Nachbin's Theorem - Exponential Type

Exponential Type

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.