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Today we are going to do a last
serious topic on the Laplace

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transform, the last topic for
which I don't have to make

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frequent and profuse apologies.
One of the things the Laplace

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transform does very well and one
of the reasons why people like

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it, engineers like it,
is that it handles functions

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with jump discontinuities very
nicely.

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Now, the OR function with a
jump discontinuity is-- Purple.

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Is a function called the unit
step function.

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I will draw a graph of it.
Even the graph is

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controversial,
but everyone is agreed that it

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zero here and one there.
What people are not agreed upon

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is its value at zero.
And some people make it zero,

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some people make it one,
and some equivocate like me.

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I will leave it undefined.
It is u of t.

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It is called the unit step.

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Because that is what it is.
And let's say we will leave u

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of zero undefined.

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If that makes you unhappy,
get over it.

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Of course, we don't always want
the jump to be at zero.

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Sometimes we will want to jump
in another place.

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If I want the function to jump,
let's say at the point a

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instead of jumping at zero,
I am going to start doing what

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everybody does.
You put in the vertical lines,

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even though I have no meaning,
whatever, but it makes the

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graph look more connected and a
little easier to read.

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So that function I will call u
sub a is the function

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which jumps at the point a.
How shall I give its

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definition?
Well, you can see it is just

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the translation by a of the unit
step function.

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So that is the way to write it,
u of t minus a.

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Now I am not done.
There is a unit box function,

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which we will draw in general
terms like this.

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It gets to a,
then it jumps up to one,

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falls down again at b and
continues onto zero.

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This happens between a and b.
And the value to which it

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arises is one.
I will call this the unit box.

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It is a function of t,
a very simple one but an

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important one.
And what would be the formula

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for the unit box function?
Well, in general,

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almost all of these functions,
as you will see when you use

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jump discontinuity,
the idea is to write them all

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cleverly using nothing but u of.

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Because it is that will have
the Laplace transformer.

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The way to write this is (u)ab.

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And, if you like,
you can treat this as the

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definition of it.
Let's make it a definition.

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Okay, three lines.
Or, better yet,

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a colon and two lines.
I am defining this to be,

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what would it be?
Make the unit step at a step up

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at a, but then I would continue
at one all the time.

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I should, therefore,
step down at b.

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Now, the way you step down is
just by taking the negative of

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the unit step function.
I step down at b by subtracting

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u sub b of t .
In other words,

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it is u of t minus a minus u of
t minus b.

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And now I have expressed it

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entirely in terms of the unit
step function.

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That will be convenient when I
want take the Laplace transform.

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What is so good about these
things?

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Well, these functions,
when you use them in

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multiplications,
they transform other functions

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in a nice way.
Not transform.

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That is not the right word.
They operate on them.

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They turn them into other
strange creatures,

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and it might be these strange
creatures that you are

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interested in.
Let me just draw you a picture.

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That will be good enough.
Suppose we have some function

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like that, f of t,
what would the function u sub

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ab,
I will put in the variable,

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t times f of t,
what function would that be?

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I am just going to draw its
graph.

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What would its graph be?
Well, in between a and b this

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function (u)ab of t has the
value one.

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All I am doing is multiplying f
of t by one.

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In short, I am not doing
anything to it at all.

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Outside of that interval,
(u)ab has the value zero,

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so that zero times f of t makes
zero.

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And, therefore,
outside of this it is zero.

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The effect of multiplying an
arbitrary function by this unit

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box function is,
you wipe away all of its graph

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except the part between a and b.
Now, that is a very useful

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thing to be able to do.
Well, that is enough of that.

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Now, let's get into the main
topic.

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That is just preliminary.
I will be using these functions

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all during the period,
but the real topic is the

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following.
Let's calculate the Laplace

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transform of the unit step
function.

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Well, this is no very big deal.
It is the integral from zero to

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infinity e to the minus s t
times y of t, dt.

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But, look, when t is bigger
than zero,

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this has the value one.
So it is the same of the

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Laplace transform of one.
In other words,

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it is one over s for
positive values of s.

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Or, to make it very clear,
the Laplace transform of one is

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exactly the same thing.
As you see, the Laplace

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transform really is not
interested in what happens when

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t is less than zero
because that is not part of the

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domain of integration,
the interval of integration.

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That is fine.
They both have Laplace

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transform of one over s.

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What is the big deal?
The big deal is,

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what is the inverse Laplace
transform of one over s?

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Will the real function please

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stand up?
Which of these two should I

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pick?
Up to now in the course we have

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been picking one just because I
never made a fuss over it and

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one was good enough.
For today one is no longer

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going to be good enough.
And we have to first

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investigate the thing in a
slightly more theoretical way

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because this problem,
I have illustrated it on the

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inverse Laplace transform of one
over x,

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but it occurs for any inverse
Laplace transform.

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Suppose I have,
in other words,

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that a function f of t has as
its Laplace transform capital F

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of s?
And now, I ask what the inverse

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Laplace transform of capital F
of s is.

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Well, of course you want to
write f of t.

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But the same thing happens.
I will draw you a picture.

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Suppose, in other words,
that here is our function f of

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t.
Well, one answer certainly is f

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of t.
That is okay.

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That is the answer we have been
using up until now.

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But, you see,
I can complete this function in

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many other ways.
Suppose I haven't told you what

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it was for s less than zero.
Any of these possibilities all

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will produce the same Laplace
transform.

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In fact, I can even make it
this.

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That is okay.
Each of these,

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f of t with any one of
these tails, all have the same

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Laplace transform.

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Because the Laplace transform,
remember the definition,

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integral zero to infinity,
e to the negative s t,

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f of t, dt

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because the Laplace transform
does not care what the function

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was doing for negative values of
t.

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Now, if we have to have a
unique answer --

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And most of the time you don't
because, in general,

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the Laplace transform is only
used for problems for future

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time.
That is the way the engineers

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and physicists and other people
who use it habitually think of

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it.
If your problem is starting now

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and going on into the future and
you don't have to know anything

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about the past,
that is a Laplace transform

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problem.
If you also have to know about

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the past, then it is a Fourier
transform problem.

