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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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I am going to translate it to
the right by a.
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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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calculated this piece of the
curve.
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It was not used when I
calculated the Laplace transform
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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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00:16:34 --> 00:16:40
In other words,
when I took the Laplace
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00:16:37 --> 00:16:43
transform, I automatically lost
all information about the
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function for negative values of
t.
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If I am later going to want
some of that information for
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00:16:48 --> 00:16:54
calculating this,
I won't have it and,
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00:16:51 --> 00:16:57
therefore, there cannot be a
formula expressing one in terms
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00:16:56 --> 00:17:02
of the other.
Now, of course,
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that cannot be the answer,
otherwise I would not have
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raised your expectations merely
to dash them.
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I don't want to do that,
of course.
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There is a formula,
of course.
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00:17:09 --> 00:17:15
It is just, I want to emphasize
that you must write it my way
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because, if you write it any
other way, you are going to get
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into the deepest trouble.
The formula is-- the good
253
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formula, the right formula --
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00:17:30 --> 00:17:36
-- accepts the given.
It says look,
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we have lost that pink part of
it.
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Therefore, I can never recover
that.
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Therefore, I won't ask for it.
The translation formula I will
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00:17:42 --> 00:17:48
ask for is not one for the
Laplace transform of f of t
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minus a,
but rather for the Laplace
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00:17:50 --> 00:17:56
transform of this thing where I
have wiped away that pink part
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from the translated function.
In other words,
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00:17:59 --> 00:18:05
the function I am talking about
now is the formula for,
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00:18:04 --> 00:18:10
I will put it over here to show
you the function what we are
265
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talking about.
It is the function f of,
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well, in terms of the pink
function it is,
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00:18:17 --> 00:18:23
I will have to reproduce some
of that picture.
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There is f of t.
f of t minus a,
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00:18:26 --> 00:18:32
then, looked like this.
And so the function I am
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00:18:32 --> 00:18:38
looking for is,
this is the thing translated,
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00:18:36 --> 00:18:42
but when I get down to the
corresponding,
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this is the point that
corresponds to that one,
273
00:18:43 --> 00:18:49
I wipe it away and just go with
zero after that.
274
00:18:48 --> 00:18:54
So this is u of t minus a times
f of t minus a times f of t
275
00:18:53 --> 00:18:59
minus a.
What is this Laplace transform?
276
00:19:00 --> 00:19:06
Now that does have a simple
answer.
277
00:19:04 --> 00:19:10
The answer is it is e to the
minus as,
278
00:19:10 --> 00:19:16
a funny exponential,
times the Laplace transform of
279
00:19:16 --> 00:19:22
the original function.
Now, this formula occurs in two
280
00:19:23 --> 00:19:29
forms.
This one is not too bad
281
00:19:26 --> 00:19:32
looking.
The trouble is,
282
00:19:29 --> 00:19:35
when you want to solve
differential equations you are
283
00:19:33 --> 00:19:39
going to be extremely puzzled
because the function that you
284
00:19:37 --> 00:19:43
will have to take to do the
calculation on will not be given
285
00:19:41 --> 00:19:47
to you in the form f of t minus
a.
286
00:19:44 --> 00:19:50
It will look sine t
or t squared or some
287
00:19:48 --> 00:19:54
polynomial in t.
It will not be written as t
288
00:19:50 --> 00:19:56
minus a.
What do you do?
289
00:19:53 --> 00:19:59
If your function does not look
like that but instead,
290
00:19:56 --> 00:20:02
in terms of symbols looks like
this, you can still use the
291
00:20:00 --> 00:20:06
formula.
Just a trivial change of
292
00:20:03 --> 00:20:09
variable means that you can
write it instead.
293
00:20:06 --> 00:20:12
Now, this is one place,
there is no way of writing the
294
00:20:10 --> 00:20:16
answer in terms of capital F of
s.
295
00:20:13 --> 00:20:19
This is one of those cases
where this notation is just no
296
00:20:17 --> 00:20:23
good anymore.
I am going to have to write it
297
00:20:20 --> 00:20:26
using the L notation.
The Laplace transform of f of,
298
00:20:24 --> 00:20:30
and wherever you see a t,
you should write t plus a.
