## Monday Math 104

The Laplace Transform
Part 5: Exponential and trigonometric functions

Next ia a very simple Laplace transform: that of exponential functions. If , we use the definition of the Laplace transform to find that
,
with  for convergence.
From this, and the linearity of the Laplace transform, we see
,
and
,
both with .

Now, in the integral we used for the exponential, we see that the integration still holds, for , if we replace a with ıω; we get
.
Thus, we use  and  to get
,
and
,
both with .

### 7 Responses to “Monday Math 104”

1. Monday Math 105 « Twisted One 151's Weblog Says:

[…] Laplace Transform Part 6: Frequency shifting, time shifting, and scaling properties Expanding upon last time’s work on the Laplace transform of the exponential, we can prove quickly the “frequency […]

2. Monday Math 107 « Twisted One 151's Weblog Says:

[…] to find the function f(t) with Laplace transform . Using , we see . Now, we recall from here and here that and , so we see that , and so we see that f(t) is times the convolution of and , where H(t) […]

3. Monday Math 109 « Twisted One 151's Weblog Says:

[…] poles are on the right side () of the s complex plane, such as for the Laplace transforms of the hyperbolic sine and cosine, then f(t) contains terms growing exponentially, and f(∞) does not exist. III. If there are […]

4. Monday Math 111 « Twisted One 151's Weblog Says:

[…] of some more complicated functions. For example, what is the Laplace transform of ? We know from here that the Laplace transform for cosine is , so Thus, via the frequency shift formula here, . […]

5. Monday Math 114 « Twisted One 151's Weblog Says:

[…] , we can use partial fractions on the above to find that , and so we see that , and using from here that the Laplace transform of an exponential is , and from here that the frequency-shifted sine and […]

6. Monday Math 115 « Twisted One 151's Weblog Says:

[…] is . So, to continue with our solution, , where we have used the frequency shift formula and the transform for the sine. Now, recalling the convolution rule, we see then that . For those who wish to know what this […]

7. Monday Math 117 « Twisted One 151's Weblog Says:

[…] constant, we turn to the initial value theorem: . Here, we see , so we find that , and we have . Recalling that , we see that our solution is simply , […]