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That is beyond the scope of
this course, you will never hear

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that word again,
but that is the difference.

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We are starting at time zero
and going forward.

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All right.
It does not care what f of t

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was doing for negative values of
t.

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And that gives us a problem
when we try to make the Laplace

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transform unique.
Now, how will I make it unique?

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Well, there is a simple way of
doing it.

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Let's agree that wherever it
makes a difference,

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and most of the time it
doesn't, but today it will,

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whenever it makes a difference
we will declare,

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we will by brute force make our
function zero for negative

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values of t.
That makes it unique.

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I am going to say that to make
it unique, now,

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how do I make f of t zero
for negative values

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of t?
The answer is multiply it by

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the unit step function.
That leaves it what it was.

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It multiplies it by one for
positive values but multiplies

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it by zero for negative values.
The answer is going to be u of

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t times f of t.
That will be the function that

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will look just that way that I
drew, but I will draw it once

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more.
It is the function that looks

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like this.
And when I do this,

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it makes the inverse Laplace
transform unique.

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Out of all the possible tails I
might have put on f of t,

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it picks the least
interesting one,

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the tail zero.

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That is a start.
But what we have to do now is

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--

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What I want is a formula.
What we are going to need is,

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as you see right even in the
beginning, if for example,

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if I want to calculate the
Laplace transform of this,

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what I would like to have is a
nice Laplace transform for the

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translate.
If you translate a function,

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how does that effect this
Laplace transform?

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In other words,
the formula I am looking for is

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--

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I want to express the Laplace
transform of f of t minus a.

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In other words,

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the function translated,
let's say a is positive,

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so I translate it to the right
along the t axis by the distance

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a.
I want a formula for this in

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terms of the Laplace transform
of the function I started with.

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Now, my first task is to
convince you that,

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though this would be very
useful and interesting,

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there cannot possibly be such a
formula.

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There is no such formula.

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Why not?
Well, I think I will explain it

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over there since there is a
little piece of board I did not

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use.
Waste not want not.

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Why can't there be such a
formula?

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What is it we are looking for?
Let's take a nice average

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function f of t.
It has a Laplace transform.

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And now I am going to translate
it.

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Let's say this is the point
negative a.

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And so the corresponding point
positive a will be around here.

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00:14:34,000 --> 00:14:40,000
I am going to translate it to
the right by a.

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00:14:38,000 --> 00:14:44,000
What is it going to look like?
Well, then it is going to start

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here and is going to look like
this dashy thing.

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That is f of t minus a.

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That is not too bad a picture.
It will do.

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I just took that curve and
shoved it to the right by a.

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Now, why is it impossible to
express the Laplace transform of

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the dashed line in terms of the
Laplace transform of the solid

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line?
The answer is this piece.

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I will write it this way.
The trouble is,

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this piece is not used for the
Laplace transform of f of t.

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Why isn't it used?

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Well, because it occurs to the
left of the vertical axis.

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It occurs for negative values
of t.

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And the Laplace transform of f
of t simply does not

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care what f of t was
doing to the left of that line,

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for negative values of t.
It does not enter into the

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integral.
It was not used when I

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00:16:01,000 --> 00:16:07,000
calculated this piece of the
curve.

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It was not used when I
calculated the Laplace transform

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00:16:07,000 --> 00:16:13,000
of f of t.
On the other hand,

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it is going to be needed.
It occurs here,

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after I shift it to the right.
It is going to be needed for

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the Laplace transform of f of t
minus a,

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because I will have to start
the integration here,

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and I will have to know what
that is.

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00:16:29,000 --> 00:16:35,000


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00:16:34,000 --> 00:16:40,000
In other words,
when I took the Laplace

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00:16:37,000 --> 00:16:43,000
transform, I automatically lost
all information about the

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00:16:41,000 --> 00:16:47,000
function for negative values of
t.

242
00:16:44,000 --> 00:16:50,000
If I am later going to want
some of that information for

243
00:16:48,000 --> 00:16:54,000
calculating this,
I won't have it and,

244
00:16:51,000 --> 00:16:57,000
therefore, there cannot be a
formula expressing one in terms

245
00:16:56,000 --> 00:17:02,000
of the other.
Now, of course,

246
00:16:59,000 --> 00:17:05,000
that cannot be the answer,
otherwise I would not have

247
00:17:02,000 --> 00:17:08,000
raised your expectations merely
to dash them.

248
00:17:05,000 --> 00:17:11,000
I don't want to do that,
of course.

249
00:17:07,000 --> 00:17:13,000
There is a formula,
of course.

250
00:17:09,000 --> 00:17:15,000
It is just, I want to emphasize
that you must write it my way

251
00:17:13,000 --> 00:17:19,000
because, if you write it any
other way, you are going to get

252
00:17:17,000 --> 00:17:23,000
into the deepest trouble.
The formula is-- the good

253
00:17:21,000 --> 00:17:27,000
formula, the right formula --

254
00:17:24,000 --> 00:17:30,000


255
00:17:30,000 --> 00:17:36,000
-- accepts the given.
It says look,

256
00:17:32,000 --> 00:17:38,000
we have lost that pink part of
it.

257
00:17:35,000 --> 00:17:41,000
Therefore, I can never recover
that.

258
00:17:37,000 --> 00:17:43,000
Therefore, I won't ask for it.
The translation formula I will

259
00:17:42,000 --> 00:17:48,000
ask for is not one for the
Laplace transform of f of t

260
00:17:46,000 --> 00:17:52,000
minus a,
but rather for the Laplace

261
00:17:50,000 --> 00:17:56,000
transform of this thing where I
have wiped away that pink part

262
00:17:54,000 --> 00:18:00,000
from the translated function.
In other words,

263
00:17:59,000 --> 00:18:05,000
the function I am talking about
now is the formula for,

264
00:18:04,000 --> 00:18:10,000
I will put it over here to show
you the function what we are

265
00:18:10,000 --> 00:18:16,000
talking about.
It is the function f of,

266
00:18:13,000 --> 00:18:19,000
well, in terms of the pink
function it is,

267
00:18:17,000 --> 00:18:23,000
I will have to reproduce some
of that picture.

