, let's calculate the Laplace
521
00:32:02 --> 00:32:08
transform of,
let's say, sine of a t
522
00:32:05 --> 00:32:11
and cosine a t.
523
00:32:08 --> 00:32:14
What do you get for that?
Well, just for a little
524
00:32:12 --> 00:32:18
variety, we could do it by using
that formula,
525
00:32:15 --> 00:32:21
and taking its real and
imaginary parts.
526
00:32:18 --> 00:32:24
Since some of you had so much
difficulty with the backwards
527
00:32:23 --> 00:32:29
Euler formula,
he is a good case where you
528
00:32:26 --> 00:32:32
could use it.
Suppose you want to calculate
529
00:32:29 --> 00:32:35
the Laplace transform of cosine
a t.
530
00:32:35 --> 00:32:41
Well, I'm going to write that
using, I want to calculate using
531
00:32:39 --> 00:32:45
complex exponentials.
The way I will do it is by
532
00:32:43 --> 00:32:49
using the backwards Euler
formula.
533
00:32:45 --> 00:32:51
So, this is e to the i a t plus
e to the minus i a t divided by
534
00:32:50 --> 00:32:56
two.
535
00:32:52 --> 00:32:58
Remember, the foreword Euler
formula would say e to the i a t
536
00:32:57 --> 00:33:03
equals cosine a t plus i sine a
t.
537
00:33:01 --> 00:33:07
That expresses the complex
538
00:33:04 --> 00:33:10
exponential in terms of sines
and cosines.
539
00:33:07 --> 00:33:13
This is the backward formula,
which just read it backwards,
540
00:33:11 --> 00:33:17
expressing cosines and sines in
terms of complex exponentials
541
00:33:15 --> 00:33:21
instead.
Both formulas are useful,
542
00:33:17 --> 00:33:23
almost equally useful,
in fact.
543
00:33:19 --> 00:33:25
And anyway, just remind you of
it, let's use this one.
544
00:33:23 --> 00:33:29
Okay, what's the Laplace
transform, then,
545
00:33:26 --> 00:33:32
of cosine a t?
Well, by linearity,
546
00:33:30 --> 00:33:36
it's equal to one half the
Laplace transform of this guy
547
00:33:35 --> 00:33:41
plus the Laplace transform of
that guy.
548
00:33:38 --> 00:33:44
And, what are those?
Well, the Laplace transform of
549
00:33:43 --> 00:33:49
e to the i a t is one over s
minus i a,
550
00:33:48 --> 00:33:54
and the Laplace
transform of the other guy is
551
00:33:53 --> 00:33:59
one divided by s plus i a.
552
00:33:56 --> 00:34:02
Now, of course,
this has become out to be a
553
00:34:00 --> 00:34:06
real function.
This is real.
554
00:34:03 --> 00:34:09
Every integral is real.
This must come out to be real.
555
00:34:07 --> 00:34:13
This looks kind of complex,
but it isn't.
556
00:34:10 --> 00:34:16
I know automatically that this
is going to be a real function.
557
00:34:15 --> 00:34:21
How I know that?
Well, mentally,
558
00:34:17 --> 00:34:23
you can combine the terms and
calculate.
559
00:34:20 --> 00:34:26
But, I know even before that.
Remember, there are two ways to
560
00:34:24 --> 00:34:30
see that something is real.
You can calculate it and see
561
00:34:28 --> 00:34:34
that its imaginary part is zero,
hack, or without any
562
00:34:32 --> 00:34:38
calculation, if you change i to
minus i,
563
00:34:36 --> 00:34:42
and you get the same thing,
it must be real.
564
00:34:41 --> 00:34:47
Now, if I change i to minus i
in this expression,
565
00:34:45 --> 00:34:51
what happens?
If I change i to minus i,
566
00:34:48 --> 00:34:54
this term turns into that one,
and this one turns into that
567
00:34:54 --> 00:35:00
one.
Conclusion: the sum of the two
568
00:34:57 --> 00:35:03
is unchanged.
And therefore,
569
00:34:59 --> 00:35:05
this is real.
Well, of course,
570
00:35:03 --> 00:35:09
in the time I took to make that
argument, I could have actually
571
00:35:07 --> 00:35:13
calculated it.
So, what the heck,
572
00:35:09 --> 00:35:15
let's calculate it?
