Nachbin's theorem

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 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. In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, given below. Exponential type A function defined on the complex plane is said to be of exponential type if there exist constants and such that in the limit of . Here, the complex variable was written as to emphasize that the limit must hold in all directions . Letting stand for the infimum of all such , one then says that the function is of exponential type . For example, let . Then one says that is of exponential type , since is the smallest number that bounds the growth of along the imaginary axis. So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than . Bounding may be defined for other functions besides the exponential function. In general, a function is a comparison function if it has a series with for all , and Comparison functions are necessarily entire, which follows from the ratio test. If is such a comparison function, one then says that is of -type if there exist constants and such that as . If is the infimum of all such one says that is of -type . Nachbin's theorem states that a function with the series is of -type if and only if 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 is of -type , 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 .