# can this be a proper Laplace transform?

Started by October 3, 2005
```sin(s)/(s^2+4),

can it be a proper Laplace transform?

why?

-----------------

I honestly claim this is not a HW problem. Thanks a lot!

```
```lucy wrote:
> sin(s)/(s^2+4),
>
> can it be a proper Laplace transform?
>
> why?
>
> -----------------
>
> I honestly claim this is not a HW problem. Thanks a lot!
>

Is it for a test?
```
```I honestly claim that this is not a test problem either. thanks

```
```"lucy" <losemind@yahoo.com> wrote in message
> I honestly claim that this is not a test problem either. thanks
>

Have you taken the comp.dsp oath first?

McC

```
```>>>>> "lucy" == lucy  <losemind@yahoo.com> writes:

lucy> sin(s)/(s^2+4),
lucy> can it be a proper Laplace transform?

What do you mean by proper Laplace transform?

Ray
```
```in article 1128377002.944857.263010@z14g2000cwz.googlegroups.com, lucy at
losemind@yahoo.com wrote on 10/03/2005 18:03:

> sin(s)/(s^2+4),
>
> can it be a proper Laplace transform?
>
> why?
>
> -----------------
>
> I honestly claim this is not a HW problem. Thanks a lot!

well since you honestly say it is not, the "constructive" way to deal with
this is to find an function x(t) that has LT of X(s) = sin(s)/(s^2+4).  to
try to do that, try substituting s = j*w ("w" is "omega") and see if

X(jw) = sin(j*w)/((j*w)^2 + 4)

= j*sinh(w)/(4 - w^2)

= j/2 * (exp(w) - exp(-w))/(4 - w^2)

now try to find if  1/(4 - w^2) is the "proper Fourier Transform" of
something.  or if 1/(s^2+4) is the L.T. of something.  then ask, what does
multiplying X(jw) by exp(w) do?

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```lucy wrote:
> sin(s)/(s^2+4),
>
> can it be a proper Laplace transform?

I don't think so. The Laplace Inversion formula doesn't
work for it.

>
> why?

```
```>>>>> "dave" == dave  <david_lawrence_petry@yahoo.com> writes:

dave> lucy wrote:
>> sin(s)/(s^2+4),
>>
>> can it be a proper Laplace transform?

dave> I don't think so. The Laplace Inversion formula doesn't
dave> work for it.

What fails?

I took a quick stab at it by replacing sin(s) with its power series.
Then we have terms of the form s^(2*k+1)/(s^2+4), for which we know
the inverse Laplace transform.  I think the resulting infinite series
converges.  But I did leave out all the terms having to do with the
initial values, so perhaps it doesn't really converge.

Ray

```
```Raymond Toy wrote:
> >>>>> "dave" == dave  <david_lawrence_petry@yahoo.com> writes:
>
>     dave> lucy wrote:
>     >> sin(s)/(s^2+4),
>     >>
>     >> can it be a proper Laplace transform?
>
>     dave> I don't think so. The Laplace Inversion formula doesn't
>     dave> work for it.
>
> What fails?
>
> I took a quick stab at it by replacing sin(s) with its power series.
> Then we have terms of the form s^(2*k+1)/(s^2+4), for which we know
> the inverse Laplace transform.

Really? What is the inverse Laplace transform of  s^5/(s^2+4) ?

"Proper" (I'm not sure what that means) Laplace transforms usually
tend to 0 as s goes to infinity.

> I think the resulting infinite series
> converges.  But I did leave out all the terms having to do with the
> initial values, so perhaps it doesn't really converge.

I suspect you left out the key parts.

```
```In article <1128377002.944857.263010@z14g2000cwz.googlegroups.com>,
lucy <losemind@yahoo.com> wrote:
>sin(s)/(s^2+4),

>can it be a proper Laplace transform?

>why?

The Laplace transform F(s) of an exponentially bounded function
f(t) (say with |f(t)| <= K exp(Bt)) converges for Re(s) > B with
|F(s)| <= K/(Re s - B).  But |sin(s)| ~ exp(|Im(s)|)/2 as
|Im(s)| -> infty.  So sin(s)/(s^2+4) is not the Laplace transform
of an exponentially bounded function.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel
University of British Columbia            Vancouver, BC, Canada

```