So, you do the high school
573
00:35:12 --> 00:35:18
thing, and it's this guy plus
that guy on top,
574
00:35:15 --> 00:35:21
which makes 2s.
I on the bottom is the product
575
00:35:18 --> 00:35:24
of those, which by now you
should know the product of two
576
00:35:22 --> 00:35:28
complex numbers.
A product of a number and its
577
00:35:25 --> 00:35:31
complex conjugate is the sum of
the squares.
578
00:35:29 --> 00:35:35
So, what's the answer?
The twos cancel,
579
00:35:31 --> 00:35:37
and the answer is that the
Laplace transform of cosine a t
580
00:35:35 --> 00:35:41
is s over s squared plus a
squared.
581
00:35:39 --> 00:35:45
And, that will be true as,
582
00:35:41 --> 00:35:47
in general, it's true up there
for positive values of s only.
583
00:35:46 --> 00:35:52
And, the sine a t,
you can calculate that in
584
00:35:49 --> 00:35:55
recitation tomorrow.
The answer to that is a divided
585
00:35:53 --> 00:35:59
by s squared plus a squared.
586
00:35:56 --> 00:36:02
You would get the same answers
if you took the real and
587
00:36:00 --> 00:36:06
imaginary parts of that
expression.
588
00:36:04 --> 00:36:10
It's another way of getting at
the recitations tomorrow;
589
00:36:08 --> 00:36:14
we'll get practice in
calculating other functions
590
00:36:11 --> 00:36:17
related to these by using these
formulas, and also from scratch
591
00:36:16 --> 00:36:22
directly from the definition of
the Laplace transform.
592
00:36:20 --> 00:36:26
Well, there are two things
which we still should do.
593
00:36:24 --> 00:36:30
The first is I want to get you
started with calculating inverse
594
00:36:29 --> 00:36:35
Laplace transforms.
And, the reason for doing that
595
00:36:33 --> 00:36:39
is, in other words,
I've started with f of t,
596
00:36:36 --> 00:36:42
and we've been focusing
on what is capital F of s?
597
00:36:40 --> 00:36:46
But, you will find that when
598
00:36:42 --> 00:36:48
you go to solve differential
equations, by far,
599
00:36:45 --> 00:36:51
the hardest part of the
procedure is you get F of s.
600
00:36:49 --> 00:36:55
The Laplace transform of the
answer, and you have to convert
601
00:36:52 --> 00:36:58
that back into the answer in
terms of t that you were looking
602
00:36:56 --> 00:37:02
for.
In other words,
603
00:36:58 --> 00:37:04
the main step in the procedure
that you are going to be using
604
00:37:01 --> 00:37:07
for solving differential
equations is,
605
00:37:04 --> 00:37:10
and the hardest part of the
step will be to calculate
606
00:37:07 --> 00:37:13
inverse laplace transforms.
Now, you think that could be
607
00:37:13 --> 00:37:19
done by tables,
but, in fact,
608
00:37:15 --> 00:37:21
it can't unless the tables are
too long to be useful.
609
00:37:19 --> 00:37:25
You have to do a certain amount
of work yourself.
610
00:37:23 --> 00:37:29
And, the certain amount of work
that you have to do yourself
611
00:37:28 --> 00:37:34
involves partial fractions
decompositions.
612
00:37:33 --> 00:37:39
And, in case you were wondering
which you are not,
613
00:37:36 --> 00:37:42
the reason you learned partial
fractions in 18.01 was not to
614
00:37:41 --> 00:37:47
learn those silly integrals,
but he learned it so that when
615
00:37:45 --> 00:37:51
you got to 18.03 you would be
able to calculate,
616
00:37:48 --> 00:37:54
solve differential equations by
using Laplace transforms.
617
00:37:53 --> 00:37:59
Sorry.
That's life.
618
00:37:54 --> 00:38:00
Now, so a certain amount of the
recitation time tomorrow will be
619
00:37:59 --> 00:38:05
devoted to reminding you how to
do partial fractions since you
620
00:38:03 --> 00:38:09
haven't done it in a while,
and I assume,
621
00:38:06 --> 00:38:12
yeah, we had that,
I think.
