- #1

Pere Callahan

- 586

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**"Generalized" Laplace transform**

Hello,

I'm having trouble proving injectivity of what might be called a "generalized" Laplace transform (not the one by Varma).

Let f be a rational function and C be a fixed closed contour in the complex plane, (such that C contains not pole of f):

The operator L is then defined by

[tex]

(Lf)(w) = \int_C{e^{zw}f(z)dz}.

[/tex]

One could define L also for a more general class of functions, but for the moment rational ones suffice. It is clear that when f is the zero function, Lf is the zero function as well. I'm wondering about the converse; for the ordinary Laplace transform we know that the zero function is in fact the only function yielding a vanishing Laplace transform. I think that should also be true in this case but I have been unable to prove it.

When we restrict the class of functions in the domain even further so as to only contain "step functions" with N steps i was able to prove that (Lf)(w)=0 for N real values of w implies that f is the zero function by using the linear independence of exponentials of different frequencies.

I would appreciate any help in doing the step () from step functions to rational functions.Pere