I was wondering: Does a given time function has several Laplace transforms? For example, take a simple exponential function:
$f(t)=e^{-at}$
When the inverse Laplace transform was done with the Residue theorem, then this time function is only depended by the poles in the Laplace domain and the derivative of the Laplace transform at these poles.
$\mathcal{L}^{-1}=Res[\frac{1}{g(s)}\cdot e^{st}, s=poles]=\frac{1}{\frac{\partial g(s)}{\partial s}}\cdot e^{st}|_{s=poles} $
$g(s)$ can be any function of s with poles. For me it seems, that as long as a Laplace function has:
- the same poles
- the same derivative at the poles
there exists several Laplace transform to one time function.
As an example to such Laplace functions:
$\frac{1}{J_0(\sqrt{s})\cdot \sinh(J_0(\sqrt{s}))}$
$\frac{1}{J_0(\sqrt{s})\cdot \tanh(J_0(\sqrt{s}))}$
Thanks for the help