LAPLACE TRANSFORM OF A COMPLEX POWER

In this post we will show that



\begin{align*} \int_0^\infty e^{- s \,t} t^{w-1} \,dt=\frac{\Gamma\left(w \right)}{s^w} \end{align*}


For \text{Re}(w), \text{Re}(s)>0, where s=\left| s \right| e^{i \,\text{arg}(s)}=\left| s \right| e^{i \,\beta} and \left| \text{arg}(s) \right| < \pi/2.


As a Corollary we also obtain the beatiful pair of integrals



\begin{align*} &\int_0^\infty t^{w-1} e^{- a \,t} \cos\left( bt\right)\,dt=\frac{\Gamma\left(w \right)}{\left(a^2+b^2\right)^{w/2}} \cos\left(w\, \arctan\frac{b}{a} \right)\\ \\ &\int_0^\infty t^{w-1} e^{- a \,t} \sin\left( bt\right)\,dt=\frac{\Gamma\left(w \right)}{\left(a^2+b^2\right)^{w/2}} \sin\left( w\,\arctan\frac{b}{a} \right) \end{align*}



Click here for the proof.

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