In mathematics, in the area of complex analysis, Nachbin's theorem 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. The theorem is named in honour of Leopoldo Nachbin.

Exponential type

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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

$|f(re^\{i\backslash theta\})|\backslash le\; Me^\{\backslash tau\; r\}$

in the limit of $r\backslash to\backslash infty$. Here, the complex variable z was written as $z=re^\{i\backslash theta\}$ 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 $f(z)=\backslash sin(\backslash pi\; z)$. Then one says that $\backslash sin(\backslash pi\; z)$ is of exponential type π, since π is the smallest number that bounds the growth of $\backslash sin(\backslash pi\; z)$ along the imaginary axis. So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than π.

Ψ type

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Bounding may be defined for other functions besides the exponential function. In general, a function $\backslash Psi(t)$ is a comparison function if it has a series

$\backslash Psi(t)=\backslash sum\_\{n=0\}^\backslash infty\; \backslash Psi\_n\; t^n$

with $\backslash Psi\_n>0$ for all n, and

$\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash frac\{\backslash Psi\_\{n+1\}\}\{\backslash Psi\_n\}\; =\; 0$

Note that comparison functions are necessarily entire, which follows from the ratio test. If $\backslash Psi(t)$ is such a comparison function, one then says that f is of Ψ-type if there exist constants M and τ such that

$\backslash left|f\backslash left(re^\{i\backslash theta\}\backslash right)\backslash right|\; \backslash le\; M\backslash Psi(\backslash tau\; r)$

as $r\backslash to\; \backslash infty$. If τ is the infimum of all such τ one says that f is of Ψ-type τ.

Nachbin's theorem

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Nachbin's theorem states that a function f(z) with the series

$f(z)=\backslash sum\_\{n=0\}^\backslash infty\; f\_n\; z^n$

is of Ψ-type τ if an only if

$\backslash limsup\; \backslash left|\; \backslash frac\{f\_n\}\{\backslash Psi\_n\}\; \backslash right|^\{1/n\}\; =\; \backslash tau$

Borel transform

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Nachbin's theorem has immediate applications in Cauchy theorem-like situations, and for integral transforms. For example, the generalized Borel transform is given by

$F(w)=\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{f\_n\}\{\backslash Psi\_n\; w^\{n+1\}\}$

If f is of Ψ-type τ, then the exterior of the domain of convergence of $F(w)$, and all of its singular points, are contained within the disk

$|w|\; \backslash le\; \backslash tau$

Furthermore, one has

$f(z)=\backslash frac\{1\}\{2\backslash pi\; i\}\; \backslash oint\_\backslash gamma\; \backslash Psi\; (zw)\; F(w)\; dw$

where the contour of integration γ encircles the disk $|w|\; \backslash le\; \backslash tau$. This generalizes the usual Borel transform for exponential type, where $\backslash Psi(t)=e^t$. The integral form for the generalized Borel transform follows as well. Let $\backslash alpha(t)$ be a function whose first derivative is bounded on the interval $[0,\backslash infty)$, so that

$\backslash frac\{1\}\{\backslash Psi\_n\}\; =\; \backslash int\_0^\backslash infty\; t^n\; d\backslash alpha(t)$

where $d\backslash alpha(t)=\backslash alpha^\{\backslash prime\}(t)dt$. Then the integral form of the generalized Borel transform is

$F(w)=\backslash frac\{1\}\{w\}\; \backslash int\_0^\backslash infty\; f\; \backslash left(\backslash frac\{t\}\{w\}\backslash right)\; d\backslash alpha(t)$

The ordinary Borel transform is regained by setting $\backslash alpha(t)=e^\{-t\}$. Note that the integral form of the Borel transform is just the Laplace transform.

References

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• L. Nachbin, "An extension of the notion of integral functions of the finite exponential type", Anais Acad. Brasil. Ciencias. 16 (1944) 143-147.
• Ralph P. Boas, Jr. and R. Creighton Buck, Polynomial Expansions of Analytic Functions (Second Printing Corrected), (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263. (Provides a statement and proof of Nachbin's theorem, as well as a general review of this topic.)

Complex analysis | Mathematical theorems