299
00:20:28 --> 00:20:34
Basically, this is the same
300
00:20:31 --> 00:20:37
formula as that one.
But I will have to stand on my
301
00:20:35 --> 00:20:41
head for one minute to try to
convince you of it.
302
00:20:38 --> 00:20:44
I won't do that now.
I would like you just to take a
303
00:20:42 --> 00:20:48
look at the formula.
You should know what it is
304
00:20:45 --> 00:20:51
called.
There are a certain number of
305
00:20:48 --> 00:20:54
idiots who call this the
exponential shift formula
306
00:20:51 --> 00:20:57
because on the right side you
multiply by an exponential,
307
00:20:55 --> 00:21:01
and that corresponds to
shifting the function.
308
00:21:00 --> 00:21:06
Unfortunately,
we have preempted that.
309
00:21:02 --> 00:21:08
We are not going to call it
this.
310
00:21:05 --> 00:21:11
I will call it what your book
calls it.
311
00:21:08 --> 00:21:14
The difficulty is there is no
universal designation for this
312
00:21:13 --> 00:21:19
formula, important as it is.
However, your book calls this
313
00:21:17 --> 00:21:23
t-axis translation formula.
Translation because I am
314
00:21:21 --> 00:21:27
translating on the t-axis.
And that is what I do to the
315
00:21:25 --> 00:21:31
function, essentially.
And this tells me what its new
316
00:21:30 --> 00:21:36
Laplace transform is.
317
00:21:33 --> 00:21:39
318
00:21:38 --> 00:21:44
The other formula,
remember it?
319
00:21:40 --> 00:21:46
The exponential shift formula,
the shift or the translation
320
00:21:44 --> 00:21:50
occurs on the s-axis.
In other words,
321
00:21:47 --> 00:21:53
the formula said that F of s
minus a,
322
00:21:51 --> 00:21:57
you do the translation in s
variable corresponded to
323
00:21:55 --> 00:22:01
multiplying this by e
to the a t.
324
00:22:00 --> 00:22:06
In other words,
the formulas are sort of dual
325
00:22:02 --> 00:22:08
to each other.
This guy translates on the left
326
00:22:05 --> 00:22:11
side and multiplies by the
exponential on the right.
327
00:22:08 --> 00:22:14
The formula that you know
translates on the right and
328
00:22:12 --> 00:22:18
multiplies by the exponential on
the left.
329
00:22:14 --> 00:22:20
What are we going to calculate?
I am trying to calculate,
330
00:22:18 --> 00:22:24
so I am trying to prove this
first formula.
331
00:22:20 --> 00:22:26
The second one will be an easy
consequence.
332
00:22:23 --> 00:22:29
I am trying to calculate the
Laplace transform of that thing.
333
00:22:27 --> 00:22:33
What is it?
Well, it is the integral from
334
00:22:31 --> 00:22:37
zero to infinity of e to the
minus st times u of t minus a,
335
00:22:35 --> 00:22:41
f of t minus a times dt.
336
00:22:39 --> 00:22:45
337
00:22:42 --> 00:22:48
That is the formula for it.
But I am trying to express it
338
00:22:46 --> 00:22:52
in terms of the Laplace
transform of f itself.
339
00:22:49 --> 00:22:55
Now, it is trying to be the
Laplace transform of f.
340
00:22:54 --> 00:23:00
The problem is that here,
a (t minus a) occurs,
341
00:22:58 --> 00:23:04
which I don't like.
I would like that to be just a
342
00:23:03 --> 00:23:09
t.
Now, in order not to confuse
343
00:23:05 --> 00:23:11
you, and this is what confused
everybody, I will set t1 equal
344
00:23:09 --> 00:23:15
to t minus a.
I will change the variable.
345
00:23:13 --> 00:23:19
This is called changing the
variable in a definite integral.
346
00:23:17 --> 00:23:23
How do you change the variable
in a definite integral?
347
00:23:21 --> 00:23:27
You do it.
Well, let's leave the limits
348
00:23:24 --> 00:23:30
for the moment.
e to the minus s
349
00:23:27 --> 00:23:33
times --
Now, t, remember you can change
350
00:23:31 --> 00:23:37
the variable forwards,
direct substitution,
351
00:23:33 --> 00:23:39
but now I have to use the
inverse substitution.
