Fourier transform of $x \mapsto e^{x}e^{-e^x}$

Question

I don't even know where to start to tackle this integral: $$\int_{\mathbb{R}} \exp\left(x - e^x-ix\xi\right)\,dx$$

Any hints or help will be greatly appreciated, thanks!

Answer 1

Let $u = e^x$ then your integral becomes $$ \int_0^\infty e^{-u} u^{-i\xi} ~d u. $$ This integral is the Gamma function. So $$ \int_0^\infty e^{-u} u^{-i\xi} ~d u = \Gamma(1-i\xi). $$ (I'm assuming $\xi$ is a real number?)