how to compute the laplace and fourier transform of this function?

Hi there,

Suppose I have a function f(t) which I knew its laplace and fourier
transform.

What is the laplace and fourier transform of the following:

exp(a*f(t))

???

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

Is there a way to evaluate the laplace transform of

heaviside((exp(x)-a)),

where the heaviside function is also called step function,

heaviside(x) = 1, when x>0, and =0, when x<0, and it has a jump from 0
to 1 at x=0...

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

Please give me some pointers! thanks a lot!

In article <1156111223.392350.157580@75g2000cwc.googlegroups.com>,
Luna Moon <lunamoonmoon@gmail.com> wrote:
>Hi there,
>
>Suppose I have a function f(t) which I knew its laplace and fourier
>transform.
>
>What is the laplace and fourier transform of the following:
>
>exp(a*f(t))
>
>???

No nice formula.  
exp(a f(t)) = sum_{n=0}^infty a^n f(t)^n/n!

Now the Fourier transform of a power of f is a "convolution power"
of the Fourier transform of f.  But in general that's making matters 
more complicated rather than less.

>----------------------------
>
>Is there a way to evaluate the laplace transform of
>
>heaviside((exp(x)-a)),
>
>where the heaviside function is also called step function,
>
>heaviside(x) = 1, when x>0, and =0, when x<0, and it has a jump from 0
>to 1 at x=0...

Yes, that's easy.  Hint: when does exp(x) - a change sign?
Do the cases a <= 0 and a > 0 separately.

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

On 20 Aug 2006 15:00:23 -0700, "Luna Moon" <lunamoonmoon@gmail.com>
wrote:

>Hi there,
>
>Suppose I have a function f(t) which I knew its laplace and fourier
>transform.
>
>What is the laplace and fourier transform of the following:
>
>exp(a*f(t))
>
>???
>
>----------------------------
>
>Is there a way to evaluate the laplace transform of
>
>heaviside((exp(x)-a)),
>
>where the heaviside function is also called step function,
>
>heaviside(x) = 1, when x>0, and =0, when x<0, and it has a jump from 0
>to 1 at x=0...
>
>---------------------------
>
>Please give me some pointers! thanks a lot!
The Heaviside transform is just like the Laplace except with the
Heaviside, the step function is already built in, as I recall. So the
inverse of the Heaviside is the response to a unit step.
Laplace requires you to specify the input function so the Laplace
transform is s times the Heaviside. See if that doesn't work.
John Polasek
John Polasek
http://www.dualspace.net

On 20 Aug 2006 15:00:23 -0700, "Luna Moon" <lunamoonmoon@gmail.com>
wrote:

>Hi there,
>
>Suppose I have a function f(t) which I knew its laplace and fourier
>transform.
>
>What is the laplace and fourier transform of the following:
>
>exp(a*f(t))
>
>???
>
>----------------------------
>
>Is there a way to evaluate the laplace transform of
>
>heaviside((exp(x)-a)),
>
>where the heaviside function is also called step function,
>
>heaviside(x) = 1, when x>0, and =0, when x<0, and it has a jump from 0
>to 1 at x=0...
>
>---------------------------
>
>Please give me some pointers! thanks a lot!
(This message did not appear when I posted it last night)
The Heaviside transform is just like the Laplace except with the
Heaviside, the step function is already built in, as I recall. So the
inverse of the Heaviside is the response to a unit step.
Laplace requires you to specify the input function so the Laplace
transform is s times the Heaviside. See if that doesn't work.
John Polasek
John Polasek
http://www.dualspace.net

Luna Moon wrote:
>
> Suppose I have a function f(t) which I knew its laplace and fourier
> transform.
>
> What is the laplace and fourier transform of the following:
>
> exp(a*f(t))
>
> ???

there is no theorem that will do this nicely.  you gotta plug in f(t)
and see what you get.


> Is there a way to evaluate the laplace transform of
>
> heaviside((exp(x)-a)),
>
> where the heaviside function is also called step function,
>
> heaviside(x) = 1, when x>0, and =0, when x<0, and it has a jump from 0
> to 1 at x=0...

this one can be simplified because of the nature of heaviside(x)

    h(x) = heaviside( exp(x) - a )  =  heaviside( x - log(a) )

the Laplace transform is

    H(s) = Laplace{ h(x) } = 1/s * exp(-s*log(a)) = 1/(s*a^s)

r b-j