Bromwich integral of $1/s^k$ with k real (non integer) and $1Is there a simple way to compute the inverse laplace transform of $1/s^k$ with k non integer using Bromwich integral (basically without using the known laplace transform of $t^n$)?
 A: $\newcommand{\+}{^{\dagger}}%
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\begin{align}
&\int_{0}^{-\infty}{\expo{st} \over \pars{-s}^{k}\expo{\ic\pi k}}
\,{\dd s \over 2\pi\ic}
+
\int_{-\infty}^{0}{\expo{st} \over \pars{-s}^{k}\expo{-\ic\pi k}}
\,{\dd s \over 2\pi\ic}
\\[3mm]&=
-\expo{-\ic\pi k}\int_{0}^{\infty}{\expo{-st} \over s^{k}}\,{\dd s \over 2\pi\ic}
+
\expo{\ic\pi k}\int_{0}^{\infty}{\expo{-st} \over s^{k}}\,{\dd s \over 2\pi\ic}
={\sin\pars{\pi k} \over \pi}\int_{0}^{\infty}s^{-k}\expo{-st}\,\dd s
\\[3mm]&=
{\sin\pars{\pi k} \over \pi}\,t^{k - 1}\int_{0}^{\infty}s^{-k}\expo{-s}\,\dd s
={\sin\pars{\pi k} \over \pi}\,t^{k - 1}\Gamma\pars{1 - k}
={\sin\pars{\pi k} \over \pi}\,t^{k - 1}\,{\pi \over \Gamma\pars{k}\sin\pars{\pi k}}
\\[3mm]&=\color{#000}{\large{t^{k - 1} \over \Gamma\pars{k}}}
\end{align}
