A function f defined on the complex plane is said to be of exponential type if there exist real-valued constants M and τ such that in the limit of. Here, the complex variablez 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 a theorem ultimately anchored in the theory offinite differences.
Formal definition
A holomorphic function is said to be of exponential type if for every there exists a real-valued constant such that for where. We say is of exponential type if is of exponential type for some. The number is the exponential type of. The limit superior here means the limit of the supremum of the ratio outside a given radius as the radius goes to infinity. This is also the limit superior of the maximum of the ratio at a given radius as the radius goes to infinity. The limit superior may exist even if the maximum at radius r does not have a limit as r goes to infinity. For example, for the function the value of at is asymptotic to and thus goes to zero as n goes to infinity, but F is nevertheless of exponential type 1, as can be seen by looking at the points.
has given a generalization of exponential type for entire functions of several complex variables. Suppose is a convex, compact, and symmetric subset of. It is known that for every such there is an associated norm with the property that In other words, is the unit ball in with respect to. The set is called the polar set and is also a convex, compact, and symmetric subset of. Furthermore, we can write We extend from to by An entire function of -complex variables is said to be of exponential type with respect to if for every there exists a real-valued constant such that for all.
Collections of functions of exponential type can form a complete uniform space, namely a Fréchet space, by the topology induced by the countable family of norms