352
00:23:37 --> 00:23:43
It's trivial,
but t is equal to t1 plus a.
353
00:23:40 --> 00:23:46
To change this I must
354
00:23:42 --> 00:23:48
substitute backwards and make
that t1 plus a.
355
00:23:45 --> 00:23:51
How about the rest of it?
Well, this becomes u of t1.
356
00:23:49 --> 00:23:55
This is f of t1.
357
00:23:51 --> 00:23:57
I have to change the dt,
too, but that's no problem.
358
00:23:55 --> 00:24:01
dt1 equals dt
because a is a constant.
359
00:24:00 --> 00:24:06
That is dt1.
And the last step is to put in
360
00:24:03 --> 00:24:09
the limits.
Now, when t is equal to zero,
361
00:24:06 --> 00:24:12
t1 has the value
negative a.
362
00:24:10 --> 00:24:16
So this has to be negative a
when t is infinity.
363
00:24:13 --> 00:24:19
Infinity minus a is still
infinity, so that is still
364
00:24:17 --> 00:24:23
infinity.
In other words,
365
00:24:19 --> 00:24:25
this changes to that.
These two things,
366
00:24:22 --> 00:24:28
whatever they are,
they have the same value.
367
00:24:25 --> 00:24:31
All I have done is changed the
variable.
368
00:24:30 --> 00:24:36
Make it change a variable.
But now, of course,
369
00:24:32 --> 00:24:38
I want to make this look
better.
370
00:24:34 --> 00:24:40
How am I going to do that?
Well, first multiply out the
371
00:24:37 --> 00:24:43
exponential and then you get a
factor e to the minus s(t1).
372
00:24:40 --> 00:24:46
That is good.
373
00:24:42 --> 00:24:48
That goes with this guy.
Now I get a factor e to the
374
00:24:44 --> 00:24:50
minus s times a from
the exponential law.
375
00:24:47 --> 00:24:53
But that does not have anything
to do with the integral.
376
00:24:51 --> 00:24:57
It is a constant as far as the
integral is concerned because it
377
00:24:54 --> 00:25:00
doesn't involve t1.
And, therefore,
378
00:24:56 --> 00:25:02
I can pull it outside of the
integral sign.
379
00:25:00 --> 00:25:06
And write that e to the minus s
times a.
380
00:25:05 --> 00:25:11
Let's write it the other way.
Times the integral of what?
381
00:25:10 --> 00:25:16
Well, e to the negative st1.
382
00:25:14 --> 00:25:20
Now, u of t1,
f of t1 times dt1.
383
00:25:18 --> 00:25:24
Still integrated from minus a
384
00:25:22 --> 00:25:28
to infinity.
And now the final step.
385
00:25:29 --> 00:25:35
This u of t1 is zero
for negative values of t.
386
00:25:34 --> 00:25:40
And, therefore,
it is equal to one for positive
387
00:25:38 --> 00:25:44
values of t.
It is equal to zero for
388
00:25:42 --> 00:25:48
negative values of t,
which means I can forget about
389
00:25:47 --> 00:25:53
the part of the integral that
goes from negative a to zero.
390
00:25:52 --> 00:25:58
I better rewrite this.
391
00:25:56 --> 00:26:02
Okay, leave that.
In other words,
392
00:26:00 --> 00:26:06
this is equal to e to the minus
as times the
393
00:26:04 --> 00:26:10
integral from zero to infinity
of e to the minus s --
394
00:26:08 --> 00:26:14
Let me do the shifty part now.
395
00:26:11 --> 00:26:17
396
00:26:17 --> 00:26:23
And this is since u of t1 is
equal to zero for
397
00:26:23 --> 00:26:29
t1 less than zero.
That is why I can replace this
398
00:26:29 --> 00:26:35
with zero.
Because from negative a to
399
00:26:33 --> 00:26:39
zero, nothing is happening.
The integrand is zero.
400
00:26:37 --> 00:26:43
And why can I get rid of it
after that?
401
00:26:40 --> 00:26:46
Well, because it is one after
that.
402
00:26:43 --> 00:26:49
And what is this thing?
This is the Laplace transform.
403
00:26:47 --> 00:26:53
No, it is not the Laplace
transform they said.
