WEBVTT
00:00:05.031 --> 00:00:06.030
PROFESSOR: Welcome back.
00:00:06.030 --> 00:00:09.180
So in this session we're going
to look at Laplace transform.
00:00:09.180 --> 00:00:13.310
And we'll start with asking you
to recall the definition that
00:00:13.310 --> 00:00:16.400
you saw in class, then to
use the definition to compute
00:00:16.400 --> 00:00:19.680
the Laplace transform of the
function 1, exponential a*t,
00:00:19.680 --> 00:00:21.140
and the delta function.
00:00:21.140 --> 00:00:23.810
For each one of these, give
the domain of definition,
00:00:23.810 --> 00:00:26.232
or convergence of the integral.
00:00:26.232 --> 00:00:27.690
For the last
question, you're asked
00:00:27.690 --> 00:00:31.450
to use the results
of question 2 to give
00:00:31.450 --> 00:00:33.650
the Laplace transform of
this linear combination
00:00:33.650 --> 00:00:34.330
of functions.
00:00:34.330 --> 00:00:36.788
In the last part, you're asked
to compute Laplace transform
00:00:36.788 --> 00:00:38.762
of cosine and sine.
00:00:38.762 --> 00:00:40.970
So why don't you pause the
video, take a few minutes,
00:00:40.970 --> 00:00:41.886
and work through that.
00:00:53.360 --> 00:00:54.397
Welcome back.
00:00:54.397 --> 00:00:55.855
So let's start with
the definition.
00:00:58.630 --> 00:01:06.370
Laplace transform of the
function s was defined
00:01:06.370 --> 00:01:11.530
as the integral from 0 minus to
infinity of the function f of t
00:01:11.530 --> 00:01:16.340
exponential minus s*t dt.
00:01:16.340 --> 00:01:18.930
So note here, the
interval of integration
00:01:18.930 --> 00:01:21.150
is 0 minus to infinity.
00:01:21.150 --> 00:01:23.750
So using this definition,
we can go ahead
00:01:23.750 --> 00:01:29.820
and compute our first
Laplace transform, L of 1.
00:01:29.820 --> 00:01:33.580
So I'm just going to
substitute 1 in that integral,
00:01:33.580 --> 00:01:40.670
which gives me basically
exponential minus s*t dt,
00:01:40.670 --> 00:01:45.980
which is just the integral of
the exponential minus s*t over
00:01:45.980 --> 00:01:50.450
minus s from 0
minus to infinity.
00:01:50.450 --> 00:01:57.260
And if I expand this,
basically, I end up with 1/s,
00:01:57.260 --> 00:02:00.780
the minus reverses the order of
integration, so I start with 0,
00:02:00.780 --> 00:02:04.840
which is 1, minus the limit
when T goes to infinity,
00:02:04.840 --> 00:02:08.930
of exponential minus s*T.
00:02:08.930 --> 00:02:12.850
So here the sign of
s becomes important.
00:02:12.850 --> 00:02:15.910
If s was positive,
then this term
00:02:15.910 --> 00:02:18.130
would go to 0 as t
goes to infinity.
00:02:18.130 --> 00:02:21.002
If s is negative, then
this term diverges,
00:02:21.002 --> 00:02:23.210
and so we're not anymore in
the domain of convergence
00:02:23.210 --> 00:02:24.690
of the Laplace integral.
00:02:24.690 --> 00:02:26.710
But really, s could
be also complex.
00:02:26.710 --> 00:02:30.070
So what we're
interested in is really
00:02:30.070 --> 00:02:31.770
the sign of the real part of s.
00:02:35.020 --> 00:02:38.110
So if the real part
of s is positive,
00:02:38.110 --> 00:02:42.700
this term is goes to 0, and
the Laplace transform of 1
00:02:42.700 --> 00:02:44.780
is just 1/s.
00:02:44.780 --> 00:02:50.500
And if the real part
of s is negative,
00:02:50.500 --> 00:02:52.270
then the Laplace diverges.