622
00:38:10 --> 00:38:16
Okay, now, they also remind you
of the most efficient method,
623
00:38:16 --> 00:38:22
which about half of you have
had, and the rest think you
624
00:38:23 --> 00:38:29
might have had,
but really aren't sure.
625
00:38:27 --> 00:38:33
So, here's the answer.
We want to find out what it's
626
00:38:33 --> 00:38:39
inverse Laplace transform is.
What you have to do,
627
00:38:39 --> 00:38:45
it normally won't be in the
tables like this.
628
00:38:42 --> 00:38:48
You have to put it in a form in
which it will be in the tables.
629
00:38:46 --> 00:38:52
As you do that,
you have to make partial
630
00:38:48 --> 00:38:54
fractions decompositions,
which, to do it quickly,
631
00:38:51 --> 00:38:57
so if you don't know what I'm
doing now, or you think you once
632
00:38:55 --> 00:39:01
knew but don't quite remember,
go to recitation tomorrow.
633
00:38:59 --> 00:39:05
To get the coefficient here,
I cover up s,
634
00:39:02 --> 00:39:08
and I put s equals zero
because that's the law.
635
00:39:07 --> 00:39:13
To get this coefficient,
I cover up s plus three and I
636
00:39:10 --> 00:39:16
put s equals a negative three
because that's what you're
637
00:39:14 --> 00:39:20
supposed to do.
Put s equal negative three,
638
00:39:17 --> 00:39:23
you get minus one third.
This is equal to that.
639
00:39:20 --> 00:39:26
In this form,
I don't know what the inverse
640
00:39:23 --> 00:39:29
Laplace form is,
but in this form,
641
00:39:25 --> 00:39:31
I certainly do know with the
inverse Laplace transform
642
00:39:29 --> 00:39:35
because the inverse Laplace
transform is linear,
643
00:39:32 --> 00:39:38
and because each of these guys
especially occurs in those
644
00:39:36 --> 00:39:42
tables.
Well, what's this?
645
00:39:40 --> 00:39:46
Well, it's whatever the Laplace
transform of,
646
00:39:43 --> 00:39:49
inverse Laplace transform of
one over s is multiplied by one
647
00:39:49 --> 00:39:55
third.
Well, the inverse Laplace
648
00:39:51 --> 00:39:57
transform of one over s is one.
So, it's one third times one.
649
00:39:56 --> 00:40:02
How about the other guy?
Minus one third,
650
00:40:00 --> 00:40:06
the inverse Laplace transform
of one over s plus three,
651
00:40:05 --> 00:40:11
that's this formula.
a is negative three,
652
00:40:09 --> 00:40:15
and that makes e to the minus
3t.
653
00:40:13 --> 00:40:19
So, if this was the Laplace
transform of the solution to the
654
00:40:18 --> 00:40:24
differential equation,
then the solution in terms of t
655
00:40:22 --> 00:40:28
was this function.
Now, you'll get lots of
656
00:40:26 --> 00:40:32
practice in that.
All I'm doing now is signaling
657
00:40:30 --> 00:40:36
that that's the most important
and difficult step of the
658
00:40:34 --> 00:40:40
procedure, and that,
please, start getting practice.
659
00:40:40 --> 00:40:46
Get up to snuff doing that
procedure.
660
00:40:42 --> 00:40:48
Okay, in the time remaining,
I want to add one formula to
661
00:40:46 --> 00:40:52
this list, and that is going to
be the Laplace transform of,
662
00:40:50 --> 00:40:56
we still haven't done
polynomials.
663
00:40:52 --> 00:40:58
And now, to polynomials,
because the Laplace transform
664
00:40:56 --> 00:41:02
is linear, all I have to do is
know what the Laplace transform
665
00:41:00 --> 00:41:06
of, the individual term of a
polynomial.
666
00:41:04 --> 00:41:10
In other words,
what the Laplace transform of t
667
00:41:07 --> 00:41:13
to the n,
where n is some positive
668
00:41:10 --> 00:41:16
integer?
Well, let's bravely start
669
00:41:13 --> 00:41:19
trying to calculate it.
Integral from zero to infinity
670
00:41:17 --> 00:41:23
t to the n e to the negative st
dt.
671
00:41:22 --> 00:41:28
Now, I think you can see that
672
00:41:26 --> 00:41:32
the method you should use is
integration by part because this
673
00:41:31 --> 00:41:37
is a product of two things,
one of which you would like to
674
00:41:35 --> 00:41:41
differentiate a lot of times,
in fact, and the other won't
675
00:41:40 --> 00:41:46
hurt to integrate it because
it's very easy to integrate.