404
00:26:51 --> 00:26:57
Because you had t1 there,
not t.
405
00:26:53 --> 00:26:59
It is the Laplace transform
because this is a dummy
406
00:26:58 --> 00:27:04
variable.
The t1 is integrated out.
407
00:27:01 --> 00:27:07
It is a dummy variable.
It doesn't matter what you call
408
00:27:05 --> 00:27:11
it.
It is still the Laplace
409
00:27:07 --> 00:27:13
transform if I make that wiggly
t or t star or tau or u.
410
00:27:12 --> 00:27:18
I can call it anything I want
and it is still the Laplace
411
00:27:16 --> 00:27:22
transform of f of t.
412
00:27:20 --> 00:27:26
413
00:27:24 --> 00:27:30
What is the answer?
That is e to the negative as
414
00:27:27 --> 00:27:33
times the Laplace transform of
the function f.
415
00:27:33 --> 00:27:39
That is what I promised you in
that formula.
416
00:27:35 --> 00:27:41
Now, how about the other
formula?
417
00:27:37 --> 00:27:43
Well, let's look at that
quickly.
418
00:27:39 --> 00:27:45
That is, as I say,
just sleight of hand.
419
00:27:42 --> 00:27:48
But since that is the formula
you will be using at least half
420
00:27:46 --> 00:27:52
the time you better learn it.
This little sleight of hand is
421
00:27:50 --> 00:27:56
also reproduced in one page of
notes that I give you,
422
00:27:53 --> 00:27:59
but maybe you will find it easy
to understand if I talk it out
423
00:27:57 --> 00:28:03
loud.
The problem now is for the
424
00:28:00 --> 00:28:06
second formula.
I am going to have to recopy
425
00:28:03 --> 00:28:09
out the first one in order to
make the argument in a form in
426
00:28:07 --> 00:28:13
which you will understand it,
I hope.
427
00:28:09 --> 00:28:15
This goes to e to the minus as
F s,
428
00:28:13 --> 00:28:19
except I am now going to write
that not in F of s;
429
00:28:16 --> 00:28:22
since I will not be able to
write the second formula using F
430
00:28:20 --> 00:28:26
of s, I am not going to write
the first formula that way
431
00:28:24 --> 00:28:30
either.
I will write it as the Laplace
432
00:28:26 --> 00:28:32
transform with f of t.
433
00:28:30 --> 00:28:36
Now, formally if somebody says,
okay, how do I calculate the
434
00:28:34 --> 00:28:40
Laplace transform of this thing?
I say put down this.
435
00:28:39 --> 00:28:45
Well, that has no t in it.
It doesn't have the f in it
436
00:28:43 --> 00:28:49
either.
Then write this.
437
00:28:45 --> 00:28:51
What formula did I do?
I looked at that and changed t
438
00:28:50 --> 00:28:56
minus a to t. Now,
how did I change t minus a?
439
00:28:55 --> 00:29:01
The way to say it is
440
00:28:58 --> 00:29:04
I changed t.
Because the t is always there.
441
00:29:02 --> 00:29:08
t to t plus a.
You get this by replace t by t
442
00:29:08 --> 00:29:14
plus a to get the right-hand
side.
443
00:29:12 --> 00:29:18
444
00:29:16 --> 00:29:22
I replace this t by t plus a,
and that turns this
445
00:29:20 --> 00:29:26
into f of t.
And that is the f of t
446
00:29:22 --> 00:29:28
that went in there.
That is the universal rule for
447
00:29:26 --> 00:29:32
doing it.
Now I am going to use that same
448
00:29:28 --> 00:29:34
rule for transforming u of t
minus a times f of t.
449
00:29:32 --> 00:29:38
See, the problem is now I have
450
00:29:36 --> 00:29:42
a function like t squared
or sine t,
451
00:29:39 --> 00:29:45
which is not written in terms
of t minus a.
452
00:29:42 --> 00:29:48
And I don't know what to do
with it.
453
00:29:44 --> 00:29:50
The answer is,
by brute force,
454
00:29:46 --> 00:29:52
write it in terms of t minus a.
455
00:29:49 --> 00:29:55
What is brute force?
Brute force is the following.
