Exponential type

In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function

for some real-valued constant

When a function is bounded in this way, it is then possible to express it as certain kinds of convergent summations over a series of other complex functions, as well as understanding when it is possible to apply techniques such as Borel summation, or, for example, to apply the Mellin transform, or to perform approximations using the Euler–Maclaurin formula.

The general case is handled by Nachbin's theorem, which defines the analogous notion of

defined on the complex plane is said to be of exponential type if there exist real-valued constants

to emphasize that the limit must hold in all directions

stand for the infimum of all such

f ( z ) = sin ⁡ ( π z )

sin ⁡ ( π z )

is the smallest number that bounds the growth of

sin ⁡ ( π z )

along the imaginary axis.

So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than

Similarly, the Euler–Maclaurin formula cannot be applied either, as it, too, expresses a theorem ultimately anchored in the theory of finite differences.

there exists a real-valued constant

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

term so we have the asymptotic expressions: and this goes to zero as

is nevertheless of exponential type 1, as can be seen by looking at the points

Stein (1957) has given a generalization of exponential type for entire functions of several complex variables.

is a convex, compact, and symmetric subset of

with the property that In other words,

The set is called the polar set and is also a convex, compact, and symmetric subset of

Furthermore, we can write We extend

-complex variables is said to be of exponential type with respect to

there exists a real-valued constant

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