676
00:41:44 --> 00:41:50
So, this factor is going to be
677
00:41:48 --> 00:41:54
the one that's to be
differentiated,
678
00:41:50 --> 00:41:56
and this is the factor that
will be pleased to integrate it.
679
00:41:54 --> 00:42:00
Let's get started and see what
we can get out of it.
680
00:41:57 --> 00:42:03
Well, this time I'm going to
be, well, I'd better be a little
681
00:42:01 --> 00:42:07
careful because there's a point
here that's tricky.
682
00:42:06 --> 00:42:12
Okay, the first step of
integration by parts is you only
683
00:42:10 --> 00:42:16
do the integration.
You don't do the
684
00:42:12 --> 00:42:18
differentiation.
Remember, the variable is t.
685
00:42:15 --> 00:42:21
The s is just a parameter.
It's just a constant.
686
00:42:19 --> 00:42:25
It's hanging around,
not knowing what to do.
687
00:42:22 --> 00:42:28
Okay, so the first step is you
don't do the differentiation.
688
00:42:26 --> 00:42:32
You only do the integration.
Evaluate it between limits,
689
00:42:30 --> 00:42:36
and then you put a minus sign
before you forget to do it.
690
00:42:36 --> 00:42:42
And then, integral zero to
infinity.
691
00:42:38 --> 00:42:44
Now you do both operations.
So, it's n t to the n
692
00:42:43 --> 00:42:49
minus one,
and you also do the
693
00:42:46 --> 00:42:52
integration.
Okay, let's consider each of
694
00:42:49 --> 00:42:55
these pieces in turn.
Now, this piece,
695
00:42:52 --> 00:42:58
well, there's no problem with
the lower limit,
696
00:42:55 --> 00:43:01
zero, because when t is equal
to zero, this factor is zero,
697
00:43:00 --> 00:43:06
and the thing disappears as
long as n is one or higher.
698
00:43:06 --> 00:43:12
So, it's minus zero here at the
lower limit.
699
00:43:10 --> 00:43:16
The question is,
what is at the upper limit?
700
00:43:15 --> 00:43:21
So, what I have to do is find
out, what is the limit?
701
00:43:20 --> 00:43:26
The limit, as t goes to
infinity, that's what's
702
00:43:25 --> 00:43:31
happening up there,
of t to the n times e to the
703
00:43:30 --> 00:43:36
negative s t divided
by minus s.
704
00:43:37 --> 00:43:43
Well, as t goes to infinity,
this goes to infinity,
705
00:43:40 --> 00:43:46
of course.
This had better go to zero
706
00:43:43 --> 00:43:49
unless I want an answer,
infinity, which won't do me any
707
00:43:46 --> 00:43:52
good.
If this goes to zero,
708
00:43:48 --> 00:43:54
s had better be positive.
So, I'd better be restricting
709
00:43:52 --> 00:43:58
myself to that case.
Okay, so let's assume that s is
710
00:43:56 --> 00:44:02
positive so that this minus s
really is a negative number.
711
00:44:01 --> 00:44:07
Okay, then I have a chance.
So, this is going to be the
712
00:44:04 --> 00:44:10
limit.
Let's write it in a more
713
00:44:06 --> 00:44:12
familiar form with that down
below.
714
00:44:08 --> 00:44:14
So, it's t to the n.
That's going to infinity.
715
00:44:12 --> 00:44:18
But, the bottom is e to the
minus s t.
716
00:44:15 --> 00:44:21
But now, it's plus s t.
And, that's going to infinity,
717
00:44:19 --> 00:44:25
too, because s is positive.
So, the two guys are racing,
718
00:44:23 --> 00:44:29
and the question is,
oh, I lost a minus s here.
719
00:44:26 --> 00:44:32
So, oh... equals minus
one over s.
720
00:44:30 --> 00:44:36
How's that?
So, the question is only,
721
00:44:33 --> 00:44:39
which guy wins?
In the race to infinity,
722
00:44:35 --> 00:44:41
which one wins,
and how do you decide?
723
00:44:38 --> 00:44:44
And, the answer,
of course, is that's the bottom
724
00:44:41 --> 00:44:47
that wins.
