I've been studying integrals of the form
$$ \int_{-a}^{b} f(t)e^{zt}dt $$
where $z$ is a complex variable. So far I've been calling them "exponential integrals" following the practice of P. Miller in his book Applied Asymptotic Analysis (see here for an example).
However, there is already a family of integrals called exponential integrals, and I'm not sure whether my and Miller's usage of the name would cause some confusion.
I've considered calling them "Fourier integrals" or "Laplace integrals", but these terms apparently have specific meanings (see, for example, this PDF), for which the variable $z$ is taken to be either purely imaginary or purely real, respectively.
It seems that either way I'd be fudging the common usage. Which would be the better choice?
Or is there another term which would be more appropriate (but just as succinct)?
Two common terms are holomorphic Fourier transform (of $f\chi_{[a,b]}$) and the Fourier-Laplace transform. For example, the Wikipedia article on Paley-Wiener theorem uses both. In your situation, calling the integral the Fourier-Laplace integral of $f$ would be very reasonable.