456
00:29:52 --> 00:29:58
I am going to put a t minus a
there if it kills me.
457
00:29:56 --> 00:30:02
t minus a plus a.
No harm in that,
458
00:30:01 --> 00:30:07
is there?
Now there is a t minus a there,
459
00:30:04 --> 00:30:10
just the way there was up
there.
460
00:30:06 --> 00:30:12
And now what is the rule?
I am just going to follow my
461
00:30:11 --> 00:30:17
nose.
What's sauce for the goose is
462
00:30:13 --> 00:30:19
sauce for the gander.
Minus as, Laplace transform of
463
00:30:17 --> 00:30:23
f of, now what am I going to
write here?
464
00:30:20 --> 00:30:26
Wherever I see a t,
I am going to change it from t
465
00:30:24 --> 00:30:30
plus a.
Here I see a t.
466
00:30:29 --> 00:30:35
I will change that
to t plus a.
467
00:30:33 --> 00:30:39
What do I have?
t plus a minus a plus a,
468
00:30:37 --> 00:30:43
well,
if you can keep count,
469
00:30:41 --> 00:30:47
what does that make?
It makes t plus a in
470
00:30:46 --> 00:30:52
the end.
471
00:30:48 --> 00:30:54
472
00:31:02 --> 00:31:08
The peace that passeth
understanding.
473
00:31:04 --> 00:31:10
Let's do some examples and
suddenly you will breathe a sigh
474
00:31:08 --> 00:31:14
of relief that this all is
doable anyway.
475
00:31:11 --> 00:31:17
Let's calculate something.
I hope I am not covering up any
476
00:31:16 --> 00:31:22
crucial, yes I am.
I am covering up the u of t's,
477
00:31:19 --> 00:31:25
but you know that by now.
Let's see.
478
00:31:22 --> 00:31:28
What should we calculate first?
What I just covered up.
479
00:31:27 --> 00:31:33
Let's calculate the Laplace
transform of (u)ab of t.
480
00:31:31 --> 00:31:37
What is that going to be?
481
00:31:34 --> 00:31:40
Well, first of all,
write out what it is in terms
482
00:31:38 --> 00:31:44
of the unit step function.
483
00:31:41 --> 00:31:47
484
00:31:45 --> 00:31:51
Remember that formula?
There.
485
00:31:47 --> 00:31:53
Now you see it.
Now you don't.
486
00:31:50 --> 00:31:56
Its Laplace transform is going
to be what?
487
00:31:54 --> 00:32:00
Well, the Laplace transform of
t minus a,
488
00:31:59 --> 00:32:05
that is a special case here
where this function is one.
489
00:32:05 --> 00:32:11
Well, that one.
Either one.
490
00:32:07 --> 00:32:13
It makes no difference.
It is simply going to be the
491
00:32:11 --> 00:32:17
Laplace transform of what f of t
would have been,
492
00:32:15 --> 00:32:21
which is --
493
00:32:17 --> 00:32:23
494
00:32:26 --> 00:32:32
See, the Laplace transform of u
of t is what?
495
00:32:30 --> 00:32:36
That's one over s,
right?
496
00:32:31 --> 00:32:37
Because this is the function
one, and we don't care the fact
497
00:32:36 --> 00:32:42
that it is zero or negative
values of t.
498
00:32:39 --> 00:32:45
That is my f of s.
And so I multiply it by e to
499
00:32:43 --> 00:32:49
the minus as times one over s.
500
00:32:46 --> 00:32:52
I am using this formula,
e to the minus as times the
501
00:32:50 --> 00:32:56
Laplace transform of the unit
step function,
502
00:32:53 --> 00:32:59
which is one over s.
How about the translation?
503
00:32:59 --> 00:33:05
That was taken care of by the
exponential factor.
504
00:33:03 --> 00:33:09
And it's minus because this is
minus.
505
00:33:06 --> 00:33:12
The same thing with the b.
This is the Laplace transform
506
00:33:10 --> 00:33:16
of the unit box function.
It looks a little hairy.
507
00:33:14 --> 00:33:20
You will learn to work with it,
don't worry about it.
508
00:33:19 --> 00:33:25
How about the Laplace transform
of --
509
00:33:23 --> 00:33:29
Okay.
