In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function. This article will provide a brief review of growth rates, including the idea of a function of exponential type. Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated. In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, given below.
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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
For example, let
Ψ type
Bounding may be defined for other functions besides the exponential function. In general, a function
with
Comparison functions are necessarily entire, which follows from the ratio test. If
as
Nachbin's theorem states that a function f(z) with the series
is of Ψ-type τ if and only if
Borel transform
Nachbin's theorem has immediate applications in Cauchy theorem-like situations, and for integral transforms. For example, the generalized Borel transform is given by
If f is of Ψ-type τ, then the exterior of the domain of convergence of
Furthermore, one has
where the contour of integration γ encircles the disk
where
The ordinary Borel transform is regained by setting
Nachbin resummation
Nachbin resummation (generalized Borel transform) can be used to sum divergent series that escape to the usual Borel summation or even to solve (asymptotically) integral equations of the form:
where f(t) may or may not be of exponential growth and the kernel K(u) has a Mellin transform. The solution can be obtained as
Fréchet space
Collections of functions of exponential type