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. 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 -type.

Basic idea

defined on the complex plane such that

. Here, the complex variable stand for the infimum is of .

along the imaginary axis. So, for this example, Carlson's theorem. Similarly, the Euler–Maclaurin formula cannot be applied either, as it, too, expresses a theorem ultimately anchored in the theory of finite differences.

Formal definition

A holomorphic function is said to be of exponential type such that

. The number

. The limit superior here means the limit of the supremum goes to infinity. For example, for the function

the value of

is dominated by the −1 goes to infinity,.

Exponential type with respect to a symmetric convex body

has given a generalization of exponential type for entire functions of several complex variables is a convex, compact, and symmetric there is an associated norm with the property that

. The set

is called the polar set and is also a convex, compact, and symmetric. Furthermore, we can write

by

such that

.

Fréchet space

can form a complete uniform space, namely a Fréchet space, by the topology induced by the countable family of normss

See also

• Paley–Wiener theorem

• Paley–Wiener space

References