268
00:18:21,000 --> 00:18:27,000
There is f of t.
f of t minus a,

269
00:18:26,000 --> 00:18:32,000
then, looked like this.
And so the function I am

270
00:18:32,000 --> 00:18:38,000
looking for is,
this is the thing translated,

271
00:18:36,000 --> 00:18:42,000
but when I get down to the
corresponding,

272
00:18:39,000 --> 00:18:45,000
this is the point that
corresponds to that one,

273
00:18:43,000 --> 00:18:49,000
I wipe it away and just go with
zero after that.

274
00:18:48,000 --> 00:18:54,000
So this is u of t minus a times
f of t minus a times f of t

275
00:18:53,000 --> 00:18:59,000
minus a.
What is this Laplace transform?

276
00:19:00,000 --> 00:19:06,000
Now that does have a simple
answer.

277
00:19:04,000 --> 00:19:10,000
The answer is it is e to the
minus as,

278
00:19:10,000 --> 00:19:16,000
a funny exponential,
times the Laplace transform of

279
00:19:16,000 --> 00:19:22,000
the original function.
Now, this formula occurs in two

280
00:19:23,000 --> 00:19:29,000
forms.
This one is not too bad

281
00:19:26,000 --> 00:19:32,000
looking.
The trouble is,

282
00:19:29,000 --> 00:19:35,000
when you want to solve
differential equations you are

283
00:19:33,000 --> 00:19:39,000
going to be extremely puzzled
because the function that you

284
00:19:37,000 --> 00:19:43,000
will have to take to do the
calculation on will not be given

285
00:19:41,000 --> 00:19:47,000
to you in the form f of t minus
a.

286
00:19:44,000 --> 00:19:50,000
It will look sine t
or t squared or some

287
00:19:48,000 --> 00:19:54,000
polynomial in t.
It will not be written as t

288
00:19:50,000 --> 00:19:56,000
minus a.
What do you do?

289
00:19:53,000 --> 00:19:59,000
If your function does not look
like that but instead,

290
00:19:56,000 --> 00:20:02,000
in terms of symbols looks like
this, you can still use the

291
00:20:00,000 --> 00:20:06,000
formula.
Just a trivial change of

292
00:20:03,000 --> 00:20:09,000
variable means that you can
write it instead.

293
00:20:06,000 --> 00:20:12,000
Now, this is one place,
there is no way of writing the

294
00:20:10,000 --> 00:20:16,000
answer in terms of capital F of
s.

295
00:20:13,000 --> 00:20:19,000
This is one of those cases
where this notation is just no

296
00:20:17,000 --> 00:20:23,000
good anymore.
I am going to have to write it

297
00:20:20,000 --> 00:20:26,000
using the L notation.
The Laplace transform of f of,

298
00:20:24,000 --> 00:20:30,000
and wherever you see a t,
you should write t plus a.

299
00:20:28,000 --> 00:20:34,000
Basically, this is the same

300
00:20:31,000 --> 00:20:37,000
formula as that one.
But I will have to stand on my

301
00:20:35,000 --> 00:20:41,000
head for one minute to try to
convince you of it.

302
00:20:38,000 --> 00:20:44,000
I won't do that now.
I would like you just to take a

303
00:20:42,000 --> 00:20:48,000
look at the formula.
You should know what it is

304
00:20:45,000 --> 00:20:51,000
called.
There are a certain number of

305
00:20:48,000 --> 00:20:54,000
idiots who call this the
exponential shift formula

306
00:20:51,000 --> 00:20:57,000
because on the right side you
multiply by an exponential,

307
00:20:55,000 --> 00:21:01,000
and that corresponds to
shifting the function.

308
00:21:00,000 --> 00:21:06,000
Unfortunately,
we have preempted that.

309
00:21:02,000 --> 00:21:08,000
We are not going to call it
this.

310
00:21:05,000 --> 00:21:11,000
I will call it what your book
calls it.

311
00:21:08,000 --> 00:21:14,000
The difficulty is there is no
universal designation for this

312
00:21:13,000 --> 00:21:19,000
formula, important as it is.
However, your book calls this

313
00:21:17,000 --> 00:21:23,000
t-axis translation formula.
Translation because I am

314
00:21:21,000 --> 00:21:27,000
translating on the t-axis.
And that is what I do to the

315
00:21:25,000 --> 00:21:31,000
function, essentially.
And this tells me what its new

316
00:21:30,000 --> 00:21:36,000
Laplace transform is.

317
00:21:33,000 --> 00:21:39,000


318
00:21:38,000 --> 00:21:44,000
The other formula,
remember it?

319
00:21:40,000 --> 00:21:46,000
The exponential shift formula,
the shift or the translation

320
00:21:44,000 --> 00:21:50,000
occurs on the s-axis.
In other words,

321
00:21:47,000 --> 00:21:53,000
the formula said that F of s
minus a,

322
00:21:51,000 --> 00:21:57,000
you do the translation in s
variable corresponded to

323
00:21:55,000 --> 00:22:01,000
multiplying this by e
to the a t.

324
00:22:00,000 --> 00:22:06,000
In other words,
the formulas are sort of dual

325
00:22:02,000 --> 00:22:08,000
to each other.
This guy translates on the left

326
00:22:05,000 --> 00:22:11,000
side and multiplies by the
exponential on the right.

327
00:22:08,000 --> 00:22:14,000
The formula that you know
translates on the right and

328
00:22:12,000 --> 00:22:18,000
multiplies by the exponential on
the left.

329
00:22:14,000 --> 00:22:20,000
What are we going to calculate?
I am trying to calculate,

330
00:22:18,000 --> 00:22:24,000
so I am trying to prove this
first formula.

331
00:22:20,000 --> 00:22:26,000
The second one will be an easy
consequence.

