In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is a result used to establish bounds on the growth rates for analytic functions.
In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, also called Nachbin summation.
This article provides 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.
defined on the complex plane is said to be of exponential type if there exist constants
Here, the complex variable
to emphasize that the limit must hold in all directions
is the smallest number that bounds the growth of
So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than
Additional function types may be defined for other bounding functions besides the exponential function.
is a comparison function if it has a series with
, and Comparison functions are necessarily entire, which follows from the ratio test.
Nachbin's theorem states that a function
if and only if This is naturally connected to the root test and can be considered a relative of the Cauchy–Hadamard theorem.
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
, then the exterior of the domain of convergence of
, and all of its singular points, are contained within the disk Furthermore, one has where the contour of integration γ encircles the disk
This generalizes the usual Borel transform for functions of exponential type, where
The integral form for the generalized Borel transform follows as well.
be a function whose first derivative is bounded on the interval
and that satisfies the defining equation where
Then the integral form of the generalized Borel transform is The ordinary Borel transform is regained by setting
Note that the integral form of the Borel transform is the Laplace transform.
Nachbin summation can be used to sum divergent series that Borel summation does not, for instance to asymptotically solve integral equations of the form: where
may or may not be of exponential type, and the kernel
The solution can be obtained using Nachbin summation as
In some cases as an extra condition we require
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