Let's use the other formula.
510
00:33:25 --> 00:33:31
What would be the Laplace
transform of u of t minus one
511
00:33:29 --> 00:33:35
times t squared, for example?
512
00:33:33 --> 00:33:39
See, if I gave this to you and
you only had the first formula,
513
00:33:38 --> 00:33:44
you would say,
hey, but there is no t minus
514
00:33:41 --> 00:33:47
one in there.
There is only t squared.
515
00:33:44 --> 00:33:50
What am I supposed to do?
516
00:33:46 --> 00:33:52
Well, some of you might dig way
back into high school and say
517
00:33:51 --> 00:33:57
every polynomial can be written
in powers of t minus one,
518
00:33:56 --> 00:34:02
that is what I will do.
That would give the right
519
00:34:02 --> 00:34:08
answer.
But in case you had forgotten
520
00:34:05 --> 00:34:11
how to do that,
you don't have to know because
521
00:34:08 --> 00:34:14
you could use the other formula
instead, which,
522
00:34:12 --> 00:34:18
by the way, is the way you do
it.
523
00:34:15 --> 00:34:21
What are we going to do?
It goes into e to the minus s.
524
00:34:19 --> 00:34:25
The a is one in this case,
525
00:34:22 --> 00:34:28
plus one.
e to the minus s times the
526
00:34:25 --> 00:34:31
Laplace transform of what
function?
527
00:34:30 --> 00:34:36
Change t to t plus one.
528
00:34:33 --> 00:34:39
The Laplace transform of t plus
one squared.
529
00:34:38 --> 00:34:44
What is that?
That is e to the minus s times
530
00:34:41 --> 00:34:47
the Laplace transform of t
squared plus 2t plus one.
531
00:34:46 --> 00:34:52
What's that?
532
00:34:49 --> 00:34:55
Well, by the formulas which I
am not bothering to write on the
533
00:34:54 --> 00:35:00
board anymore because you know
them, it is e to the minus s
534
00:34:59 --> 00:35:05
times --
Laplace transform of t squared
535
00:35:04 --> 00:35:10
is two factorial over s cubed.
536
00:35:07 --> 00:35:13
Remember you always have to
537
00:35:09 --> 00:35:15
raise the exponent by one.
This is two factorial,
538
00:35:12 --> 00:35:18
but that is the same as two.
Plus two.
539
00:35:15 --> 00:35:21
This two comes from there.
The Laplace transform of t is
540
00:35:19 --> 00:35:25
one over s squared.
541
00:35:21 --> 00:35:27
And, finally,
the Laplace transform of one is
542
00:35:24 --> 00:35:30
one over s.
You mean all that mess from
543
00:35:30 --> 00:35:36
this simple-looking function?
This function is not so simple.
544
00:35:34 --> 00:35:40
What is its graph?
What is it we are calculating
545
00:35:38 --> 00:35:44
the Laplace transform of?
Well, it is the function t
546
00:35:42 --> 00:35:48
squared.
But multiplying it by that
547
00:35:45 --> 00:35:51
factor u of t minus one
means that the only part of
548
00:35:50 --> 00:35:56
it I am using is this part,
because u of t minus one is one
549
00:35:55 --> 00:36:01
when t is
bigger than one.
550
00:36:00 --> 00:36:06
But when t is less than one it
is zero.
551
00:36:02 --> 00:36:08
That function doesn't look all
that simple to me.
552
00:36:05 --> 00:36:11
And that is why its Laplace
transform has three terms in it
553
00:36:09 --> 00:36:15
with this exponential factor.
Well, it is a discontinuous
554
00:36:13 --> 00:36:19
function.
And it gets discontinuous at a
555
00:36:16 --> 00:36:22
very peculiar spot.
You have to expect that.
556
00:36:19 --> 00:36:25
Where in this does it tell you
it becomes discontinuous at one?
557
00:36:23 --> 00:36:29
It is because this is e to the
minus one times s.
558
00:36:29 --> 00:36:35
This tells you where the
discontinuity occurs.
559
00:36:32 --> 00:36:38
The rest of it is just stuff
you have to take because it is
560
00:36:37 --> 00:36:43
the function t squared.
It's what it is.
561
00:36:41 --> 00:36:47
All right.