332
00:22:23,000 --> 00:22:29,000
I am trying to calculate the
Laplace transform of that thing.

333
00:22:27,000 --> 00:22:33,000
What is it?
Well, it is the integral from

334
00:22:31,000 --> 00:22:37,000
zero to infinity of e to the
minus st times u of t minus a,

335
00:22:35,000 --> 00:22:41,000
f of t minus a times dt.

336
00:22:39,000 --> 00:22:45,000


337
00:22:42,000 --> 00:22:48,000
That is the formula for it.
But I am trying to express it

338
00:22:46,000 --> 00:22:52,000
in terms of the Laplace
transform of f itself.

339
00:22:49,000 --> 00:22:55,000
Now, it is trying to be the
Laplace transform of f.

340
00:22:54,000 --> 00:23:00,000
The problem is that here,
a (t minus a) occurs,

341
00:22:58,000 --> 00:23:04,000
which I don't like.
I would like that to be just a

342
00:23:03,000 --> 00:23:09,000
t.
Now, in order not to confuse

343
00:23:05,000 --> 00:23:11,000
you, and this is what confused
everybody, I will set t1 equal

344
00:23:09,000 --> 00:23:15,000
to t minus a.
I will change the variable.

345
00:23:13,000 --> 00:23:19,000
This is called changing the
variable in a definite integral.

346
00:23:17,000 --> 00:23:23,000
How do you change the variable
in a definite integral?

347
00:23:21,000 --> 00:23:27,000
You do it.
Well, let's leave the limits

348
00:23:24,000 --> 00:23:30,000
for the moment.
e to the minus s

349
00:23:27,000 --> 00:23:33,000
times --
Now, t, remember you can change

350
00:23:31,000 --> 00:23:37,000
the variable forwards,
direct substitution,

351
00:23:33,000 --> 00:23:39,000
but now I have to use the
inverse substitution.

352
00:23:37,000 --> 00:23:43,000
It's trivial,
but t is equal to t1 plus a.

353
00:23:40,000 --> 00:23:46,000
To change this I must

354
00:23:42,000 --> 00:23:48,000
substitute backwards and make
that t1 plus a.

355
00:23:45,000 --> 00:23:51,000
How about the rest of it?
Well, this becomes u of t1.

356
00:23:49,000 --> 00:23:55,000
This is f of t1.

357
00:23:51,000 --> 00:23:57,000
I have to change the dt,
too, but that's no problem.

358
00:23:55,000 --> 00:24:01,000
dt1 equals dt
because a is a constant.

359
00:24:00,000 --> 00:24:06,000
That is dt1.
And the last step is to put in

360
00:24:03,000 --> 00:24:09,000
the limits.
Now, when t is equal to zero,

361
00:24:06,000 --> 00:24:12,000
t1 has the value
negative a.

362
00:24:10,000 --> 00:24:16,000
So this has to be negative a
when t is infinity.

363
00:24:13,000 --> 00:24:19,000
Infinity minus a is still
infinity, so that is still

364
00:24:17,000 --> 00:24:23,000
infinity.
In other words,

365
00:24:19,000 --> 00:24:25,000
this changes to that.
These two things,

366
00:24:22,000 --> 00:24:28,000
whatever they are,
they have the same value.

367
00:24:25,000 --> 00:24:31,000
All I have done is changed the
variable.

368
00:24:30,000 --> 00:24:36,000
Make it change a variable.
But now, of course,

369
00:24:32,000 --> 00:24:38,000
I want to make this look
better.

370
00:24:34,000 --> 00:24:40,000
How am I going to do that?
Well, first multiply out the

371
00:24:37,000 --> 00:24:43,000
exponential and then you get a
factor e to the minus s(t1).

372
00:24:40,000 --> 00:24:46,000
That is good.

373
00:24:42,000 --> 00:24:48,000
That goes with this guy.
Now I get a factor e to the

374
00:24:44,000 --> 00:24:50,000
minus s times a from
the exponential law.

375
00:24:47,000 --> 00:24:53,000
But that does not have anything
to do with the integral.

376
00:24:51,000 --> 00:24:57,000
It is a constant as far as the
integral is concerned because it

377
00:24:54,000 --> 00:25:00,000
doesn't involve t1.
And, therefore,

378
00:24:56,000 --> 00:25:02,000
I can pull it outside of the
integral sign.

379
00:25:00,000 --> 00:25:06,000
And write that e to the minus s
times a.

380
00:25:05,000 --> 00:25:11,000
Let's write it the other way.
Times the integral of what?

381
00:25:10,000 --> 00:25:16,000
Well, e to the negative st1.

382
00:25:14,000 --> 00:25:20,000
Now, u of t1,
f of t1 times dt1.

383
00:25:18,000 --> 00:25:24,000
Still integrated from minus a

384
00:25:22,000 --> 00:25:28,000
to infinity.
And now the final step.

385
00:25:29,000 --> 00:25:35,000
This u of t1 is zero
for negative values of t.

386
00:25:34,000 --> 00:25:40,000
And, therefore,
it is equal to one for positive

387
00:25:38,000 --> 00:25:44,000
values of t.
It is equal to zero for

388
00:25:42,000 --> 00:25:48,000
negative values of t,
which means I can forget about

389
00:25:47,000 --> 00:25:53,000
the part of the integral that
goes from negative a to zero.

390
00:25:52,000 --> 00:25:58,000
I better rewrite this.

391
00:25:56,000 --> 00:26:02,000
Okay, leave that.
In other words,

392
00:26:00,000 --> 00:26:06,000
this is equal to e to the minus
as times the

393
00:26:04,000 --> 00:26:10,000
integral from zero to infinity
of e to the minus s --

394
00:26:08,000 --> 00:26:14,000
Let me do the shifty part now.

395
00:26:11,000 --> 00:26:17,000


396
00:26:17,000 --> 00:26:23,000
And this is since u of t1 is
equal to zero for

397
00:26:23,000 --> 00:26:29,000
t1 less than zero.
That is why I can replace this

398
00:26:29,000 --> 00:26:35,000
with zero.
Because from negative a to

399
00:26:33,000 --> 00:26:39,000
zero, nothing is happening.
The integrand is zero.

