80 Laplace transform of exponentials and powers

Let ${\nu }_j>-1$ and let $p_j\in 𝐂$ for $j=1,\mathrm{\dots },N$. Then

$$f(x)=\sum _{j=1}^N{a}_j{x}^{{\nu }_j}{e}^{p_{j}x}$$

satisfies $(E)$ for $\beta >\max \{\mathrm{\Re }{p}_j\}$ and $ℒ(f)(s)$ is holomorphic for $\mathrm{\Re }s>\beta$. Proof. Let $\mathrm{\Gamma }(\alpha )={\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}{e}^{-t}𝑑t$ for $\alpha >0$. By direct calculation, we have

$$ℒ(f)(s)=\sum _{j=1}^N{a}_j\frac{\mathrm{\Gamma }({\nu }_j+1)}{{(s-p_j)}^{{\nu }_j+1}}.$$