The special function ش , III

Andrea Ossicini

We prove the following exact symbolic formula of the special function <^s>, in the entire s-complex plane with the negative real axis (including the origin) removed, with a double Laplace transform: $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{<^s>}\;(s) = L\left\{ \right. 2 \cdot \delta \left( t \right) + L \left\{ \right. \frac{1}{2\pi i}\cdot $[$\;\mbox{<^s>}\;(t \cdot e^{\; - i\pi }) - \mbox{<^s>}\; (t \cdot e^{\;i\pi })$]$\left. \right\} \left. \right\} $ \noindent where $\delta \left( t \right)$ stands for the distribution of Dirac and $e$ represents the Euler's number.