400
00:26:37,000 --> 00:26:43,000
And why can I get rid of it
after that?

401
00:26:40,000 --> 00:26:46,000
Well, because it is one after
that.

402
00:26:43,000 --> 00:26:49,000
And what is this thing?
This is the Laplace transform.

403
00:26:47,000 --> 00:26:53,000
No, it is not the Laplace
transform they said.

404
00:26:51,000 --> 00:26:57,000
Because you had t1 there,
not t.

405
00:26:53,000 --> 00:26:59,000
It is the Laplace transform
because this is a dummy

406
00:26:58,000 --> 00:27:04,000
variable.
The t1 is integrated out.

407
00:27:01,000 --> 00:27:07,000
It is a dummy variable.
It doesn't matter what you call

408
00:27:05,000 --> 00:27:11,000
it.
It is still the Laplace

409
00:27:07,000 --> 00:27:13,000
transform if I make that wiggly
t or t star or tau or u.

410
00:27:12,000 --> 00:27:18,000
I can call it anything I want
and it is still the Laplace

411
00:27:16,000 --> 00:27:22,000
transform of f of t.

412
00:27:20,000 --> 00:27:26,000


413
00:27:24,000 --> 00:27:30,000
What is the answer?
That is e to the negative as

414
00:27:27,000 --> 00:27:33,000
times the Laplace transform of
the function f.

415
00:27:33,000 --> 00:27:39,000
That is what I promised you in
that formula.

416
00:27:35,000 --> 00:27:41,000
Now, how about the other
formula?

417
00:27:37,000 --> 00:27:43,000
Well, let's look at that
quickly.

418
00:27:39,000 --> 00:27:45,000
That is, as I say,
just sleight of hand.

419
00:27:42,000 --> 00:27:48,000
But since that is the formula
you will be using at least half

420
00:27:46,000 --> 00:27:52,000
the time you better learn it.
This little sleight of hand is

421
00:27:50,000 --> 00:27:56,000
also reproduced in one page of
notes that I give you,

422
00:27:53,000 --> 00:27:59,000
but maybe you will find it easy
to understand if I talk it out

423
00:27:57,000 --> 00:28:03,000
loud.
The problem now is for the

424
00:28:00,000 --> 00:28:06,000
second formula.
I am going to have to recopy

425
00:28:03,000 --> 00:28:09,000
out the first one in order to
make the argument in a form in

426
00:28:07,000 --> 00:28:13,000
which you will understand it,
I hope.

427
00:28:09,000 --> 00:28:15,000
This goes to e to the minus as
F s,

428
00:28:13,000 --> 00:28:19,000
except I am now going to write
that not in F of s;

429
00:28:16,000 --> 00:28:22,000
since I will not be able to
write the second formula using F

430
00:28:20,000 --> 00:28:26,000
of s, I am not going to write
the first formula that way

431
00:28:24,000 --> 00:28:30,000
either.
I will write it as the Laplace

432
00:28:26,000 --> 00:28:32,000
transform with f of t.

433
00:28:30,000 --> 00:28:36,000
Now, formally if somebody says,
okay, how do I calculate the

434
00:28:34,000 --> 00:28:40,000
Laplace transform of this thing?
I say put down this.

435
00:28:39,000 --> 00:28:45,000
Well, that has no t in it.
It doesn't have the f in it

436
00:28:43,000 --> 00:28:49,000
either.
Then write this.

437
00:28:45,000 --> 00:28:51,000
What formula did I do?
I looked at that and changed t

438
00:28:50,000 --> 00:28:56,000
minus a to t. Now,
how did I change t minus a?

439
00:28:55,000 --> 00:29:01,000
The way to say it is

440
00:28:58,000 --> 00:29:04,000
I changed t.
Because the t is always there.

441
00:29:02,000 --> 00:29:08,000
t to t plus a.
You get this by replace t by t

442
00:29:08,000 --> 00:29:14,000
plus a to get the right-hand
side.

443
00:29:12,000 --> 00:29:18,000


444
00:29:16,000 --> 00:29:22,000
I replace this t by t plus a,
and that turns this

445
00:29:20,000 --> 00:29:26,000
into f of t.
And that is the f of t

446
00:29:22,000 --> 00:29:28,000
that went in there.
That is the universal rule for

447
00:29:26,000 --> 00:29:32,000
doing it.
Now I am going to use that same

448
00:29:28,000 --> 00:29:34,000
rule for transforming u of t
minus a times f of t.

449
00:29:32,000 --> 00:29:38,000
See, the problem is now I have

450
00:29:36,000 --> 00:29:42,000
a function like t squared
or sine t,

451
00:29:39,000 --> 00:29:45,000
which is not written in terms
of t minus a.

452
00:29:42,000 --> 00:29:48,000
And I don't know what to do
with it.

453
00:29:44,000 --> 00:29:50,000
The answer is,
by brute force,

454
00:29:46,000 --> 00:29:52,000
write it in terms of t minus a.

455
00:29:49,000 --> 00:29:55,000
What is brute force?
Brute force is the following.

456
00:29:52,000 --> 00:29:58,000
I am going to put a t minus a
there if it kills me.

457
00:29:56,000 --> 00:30:02,000
t minus a plus a.
No harm in that,

458
00:30:01,000 --> 00:30:07,000
is there?
Now there is a t minus a there,

459
00:30:04,000 --> 00:30:10,000
just the way there was up
there.

460
00:30:06,000 --> 00:30:12,000
And now what is the rule?
I am just going to follow my

461
00:30:11,000 --> 00:30:17,000
nose.
What's sauce for the goose is

462
00:30:13,000 --> 00:30:19,000
sauce for the gander.
Minus as, Laplace transform of

463
00:30:17,000 --> 00:30:23,000
f of, now what am I going to
write here?

464
00:30:20,000 --> 00:30:26,000
Wherever I see a t,
I am going to change it from t

465
00:30:24,000 --> 00:30:30,000
plus a.
Here I see a t.