The exponential always wins,
725
00:44:43 --> 00:44:49
and it's because of L'Hopital's
rule.
726
00:44:45 --> 00:44:51
You differentiate top and
bottom.
727
00:44:47 --> 00:44:53
Nothing much happens to the
bottom.
728
00:44:49 --> 00:44:55
It gets another factor of s,
but the top goes down to t to
729
00:44:53 --> 00:44:59
the n minus one.
L'Hopital it again,
730
00:44:56 --> 00:45:02
and again, and again,
and again, and again until
731
00:44:59 --> 00:45:05
finally you've reduced the top
to t to the zero where it's
732
00:45:03 --> 00:45:09
defenseless and just sitting
there, and nothing's happened to
733
00:45:06 --> 00:45:12
the bottom.
It's still got e to the s t.
734
00:45:11 --> 00:45:17
and that goes to infinity.
735
00:45:14 --> 00:45:20
So, the answer is,
this is zero by n applications
736
00:45:19 --> 00:45:25
of L'Hopital's rule.
Or, if you're very clever,
737
00:45:23 --> 00:45:29
you can do it in one,
but I won't tell you how.
738
00:45:27 --> 00:45:33
So, the answer is that this is
zero.
739
00:45:32 --> 00:45:38
At the upper limit,
it's also zero at least if s is
740
00:45:35 --> 00:45:41
positive, which is the case
we're considering.
741
00:45:38 --> 00:45:44
That leaves the rest of this.
All right, let's pull the
742
00:45:41 --> 00:45:47
constants out front.
That's plus.
743
00:45:43 --> 00:45:49
Two negatives make a plus.
n over s,
744
00:45:46 --> 00:45:52
now, what's left?
The integral from zero to
745
00:45:49 --> 00:45:55
infinity of t to the n minus
one, e to the minus s t dt.
746
00:45:53 --> 00:45:59
747
00:45:57 --> 00:46:03
But, what on Earth is that?
That is n over s times
748
00:46:00 --> 00:46:06
the Laplace transform of t to
the n minus one.
749
00:46:04 --> 00:46:10
We got a reduction for it.
We don't get the answer in one
750
00:46:08 --> 00:46:14
step.
But, we get a reduction
751
00:46:11 --> 00:46:17
formula.
And, it says that the Laplace
752
00:46:13 --> 00:46:19
transform, let me write it this
way for once.
753
00:46:16 --> 00:46:22
The first way is now better,
is equal to n over s times the
754
00:46:20 --> 00:46:26
Laplace transform of n minus t
to the n minus one.
755
00:46:24 --> 00:46:30
Okay, the next step,
756
00:46:27 --> 00:46:33
this would be n over s times n
minus one over s times the
757
00:46:30 --> 00:46:36
Laplace transform of t to the n
minus two.
758
00:46:34 --> 00:46:40
If I can continue,
759
00:46:38 --> 00:46:44
I finally get in the top n
times n minus one times all the
760
00:46:44 --> 00:46:50
way down to one divided by the
same number of s's,
761
00:46:48 --> 00:46:54
n of them, times the Laplace
transform of t to the zero,
762
00:46:53 --> 00:46:59
finally.
See, one, zero,
763
00:46:55 --> 00:47:01
n minus one.
And so, what's the final
764
00:46:59 --> 00:47:05
answer?
It is n factorial over s to the
765
00:47:02 --> 00:47:08
what power?
Well, the Laplace transform of
766
00:47:07 --> 00:47:13
this is one over s.
So, the answer is it's s to the
767
00:47:13 --> 00:47:19
n plus one,
n of them here plus an extra
768
00:47:18 --> 00:47:24
one coming from the one over s
here.
769
00:47:21 --> 00:47:27
And, that's the answer.
The Laplace transform of t to
770
00:47:26 --> 00:47:32
the n, oddly enough,
771
00:47:29 --> 00:47:35
is more complicated,
and looks a little different
772
00:47:33 --> 00:47:39
from these.
It's n factorial over s to the
773
00:47:37 --> 00:47:43
n plus one.
774
00:47:40 --> 00:47:46
And, with that,
you can now calculate the
775
00:47:43 --> 00:47:49
Laplace transform of anything in
sight, and tomorrow you will.