 Mellin transform of call prices

For $C(k)=\mathbb{E}(e^{\xi }-e^k{)}^{+}$ and $\mathrm{Re}z>1$ in the exponential moment domain, the moment-generating function satisfies $\mathbb{E}{e}^{z\xi }={\int }_{\mathbb{R}}C(k)z(z-1)e^{(z-1)k}dk$.