E-functions are power series which solve a linear differential equation, and whose coefficients are algebraic numbers subject to a growth condition of arithmetic nature. The exponential series is an example of an E-function, as well as Bessel functions with integer parameter. Siegel introduced these functions in 1929 with the goal of generalising the Hermite-Lindemann-Weierstrass theorem on the transcendence of the values of the exponential function at algebraic numbers.
In a joint work with Javier Fresán, we found that certain exponential integrals associated with algebraic varieties give rise to differential equations whose solutions can be expressed in terms of E-functions. I will present some of this work, and talk about the inverse problem of representing a given E-function, or rather the associated differential equation, by an exponential integral, and relating these objects thereby to algebraic geometry.