466
00:30:29,000 --> 00:30:35,000
I will change that
to t plus a.

467
00:30:33,000 --> 00:30:39,000
What do I have?
t plus a minus a plus a,

468
00:30:37,000 --> 00:30:43,000
well,
if you can keep count,

469
00:30:41,000 --> 00:30:47,000
what does that make?
It makes t plus a in

470
00:30:46,000 --> 00:30:52,000
the end.

471
00:30:48,000 --> 00:30:54,000


472
00:31:02,000 --> 00:31:08,000
The peace that passeth
understanding.

473
00:31:04,000 --> 00:31:10,000
Let's do some examples and
suddenly you will breathe a sigh

474
00:31:08,000 --> 00:31:14,000
of relief that this all is
doable anyway.

475
00:31:11,000 --> 00:31:17,000
Let's calculate something.
I hope I am not covering up any

476
00:31:16,000 --> 00:31:22,000
crucial, yes I am.
I am covering up the u of t's,

477
00:31:19,000 --> 00:31:25,000
but you know that by now.
Let's see.

478
00:31:22,000 --> 00:31:28,000
What should we calculate first?
What I just covered up.

479
00:31:27,000 --> 00:31:33,000
Let's calculate the Laplace
transform of (u)ab of t.

480
00:31:31,000 --> 00:31:37,000
What is that going to be?

481
00:31:34,000 --> 00:31:40,000
Well, first of all,
write out what it is in terms

482
00:31:38,000 --> 00:31:44,000
of the unit step function.

483
00:31:41,000 --> 00:31:47,000


484
00:31:45,000 --> 00:31:51,000
Remember that formula?
There.

485
00:31:47,000 --> 00:31:53,000
Now you see it.
Now you don't.

486
00:31:50,000 --> 00:31:56,000
Its Laplace transform is going
to be what?

487
00:31:54,000 --> 00:32:00,000
Well, the Laplace transform of
t minus a,

488
00:31:59,000 --> 00:32:05,000
that is a special case here
where this function is one.

489
00:32:05,000 --> 00:32:11,000
Well, that one.
Either one.

490
00:32:07,000 --> 00:32:13,000
It makes no difference.
It is simply going to be the

491
00:32:11,000 --> 00:32:17,000
Laplace transform of what f of t
would have been,

492
00:32:15,000 --> 00:32:21,000
which is --

493
00:32:17,000 --> 00:32:23,000


494
00:32:26,000 --> 00:32:32,000
See, the Laplace transform of u
of t is what?

495
00:32:30,000 --> 00:32:36,000
That's one over s,
right?

496
00:32:31,000 --> 00:32:37,000
Because this is the function
one, and we don't care the fact

497
00:32:36,000 --> 00:32:42,000
that it is zero or negative
values of t.

498
00:32:39,000 --> 00:32:45,000
That is my f of s.
And so I multiply it by e to

499
00:32:43,000 --> 00:32:49,000
the minus as times one over s.

500
00:32:46,000 --> 00:32:52,000
I am using this formula,
e to the minus as times the

501
00:32:50,000 --> 00:32:56,000
Laplace transform of the unit
step function,

502
00:32:53,000 --> 00:32:59,000
which is one over s.
How about the translation?

503
00:32:59,000 --> 00:33:05,000
That was taken care of by the
exponential factor.

504
00:33:03,000 --> 00:33:09,000
And it's minus because this is
minus.

505
00:33:06,000 --> 00:33:12,000
The same thing with the b.
This is the Laplace transform

506
00:33:10,000 --> 00:33:16,000
of the unit box function.
It looks a little hairy.

507
00:33:14,000 --> 00:33:20,000
You will learn to work with it,
don't worry about it.

508
00:33:19,000 --> 00:33:25,000
How about the Laplace transform
of --

509
00:33:23,000 --> 00:33:29,000
Okay.
Let's use the other formula.

510
00:33:25,000 --> 00:33:31,000
What would be the Laplace
transform of u of t minus one

511
00:33:29,000 --> 00:33:35,000
times t squared, for example?

512
00:33:33,000 --> 00:33:39,000
See, if I gave this to you and
you only had the first formula,

513
00:33:38,000 --> 00:33:44,000
you would say,
hey, but there is no t minus

514
00:33:41,000 --> 00:33:47,000
one in there.
There is only t squared.

515
00:33:44,000 --> 00:33:50,000
What am I supposed to do?

516
00:33:46,000 --> 00:33:52,000
Well, some of you might dig way
back into high school and say

517
00:33:51,000 --> 00:33:57,000
every polynomial can be written
in powers of t minus one,

518
00:33:56,000 --> 00:34:02,000
that is what I will do.
That would give the right

519
00:34:02,000 --> 00:34:08,000
answer.
But in case you had forgotten

520
00:34:05,000 --> 00:34:11,000
how to do that,
you don't have to know because

521
00:34:08,000 --> 00:34:14,000
you could use the other formula
instead, which,

522
00:34:12,000 --> 00:34:18,000
by the way, is the way you do
it.

523
00:34:15,000 --> 00:34:21,000
What are we going to do?
It goes into e to the minus s.

524
00:34:19,000 --> 00:34:25,000
The a is one in this case,

525
00:34:22,000 --> 00:34:28,000
plus one.
e to the minus s times the

526
00:34:25,000 --> 00:34:31,000
Laplace transform of what
function?

527
00:34:30,000 --> 00:34:36,000
Change t to t plus one.

528
00:34:33,000 --> 00:34:39,000
The Laplace transform of t plus
one squared.

529
00:34:38,000 --> 00:34:44,000
What is that?
That is e to the minus s times

530
00:34:41,000 --> 00:34:47,000
the Laplace transform of t
squared plus 2t plus one.

531
00:34:46,000 --> 00:34:52,000
What's that?