00:02:55.680 --> 00:02:58.490
So the domain of convergence
in which you want to be on
00:02:58.490 --> 00:03:01.310
is the one where the real
part of s is positive.
00:03:01.310 --> 00:03:01.810
OK.
00:03:01.810 --> 00:03:03.840
So let's move to the second one.
00:03:03.840 --> 00:03:12.510
The second one is a Laplace
of exponential of a*t.
00:03:12.510 --> 00:03:17.565
So we can move a bit
faster now, and just
00:03:17.565 --> 00:03:18.690
merge the two exponentials.
00:03:25.410 --> 00:03:31.470
Exponential minus 0 to
infinity-- 0 to infinity
00:03:31.470 --> 00:03:33.860
of this exponential.
00:03:33.860 --> 00:03:43.310
Clearly this is just, again, a
case of exponential integration
00:03:43.310 --> 00:03:44.780
between the two bounds.
00:03:44.780 --> 00:03:47.340
And here again we're
going to hint a problem
00:03:47.340 --> 00:03:51.050
with the domain of
convergence where we need-- so
00:03:51.050 --> 00:03:52.970
let me just write these again.
00:03:52.970 --> 00:03:58.250
So we're going to
have here a minus--
00:03:58.250 --> 00:04:01.235
so we have our a minus s.
00:04:05.520 --> 00:04:09.530
So we have the
limit again when T
00:04:09.530 --> 00:04:14.710
goes to infinity of exponential
minus s plus a capital
00:04:14.710 --> 00:04:16.850
T minus 1.
00:04:21.440 --> 00:04:24.750
And here, again, we need
to impose the condition
00:04:24.750 --> 00:04:28.370
that the real part
of a minus s be
00:04:28.370 --> 00:04:34.570
negative to have the domain of
convergence of the integral.
00:04:34.570 --> 00:04:38.940
And then we're left with
the Laplace integral
00:04:38.940 --> 00:04:42.700
being 1 over s minus a.
00:04:42.700 --> 00:04:47.545
If the real part is positive,
then we have divergence.
00:04:50.340 --> 00:04:53.310
So the domain of
convergence of this Laplace
00:04:53.310 --> 00:04:56.360
is the one defined by
the real part of a less
00:04:56.360 --> 00:04:58.061
than the real part of s.
00:04:58.061 --> 00:04:58.560
OK.
00:04:58.560 --> 00:05:01.190
So let's do the last one.
00:05:01.190 --> 00:05:04.974
The last one is the Laplace
transform of the delta function
00:05:04.974 --> 00:05:05.765
that we saw before.
00:05:09.220 --> 00:05:17.705
That's 0 minus to infinity
delta exponential minus s*t dt.
00:05:17.705 --> 00:05:19.540
So from the previous
recitations,
00:05:19.540 --> 00:05:23.270
we saw that on this domain,
from 0 minus to infinity,
00:05:23.270 --> 00:05:27.380
the delta is 0 everywhere
except at 0, where it basically
00:05:27.380 --> 00:05:31.550
assigned a value of this
function at t equal to 0.
00:05:31.550 --> 00:05:34.630
So basically we're just
left with exponential of 0,
00:05:34.630 --> 00:05:35.950
which is 1.
00:05:35.950 --> 00:05:38.720
And this computation
was really easy,
00:05:38.720 --> 00:05:41.890
due to the properties
of the delta function.
00:05:41.890 --> 00:05:45.970
So that ends roughly this first
part, except that you can also
00:05:45.970 --> 00:05:47.900
notice here that the
domain of convergence
00:05:47.900 --> 00:05:51.940
for the Laplace
for delta is all s.
00:05:51.940 --> 00:05:52.815
There's no condition.
00:05:59.210 --> 00:06:02.510
So the last part,
next question, asked
00:06:02.510 --> 00:06:05.490
us to compute the
Laplace transform
00:06:05.490 --> 00:06:07.970
of a linear combination
of functions.
00:06:07.970 --> 00:06:18.200
So this one is 7 plus
8 exponential 2t plus 9
00:06:18.200 --> 00:06:20.470
exponential minus 3t.
