Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals

This article is a short elementary review of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of analysis. Some new properties are included, and several analysis-related applications are mentioned. At the end of the paper one application is described in details—certain Fourier integrals involving Γ(a+it) and Γ(a+it)Γ(b−it) are evaluated in terms of Stirling numbers.