532
00:34:49,000 --> 00:34:55,000
Well, by the formulas which I
am not bothering to write on the

533
00:34:54,000 --> 00:35:00,000
board anymore because you know
them, it is e to the minus s

534
00:34:59,000 --> 00:35:05,000
times --
Laplace transform of t squared

535
00:35:04,000 --> 00:35:10,000
is two factorial over s cubed.

536
00:35:07,000 --> 00:35:13,000
Remember you always have to

537
00:35:09,000 --> 00:35:15,000
raise the exponent by one.
This is two factorial,

538
00:35:12,000 --> 00:35:18,000
but that is the same as two.
Plus two.

539
00:35:15,000 --> 00:35:21,000
This two comes from there.
The Laplace transform of t is

540
00:35:19,000 --> 00:35:25,000
one over s squared.

541
00:35:21,000 --> 00:35:27,000
And, finally,
the Laplace transform of one is

542
00:35:24,000 --> 00:35:30,000
one over s.
You mean all that mess from

543
00:35:30,000 --> 00:35:36,000
this simple-looking function?
This function is not so simple.

544
00:35:34,000 --> 00:35:40,000
What is its graph?
What is it we are calculating

545
00:35:38,000 --> 00:35:44,000
the Laplace transform of?
Well, it is the function t

546
00:35:42,000 --> 00:35:48,000
squared.
But multiplying it by that

547
00:35:45,000 --> 00:35:51,000
factor u of t minus one
means that the only part of

548
00:35:50,000 --> 00:35:56,000
it I am using is this part,
because u of t minus one is one

549
00:35:55,000 --> 00:36:01,000
when t is
bigger than one.

550
00:36:00,000 --> 00:36:06,000
But when t is less than one it
is zero.

551
00:36:02,000 --> 00:36:08,000
That function doesn't look all
that simple to me.

552
00:36:05,000 --> 00:36:11,000
And that is why its Laplace
transform has three terms in it

553
00:36:09,000 --> 00:36:15,000
with this exponential factor.
Well, it is a discontinuous

554
00:36:13,000 --> 00:36:19,000
function.
And it gets discontinuous at a

555
00:36:16,000 --> 00:36:22,000
very peculiar spot.
You have to expect that.

556
00:36:19,000 --> 00:36:25,000
Where in this does it tell you
it becomes discontinuous at one?

557
00:36:23,000 --> 00:36:29,000
It is because this is e to the
minus one times s.

558
00:36:29,000 --> 00:36:35,000
This tells you where the
discontinuity occurs.

559
00:36:32,000 --> 00:36:38,000
The rest of it is just stuff
you have to take because it is

560
00:36:37,000 --> 00:36:43,000
the function t squared.
It's what it is.

561
00:36:41,000 --> 00:36:47,000
All right.
I think most of you are going

562
00:36:44,000 --> 00:36:50,000
to encounter the worst troubles
when you try to calculate

563
00:36:49,000 --> 00:36:55,000
inverse Laplace transforms,
so let me try to explain how

564
00:36:53,000 --> 00:36:59,000
that is done.
I will give you a simple

565
00:36:56,000 --> 00:37:02,000
example first.
And then I will try to give you

566
00:37:01,000 --> 00:37:07,000
a slightly more complicated one.
But even the simple one won't

567
00:37:06,000 --> 00:37:12,000
make your head ache.
We are going to calculate the

568
00:37:11,000 --> 00:37:17,000
inverse Laplace transform of
this guy, one plus e to the

569
00:37:16,000 --> 00:37:22,000
negative pi s --

570
00:37:18,000 --> 00:37:24,000


571
00:37:24,000 --> 00:37:30,000
-- divided by s squared plus
one.

572
00:37:30,000 --> 00:37:36,000


573
00:37:35,000 --> 00:37:41,000
All right.
Now, the first thing you must

574
00:37:37,000 --> 00:37:43,000
do is as soon as you see
exponential factors in there

575
00:37:40,000 --> 00:37:46,000
like that you know that these
functions, the answer is going

576
00:37:44,000 --> 00:37:50,000
to be a discontinuous function.
And you have got to separate

577
00:37:48,000 --> 00:37:54,000
out the different pieces of it
that go with the different

578
00:37:52,000 --> 00:37:58,000
exponentials.
Because the way the formula

579
00:37:54,000 --> 00:38:00,000
works, it has to be used
differently for each value of a.

580
00:37:59,000 --> 00:38:05,000
Now, in this case,
there is only one value of a

581
00:38:02,000 --> 00:38:08,000
that occurs.
Negative pi.

582
00:38:03,000 --> 00:38:09,000
But it does mean that we are
going to have to begin by

583
00:38:07,000 --> 00:38:13,000
separating out the thing into
one over s squared plus one

584
00:38:11,000 --> 00:38:17,000
and this other
factor e to the negative pi s

585
00:38:15,000 --> 00:38:21,000
divided by s squared plus one.

586
00:38:19,000 --> 00:38:25,000
Now all I have to do is take
the inverse Laplace transform of

587
00:38:23,000 --> 00:38:29,000
each piece.
The inverse Laplace transform

588
00:38:26,000 --> 00:38:32,000
of one over s squared plus one
is --

589
00:38:32,000 --> 00:38:38,000
Well, up to now we have been
saying its sine t,

590
00:38:36,000 --> 00:38:42,000
right?
If you say it is sine t you are

591
00:38:38,000 --> 00:38:44,000
going to get into trouble.
How come?

592
00:38:41,000 --> 00:38:47,000
We didn't get into trouble
before.

593
00:38:43,000 --> 00:38:49,000
Yes, but that was because there
were no exponentials in the

594
00:38:47,000 --> 00:38:53,000
expression.
When there are exponentials you

595
00:38:50,000 --> 00:38:56,000
have to be more careful.
Make the inverse transform

596
00:38:54,000 --> 00:39:00,000
unique.
Make it not sine t,

597
00:38:56,000 --> 00:39:02,000
but u of t sine t.

598
00:39:00,000 --> 00:39:06,000
You will see why in just a
moment.

599
00:39:02,000 --> 00:39:08,000
If this weren't there then sine
t would be perfectly okay.