00:06:23.570 --> 00:06:26.760
So here, as you saw the
Laplace is an integral,
00:06:26.760 --> 00:06:30.590
and so the Laplace transform
of this linear combination
00:06:30.590 --> 00:06:33.600
of functions is the linear
combination of the Laplace
00:06:33.600 --> 00:06:35.880
transform of the
functions individually.
00:06:35.880 --> 00:06:44.120
And so we can just rewrite
this as 7 Laplace of 1
00:06:44.120 --> 00:06:50.920
plus 8 Laplace of
exponential 2t plus 9 Laplace
00:06:50.920 --> 00:06:54.040
of exponential minus 3t.
00:06:54.040 --> 00:06:56.920
And here we can see how we
can recycle the results from
00:06:56.920 --> 00:06:59.540
the previous part, as we
computed the Laplace transform
00:06:59.540 --> 00:07:02.649
of 1, and we computed the
Laplace transform exponential
00:07:02.649 --> 00:07:04.940
a*t, which we're going to be
able to apply in these two
00:07:04.940 --> 00:07:05.830
cases.
00:07:05.830 --> 00:07:08.590
So we can write the
results directly here.
00:07:08.590 --> 00:07:11.030
And I'm just going to not
rewrite everything, just
00:07:11.030 --> 00:07:12.550
include it.
00:07:12.550 --> 00:07:15.580
So the Laplace of 1, we
found it earlier to be 1/s.
00:07:19.030 --> 00:07:22.080
The Laplace of exponential
2t we just found here,
00:07:22.080 --> 00:07:24.030
and it would be s minus 2.
00:07:28.720 --> 00:07:33.310
The Laplace of exponential minus
3t would be s minus minus 3,
00:07:33.310 --> 00:07:35.950
so it's s plus 3 with the 9.
00:07:38.750 --> 00:07:41.630
And we're done.
00:07:41.630 --> 00:07:43.680
So for the last
part, you're asked
00:07:43.680 --> 00:07:46.640
to compute the Laplace
transform of cosine and sine,
00:07:46.640 --> 00:07:49.140
and you should know
these by heart.
00:07:49.140 --> 00:07:55.280
But just a trick
to remember it--
00:07:55.280 --> 00:07:57.059
I just want to
remind you, again,
00:07:57.059 --> 00:07:59.100
of the linearity and the
fact that you could also
00:07:59.100 --> 00:08:00.016
use the Euler formula.
00:08:03.000 --> 00:08:06.180
Given what we just
derived, you could also
00:08:06.180 --> 00:08:12.100
write this, again due to the
linearity of the integral
00:08:12.100 --> 00:08:14.780
as a linear combination
of Laplace of cosine
00:08:14.780 --> 00:08:16.270
and sine here.
00:08:16.270 --> 00:08:19.890
And we know that Laplace of the
exponential a*t is just what we
00:08:19.890 --> 00:08:21.290
computed here.
00:08:21.290 --> 00:08:25.590
So that would be
s minus i*omega,
00:08:25.590 --> 00:08:32.090
which you can just
rewrite like this.
00:08:32.090 --> 00:08:34.039
And then identify
just the real part
00:08:34.039 --> 00:08:35.622
with the real part,
the imaginary part
00:08:35.622 --> 00:08:38.549
with the imaginary part,
which finishes our problem.
00:08:38.549 --> 00:08:40.760
And this is a quick
way of checking
00:08:40.760 --> 00:08:42.250
that you have that right.
00:08:42.250 --> 00:08:51.631
To give you the Laplace
transforms of cosine and sine.
00:08:55.780 --> 00:08:57.470
So that ends the
problem for today.
00:08:57.470 --> 00:09:00.530
The key point was just
remembering the definition
00:09:00.530 --> 00:09:02.210
of the Laplace
transform, and then
00:09:02.210 --> 00:09:05.170
learning how to use it
for different cases,
00:09:05.170 --> 00:09:07.938
and identify the
domains of convergence.