Exponential Type - Basic Idea

Basic Idea

A function f(z) defined on the complex plane is said to be of exponential type if there exist constants M and τ such that

in the limit of . Here, the complex variable z was written as to emphasize that the limit must hold in all directions θ. Letting τ stand for the infimum of all such τ, one then says that the function f is of exponential type τ.

For example, let . Then one says that is of exponential type π, since π is the smallest number that bounds the growth of along the imaginary axis. So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than π. Similarly, the Euler-MacLaurin formula cannot be applied either, as it, too, expresses an theorem ultimately anchored in the theory of finite differences.