I would like to ask you to help me determining the following inverse Laplace transform:
$\mathcal{L}^{-1}\left( \frac{\left( s - |\lambda| \right)^{\alpha} }{\left( s + |\lambda|\right)^{\alpha+1}} \right)$ = ?
where $\alpha$ is a positive real number, $\lambda$ is a real number, and the inverse Laplace transform is square integrable.
What I do not know, is the answer.
What I know is as follows:
- According to Oberhettinger, p. 239, Eq. (4.15)
$\mathcal{L}^{-1}\left( \frac{\left( s - |\lambda| \right)^{\nu} }{\left( s + |\lambda|\right)^{\mu}} \right) = (2 \lambda)^{(-\nu+\mu)/2} (\Gamma(\nu))^{-1} x^{(-\mu+\nu)/2-1} M_{(\mu+\nu)/2 , (\mu-\nu-1)/2}(2\lambda x) $
if $Re\left(\mu-\nu \right)>0$
This solution is, however, not square integrable.
If $\alpha \in N$, then we get the Laguerre functions.
I implemented Peter den Iseger's inverse Laplace transform algorithm to check, if the solution exists. For the Bromwich integral I chose a path $s: s= 0.089 + i y, y = -\infty ... \infty$. I experienced, that
=> for $\alpha \in N$ is the real part, indeed, the Laguerre function of order $n$, and the imaginary part is zero ( modulo num noise)
=> for $\alpha \in R, \notin N$ I get the following set of functions, parameterised with $\alpha$:
Result of numerical inverse Laplace transform of the problem
As we see, indeed, the solutions are square integrable and converge to the Laguerre functions of integer order, if $\alpha \rightarrow \alpha_N$
Thx in advance.