600
00:39:07,000 --> 00:39:13,000
With that factor there,
you have got to put in the u of

601
00:39:11,000 --> 00:39:17,000
t, otherwise you won't
be able to get the formula to

602
00:39:16,000 --> 00:39:22,000
work right.
In other words,

603
00:39:18,000 --> 00:39:24,000
I must use this particular one
that I picked out to make it

604
00:39:23,000 --> 00:39:29,000
unique at the beginning of the
period.

605
00:39:26,000 --> 00:39:32,000
Otherwise, it just won't work.
Now, I know that is fine.

606
00:39:31,000 --> 00:39:37,000
But now what is the inverse
Laplace transform of e to the

607
00:39:35,000 --> 00:39:41,000
minus pi?
In other words,

608
00:39:38,000 --> 00:39:44,000
it is the same function,
except I am now multiplying it

609
00:39:41,000 --> 00:39:47,000
by e to the negative pi s.

610
00:39:44,000 --> 00:39:50,000
Well, now I will use that
formula.

611
00:39:46,000 --> 00:39:52,000
My f of s is one over s squared
plus one,

612
00:39:50,000 --> 00:39:56,000
and that corresponds it to
sine t.

613
00:39:54,000 --> 00:40:00,000
If I multiply it by e to the
minus pi s,

614
00:39:57,000 --> 00:40:03,000
just copy it down.
It now corresponds,

615
00:40:02,000 --> 00:40:08,000
the inverse Laplace transform,
to what the left side says it

616
00:40:08,000 --> 00:40:14,000
does.
u of t minus pi

617
00:40:12,000 --> 00:40:18,000
times, in other words,
this corresponds to that.

618
00:40:17,000 --> 00:40:23,000
Then if I multiply it by e to
the minus pi s,

619
00:40:23,000 --> 00:40:29,000
it corresponds to change the t
to t minus pi.

620
00:40:31,000 --> 00:40:37,000


621
00:40:38,000 --> 00:40:44,000
What is the answer?
The answer is you sum these two

622
00:40:43,000 --> 00:40:49,000
pieces.
The first piece is u of time

623
00:40:47,000 --> 00:40:53,000
sine t.
The second piece is u of t

624
00:40:53,000 --> 00:40:59,000
minus pi sine t of minus pi.

625
00:41:00,000 --> 00:41:06,000
Now, if you leave the answer in
that form it is technically

626
00:41:05,000 --> 00:41:11,000
correct, but you are going to
lose a lot of credit.

627
00:41:09,000 --> 00:41:15,000
You have to transform it to
make it look good.

628
00:41:13,000 --> 00:41:19,000
You have to make it
intelligible.

629
00:41:16,000 --> 00:41:22,000
You are not allowed to leave it
in that form.

630
00:41:20,000 --> 00:41:26,000
What could we do to it?
Well, you see,

631
00:41:24,000 --> 00:41:30,000
this part of it is interesting
whenever t is positive.

632
00:41:30,000 --> 00:41:36,000
This part of it is only
interesting when t is greater

633
00:41:34,000 --> 00:41:40,000
than or equal to pi because
this is zero.

634
00:41:37,000 --> 00:41:43,000
Before that this is zero.
What you have to do is make

635
00:41:41,000 --> 00:41:47,000
cases.
Let's call the answer f of t.

636
00:41:44,000 --> 00:41:50,000
The function has to be

637
00:41:47,000 --> 00:41:53,000
presented in what is called the
cases format.

638
00:41:50,000 --> 00:41:56,000
That is what it is called when
you type in tech,

639
00:41:54,000 --> 00:42:00,000
which I think a certain number
of you can do anyway.

640
00:42:00,000 --> 00:42:06,000
You have to make cases.
The first case is what happened

641
00:42:04,000 --> 00:42:10,000
between zero and pi?
Well, between zero and pi,

642
00:42:08,000 --> 00:42:14,000
only this term is operational.
The other one is zero because

643
00:42:12,000 --> 00:42:18,000
of that factor.
Therefore, between zero and pi

644
00:42:16,000 --> 00:42:22,000
the function looks like,
now, I don't have to put in the

645
00:42:21,000 --> 00:42:27,000
u of t because that is
equal to one.

646
00:42:24,000 --> 00:42:30,000
It is equal to sine t
between zero and pi.

647
00:42:30,000 --> 00:42:36,000
What is it equal to bigger than
pi?

648
00:42:32,000 --> 00:42:38,000
Well, the first factor,
the first term still obtains,

649
00:42:36,000 --> 00:42:42,000
so I have to include that.
But now I have to add the

650
00:42:40,000 --> 00:42:46,000
second one.
Well, what is the second term?

651
00:42:43,000 --> 00:42:49,000
I don't include the t minus
pi, u of t minus pi

652
00:42:48,000 --> 00:42:54,000
anymore because
that is now one.

653
00:42:51,000 --> 00:42:57,000
That has the value one.
It is sine of t minus pi.

654
00:42:55,000 --> 00:43:01,000
But what is sine of t minus pi?

655
00:43:00,000 --> 00:43:06,000
You take the sine curve and you
translate it to the right by pi.

656
00:43:07,000 --> 00:43:13,000
So what happens to it?
It turns into this curve.

657
00:43:13,000 --> 00:43:19,000
In other words,
it turns into the curve,

658
00:43:18,000 --> 00:43:24,000
what curve is that?
Minus sine t.

659
00:43:23,000 --> 00:43:29,000


660
00:43:30,000 --> 00:43:36,000
The other factor,
this factor is one and this

661
00:43:33,000 --> 00:43:39,000
becomes negative sine t.

662
00:43:37,000 --> 00:43:43,000


663
00:43:42,000 --> 00:43:48,000
And so the final answer is f of
t is equal to sine t

664
00:43:50,000 --> 00:43:56,000
between zero and pi and
zero for t greater than or equal

665
00:43:58,000 --> 00:44:04,000
to pi.
That is the right form of the

666
00:44:03,000 --> 00:44:09,000
answer.