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PROFESSOR: Welcome back.

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So in this session we're going
to look at Laplace transform.

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And we'll start with asking you
to recall the definition that

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you saw in class, then to
use the definition to compute

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the Laplace transform of the
function 1, exponential a*t,

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and the delta function.

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For each one of these, give
the domain of definition,

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or convergence of the integral.

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For the last
question, you're asked

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to use the results
of question 2 to give

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the Laplace transform of
this linear combination

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of functions.

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In the last part, you're asked
to compute Laplace transform

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of cosine and sine.

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So why don't you pause the
video, take a few minutes,

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and work through that.

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Welcome back.

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So let's start with
the definition.

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Laplace transform of the
function s was defined

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as the integral from 0 minus to
infinity of the function f of t

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exponential minus s*t dt.

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So note here, the
interval of integration

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is 0 minus to infinity.

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So using this definition,
we can go ahead

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and compute our first
Laplace transform, L of 1.

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So I'm just going to
substitute 1 in that integral,

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which gives me basically
exponential minus s*t dt,

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which is just the integral of
the exponential minus s*t over

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minus s from 0
minus to infinity.

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And if I expand this,
basically, I end up with 1/s,

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the minus reverses the order of
integration, so I start with 0,

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which is 1, minus the limit
when T goes to infinity,

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of exponential minus s*T.

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So here the sign of
s becomes important.

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If s was positive,
then this term

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would go to 0 as t
goes to infinity.

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If s is negative, then
this term diverges,

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and so we're not anymore in
the domain of convergence

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of the Laplace integral.

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But really, s could
be also complex.

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So what we're
interested in is really

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the sign of the real part of s.

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So if the real part
of s is positive,

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this term is goes to 0, and
the Laplace transform of 1

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is just 1/s.

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And if the real part
of s is negative,

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then the Laplace diverges.

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So the domain of convergence
in which you want to be on

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is the one where the real
part of s is positive.

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OK.

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So let's move to the second one.

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The second one is a Laplace
of exponential of a*t.

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So we can move a bit
faster now, and just

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merge the two exponentials.

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Exponential minus 0 to
infinity-- 0 to infinity

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of this exponential.

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Clearly this is just, again, a
case of exponential integration

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between the two bounds.

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And here again we're
going to hint a problem

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with the domain of
convergence where we need-- so

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let me just write these again.

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So we're going to
have here a minus--

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so we have our a minus s.

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So we have the
limit again when T

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goes to infinity of exponential
minus s plus a capital

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T minus 1.

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And here, again, we need
to impose the condition

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that the real part
of a minus s be

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negative to have the domain of
convergence of the integral.

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And then we're left with
the Laplace integral

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being 1 over s minus a.

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If the real part is positive,
then we have divergence.

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So the domain of
convergence of this Laplace

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is the one defined by
the real part of a less

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than the real part of s.

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OK.

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So let's do the last one.

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The last one is the Laplace
transform of the delta function

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that we saw before.

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That's 0 minus to infinity
delta exponential minus s*t dt.

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So from the previous
recitations,

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we saw that on this domain,
from 0 minus to infinity,

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the delta is 0 everywhere
except at 0, where it basically

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assigned a value of this
function at t equal to 0.

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So basically we're just
left with exponential of 0,

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which is 1.

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And this computation
was really easy,

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due to the properties
of the delta function.

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So that ends roughly this first
part, except that you can also

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notice here that the
domain of convergence

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for the Laplace
for delta is all s.

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There's no condition.

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So the last part,
next question, asked

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us to compute the
Laplace transform

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of a linear combination
of functions.

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So this one is 7 plus
8 exponential 2t plus 9

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exponential minus 3t.

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So here, as you saw the
Laplace is an integral,

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and so the Laplace transform
of this linear combination

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of functions is the linear
combination of the Laplace

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transform of the
functions individually.

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And so we can just rewrite
this as 7 Laplace of 1

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plus 8 Laplace of
exponential 2t plus 9 Laplace

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of exponential minus 3t.

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And here we can see how we
can recycle the results from

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the previous part, as we
computed the Laplace transform

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of 1, and we computed the
Laplace transform exponential

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a*t, which we're going to be
able to apply in these two

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cases.

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So we can write the
results directly here.

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And I'm just going to not
rewrite everything, just

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include it.

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So the Laplace of 1, we
found it earlier to be 1/s.

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The Laplace of exponential
2t we just found here,

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and it would be s minus 2.

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The Laplace of exponential minus
3t would be s minus minus 3,

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so it's s plus 3 with the 9.

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And we're done.

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So for the last
part, you're asked

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to compute the Laplace
transform of cosine and sine,

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and you should know
these by heart.

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But just a trick
to remember it--

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I just want to
remind you, again,

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of the linearity and the
fact that you could also

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use the Euler formula.

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Given what we just
derived, you could also

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write this, again due to the
linearity of the integral

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as a linear combination
of Laplace of cosine

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and sine here.

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And we know that Laplace of the
exponential a*t is just what we

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computed here.

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So that would be
s minus i*omega,

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which you can just
rewrite like this.

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And then identify
just the real part

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with the real part,
the imaginary part

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with the imaginary part,
which finishes our problem.

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And this is a quick
way of checking

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that you have that right.

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To give you the Laplace
transforms of cosine and sine.

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So that ends the
problem for today.

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The key point was just
remembering the definition

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of the Laplace
transform, and then

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learning how to use it
for different cases,

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and identify the
domains of convergence.