I think most of you are going
562
00:36:44 --> 00:36:50
to encounter the worst troubles
when you try to calculate
563
00:36:49 --> 00:36:55
inverse Laplace transforms,
so let me try to explain how
564
00:36:53 --> 00:36:59
that is done.
I will give you a simple
565
00:36:56 --> 00:37:02
example first.
And then I will try to give you
566
00:37:01 --> 00:37:07
a slightly more complicated one.
But even the simple one won't
567
00:37:06 --> 00:37:12
make your head ache.
We are going to calculate the
568
00:37:11 --> 00:37:17
inverse Laplace transform of
this guy, one plus e to the
569
00:37:16 --> 00:37:22
negative pi s --
570
00:37:18 --> 00:37:24
571
00:37:24 --> 00:37:30
-- divided by s squared plus
one.
572
00:37:30 --> 00:37:36
573
00:37:35 --> 00:37:41
All right.
Now, the first thing you must
574
00:37:37 --> 00:37:43
do is as soon as you see
exponential factors in there
575
00:37:40 --> 00:37:46
like that you know that these
functions, the answer is going
576
00:37:44 --> 00:37:50
to be a discontinuous function.
And you have got to separate
577
00:37:48 --> 00:37:54
out the different pieces of it
that go with the different
578
00:37:52 --> 00:37:58
exponentials.
Because the way the formula
579
00:37:54 --> 00:38:00
works, it has to be used
differently for each value of a.
580
00:37:59 --> 00:38:05
Now, in this case,
there is only one value of a
581
00:38:02 --> 00:38:08
that occurs.
Negative pi.
582
00:38:03 --> 00:38:09
But it does mean that we are
going to have to begin by
583
00:38:07 --> 00:38:13
separating out the thing into
one over s squared plus one
584
00:38:11 --> 00:38:17
and this other
factor e to the negative pi s
585
00:38:15 --> 00:38:21
divided by s squared plus one.
586
00:38:19 --> 00:38:25
Now all I have to do is take
the inverse Laplace transform of
587
00:38:23 --> 00:38:29
each piece.
The inverse Laplace transform
588
00:38:26 --> 00:38:32
of one over s squared plus one
is --
589
00:38:32 --> 00:38:38
Well, up to now we have been
saying its sine t,
590
00:38:36 --> 00:38:42
right?
If you say it is sine t you are
591
00:38:38 --> 00:38:44
going to get into trouble.
How come?
592
00:38:41 --> 00:38:47
We didn't get into trouble
before.
593
00:38:43 --> 00:38:49
Yes, but that was because there
were no exponentials in the
594
00:38:47 --> 00:38:53
expression.
When there are exponentials you
595
00:38:50 --> 00:38:56
have to be more careful.
Make the inverse transform
596
00:38:54 --> 00:39:00
unique.
Make it not sine t,
597
00:38:56 --> 00:39:02
but u of t sine t.
598
00:39:00 --> 00:39:06
You will see why in just a
moment.
599
00:39:02 --> 00:39:08
If this weren't there then sine
t would be perfectly okay.
600
00:39:07 --> 00:39:13
With that factor there,
you have got to put in the u of
601
00:39:11 --> 00:39:17
t, otherwise you won't
be able to get the formula to
602
00:39:16 --> 00:39:22
work right.
In other words,
603
00:39:18 --> 00:39:24
I must use this particular one
that I picked out to make it
604
00:39:23 --> 00:39:29
unique at the beginning of the
period.
605
00:39:26 --> 00:39:32
Otherwise, it just won't work.
Now, I know that is fine.
606
00:39:31 --> 00:39:37
But now what is the inverse
Laplace transform of e to the
607
00:39:35 --> 00:39:41
minus pi?
In other words,
608
00:39:38 --> 00:39:44
it is the same function,
except I am now multiplying it
609
00:39:41 --> 00:39:47
by e to the negative pi s.
610
00:39:44 --> 00:39:50
Well, now I will use that
formula.
611
00:39:46 --> 00:39:52
My f of s is one over s squared
plus one,
612
00:39:50 --> 00:39:56
and that corresponds it to
sine t.
613
00:39:54 --> 00:40:00
If I multiply it by e to the
minus pi s,
614
00:39:57 --> 00:40:03
just copy it down.
