complex integration, residues, inverse Laplace transform, calculus

Question

Dear Mathematicians,

I kindly ask your expertise on complex integration. The problem is the last step in the solution to a differential equation, using an inverse Laplace transform.

I know that the ILT is equal to the sum of the residues of the poles of the Laplace transform. My solution is a function of the form $f(s) = \frac{p(s)}{q(s)}$. The residue of a simple pole s0 of $f(s)$ is $\frac{p(s_0)}{q'(s_0)}$, $q'(s_0)$ being the derivative with respect to the Laplace variable. But do I have to proceed if $q'(s_0)$ is zero, possibly also for higher derivatives ?

It would be marvellous if you could also state a reference for the solution as I need to include the solution in a scientific report.

Thank you for sharing,

Frank

Answer 1

Example $2$ in this article on "Laplace inverse transforms via residue theory" answers your exact question.

In general, if a function $f(z)$ has a pole of order $n$ at $a$, then its residue at this pole is given by:

$$\mbox{Res}_{z=a}f = \frac {1}{(n-1)!}\lim_{z\rightarrow a} \frac{d^{n-1}}{dz^{n-1}} [(z-a)^n f(z)].$$