I'm trying to compute the inverse Laplace transform of $f(s) = s^c/(N + s^{ir} )$ where $c,N \in \mathbb{C}$ and $r \in \mathbb{R}^+$ using the Bromwich integral $$ F(t) = \frac{1}{2 \pi i} \int_{- i \infty + \epsilon}^{i \infty+ \epsilon}\,\, ds \frac{s^c e^{st} }{N + s^{ir} } $$ with the contour deformed around a branch cut along the negative real axis from infinity to the branch point $s=0$. Note that $N$ is a complicated expression that changes with choice of $c$, and also that $c$ and $r$ are dependent.

Mathematica will not compute it via the builtin InverseLaplaceTransform function.

I've tried to compute it by hand, decomposing the contour into a small circle $\{ |s|= \delta \ll 1 , -\pi < \theta \leq \pi\}$ around the branch point and the branch cut $\theta=\pi$ with $|s|$ running from $\delta$ to $\infty$. Mathematica can do these integrals if I express the exponential $e^{st}$ as an infinite sum, giving a rather intractable sum of hypergeometric functions ${}_2 F_{1}$.

If the integral cannot be done with standard analytic methods, it would be fine to do it numerically, but it would be really nice to have at least the scaling of $F$ for large $t$ analytically. I've numerically computed some values of $N$ and it appears that $\vert{N+s^{ir}}\vert$ is strictly positive which could aid with a proof of boundedness if true for all $c$, $N$, and $r$.


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