It now corresponds,
615
00:40:02 --> 00:40:08
the inverse Laplace transform,
to what the left side says it
616
00:40:08 --> 00:40:14
does.
u of t minus pi
617
00:40:12 --> 00:40:18
times, in other words,
this corresponds to that.
618
00:40:17 --> 00:40:23
Then if I multiply it by e to
the minus pi s,
619
00:40:23 --> 00:40:29
it corresponds to change the t
to t minus pi.
620
00:40:31 --> 00:40:37
621
00:40:38 --> 00:40:44
What is the answer?
The answer is you sum these two
622
00:40:43 --> 00:40:49
pieces.
The first piece is u of time
623
00:40:47 --> 00:40:53
sine t.
The second piece is u of t
624
00:40:53 --> 00:40:59
minus pi sine t of minus pi.
625
00:41:00 --> 00:41:06
Now, if you leave the answer in
that form it is technically
626
00:41:05 --> 00:41:11
correct, but you are going to
lose a lot of credit.
627
00:41:09 --> 00:41:15
You have to transform it to
make it look good.
628
00:41:13 --> 00:41:19
You have to make it
intelligible.
629
00:41:16 --> 00:41:22
You are not allowed to leave it
in that form.
630
00:41:20 --> 00:41:26
What could we do to it?
Well, you see,
631
00:41:24 --> 00:41:30
this part of it is interesting
whenever t is positive.
632
00:41:30 --> 00:41:36
This part of it is only
interesting when t is greater
633
00:41:34 --> 00:41:40
than or equal to pi because
this is zero.
634
00:41:37 --> 00:41:43
Before that this is zero.
What you have to do is make
635
00:41:41 --> 00:41:47
cases.
Let's call the answer f of t.
636
00:41:44 --> 00:41:50
The function has to be
637
00:41:47 --> 00:41:53
presented in what is called the
cases format.
638
00:41:50 --> 00:41:56
That is what it is called when
you type in tech,
639
00:41:54 --> 00:42:00
which I think a certain number
of you can do anyway.
640
00:42:00 --> 00:42:06
You have to make cases.
The first case is what happened
641
00:42:04 --> 00:42:10
between zero and pi?
Well, between zero and pi,
642
00:42:08 --> 00:42:14
only this term is operational.
The other one is zero because
643
00:42:12 --> 00:42:18
of that factor.
Therefore, between zero and pi
644
00:42:16 --> 00:42:22
the function looks like,
now, I don't have to put in the
645
00:42:21 --> 00:42:27
u of t because that is
equal to one.
646
00:42:24 --> 00:42:30
It is equal to sine t
between zero and pi.
647
00:42:30 --> 00:42:36
What is it equal to bigger than
pi?
648
00:42:32 --> 00:42:38
Well, the first factor,
the first term still obtains,
649
00:42:36 --> 00:42:42
so I have to include that.
But now I have to add the
650
00:42:40 --> 00:42:46
second one.
Well, what is the second term?
651
00:42:43 --> 00:42:49
I don't include the t minus
pi, u of t minus pi
652
00:42:48 --> 00:42:54
anymore because
that is now one.
653
00:42:51 --> 00:42:57
That has the value one.
It is sine of t minus pi.
654
00:42:55 --> 00:43:01
But what is sine of t minus pi?
655
00:43:00 --> 00:43:06
You take the sine curve and you
translate it to the right by pi.
656
00:43:07 --> 00:43:13
So what happens to it?
It turns into this curve.
657
00:43:13 --> 00:43:19
In other words,
it turns into the curve,
658
00:43:18 --> 00:43:24
what curve is that?
Minus sine t.
659
00:43:23 --> 00:43:29
660
00:43:30 --> 00:43:36
The other factor,
this factor is one and this
661
00:43:33 --> 00:43:39
becomes negative sine t.
662
00:43:37 --> 00:43:43
663
00:43:42 --> 00:43:48
And so the final answer is f of
t is equal to sine t
664
00:43:50 --> 00:43:56
between zero and pi and
zero for t greater than or equal
665
00:43:58 --> 00:44:04
to pi.
That is the right form of the
666
00:44:03 --> 00:44:09
answer.