Summability methods nFrom Wikipedia, the free encyclopediaExponential type redirects here. For exponential types in type theory and programming languages, see Functiontype.In mathematics, in the area of complex analysis, Nachbins theorem (named after Leopoldo Nachbin) is commonlyused to establish a bound on the growth rates for an analytic function. This article will provide a brief review ofgrowth rates, including the idea of a function of exponential type. Classication of growth rates based on type helpprovide a ner tool than big O or Landau notation, since a number of theorems about the analytic structure of thebounded function and its integral transforms can be stated. In particular, Nachbins theorem may be used to give thedomain of convergence of the generalized Borel transform, given below.1 Exponential typeMain article: Exponential typeA function f(z) dened on the complex plane is said to be of exponential type if there exist constants M and suchthat|f(rei)| Merin the limit of r . Here, the complex variable z was written as z= reito emphasize that the limit must holdin all directions .Letting stand for the inmum of all such , one then says that the function f is of exponentialtype .For example, let f(z) = sin(z) . Then one says that sin(z) is of exponential type , since is the smallest numberthat bounds the growth of sin(z) along the imaginary axis. So, for this example, Carlsons theorem cannot apply,as it requires functions of exponential type less than .2 typeBounding may be dened for other functions besides the exponential function. In general, a function(t) is acomparison function if it has a series(t) =n=0ntnwith n> 0 for all n, andlimnn+1n= 0.Comparison functions are necessarily entire, which follows from the ratio test. If (t) is such a comparison function,one then says that f is of -type if there exist constants M and such that12 3 BOREL TRANSFORM

f(rei)

M(r)as r . If is the inmum of all such one says that f is of -type .Nachbins theorem states that a function f(z) with the seriesf(z) =n=0fnznis of -type if and only iflimsupn

fnn

1/n= .3 Borel transformNachbins theorem has immediate applications in Cauchy theorem-like situations, and for integral transforms. Forexample, the generalized Borel transform is given byF(w) =n=0fnnwn+1.If f is of -type , then the exterior of the domain of convergence ofF(w) , and all of its singular points, arecontained within the disk|w| .Furthermore, one hasf(z) =12i

(zw)F(w) dwwhere the contour of integration encircles the disk |w| . This generalizes the usual Borel transform forexponential type, where (t) = et. The integral form for the generalized Borel transform follows as well. Let (t)be a function whose rst derivative is bounded on the interval [0, ) , so that1n=0tnd(t)where d(t) = (t) dt . Then the integral form of the generalized Borel transform isF(w) =1w0f( tw)d(t).The ordinary Borel transform is regained by setting (t) = et. Note that the integral form of the Borel transformis just the Laplace transform.34 Nachbin resummationNachbin resummation (generalized Borel transform) can be used to sumdivergent series that escape to the usual Borelsummation or even to solve (asymptotically) integral equations of the form:g(s) = s0K(st)f(t) dtwhere f(t) may or may not be of exponential growth and the kernel K(u) has a Mellin transform. The solution, pointedout by L. Nachbin himself, can be obtained as f(x)=n=0anM(n+1)xnwith g(s)=n=0 ansnand M(n) isthe Mellin transform of K(u). an example of this is the Gram series (x) n=1logn(x)nn!(n+1).5 Frchet spaceCollections of functions of exponential type can form a complete uniform space, namely a Frchet space, by thetopology induced by the countable family of normsfn= supzCexp[(+1n)|z|]|f(z)|.6 See alsoDivergent seriesEuler summationCesro summationLambert summationPhragmnLindelf principleAbelian and tauberian theoremsVan Wijngaarden transformation7 ReferencesL. Nachbin, An extension of the notion of integral functions of the nite exponential type, Anais Acad. Brasil.Ciencias. 16 (1944) 143147.Ralph P. Boas, Jr. and R. Creighton Buck,Polynomial Expansions of Analytic Functions (Second PrintingCorrected), (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of CongressCard Number 63-23263. (Provides a statement and proof of Nachbins theorem, as well as a general review ofthis topic.)A.F. Leont'ev (2001), Function of exponential type, in Hazewinkel, Michiel, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4A.F. Leont'ev (2001), Borel transform, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4Garcia J. Borel Resummation & the Solution of Integral Equations Prespacetime Journal n 4 Vol 4. 2013http://prespacetime.com/index.php/pst/issue/view/42/showToc4 8 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES8 Text and image sources, contributors, and licenses8.1 Text Nachbins theoremSource: https://en.wikipedia.org/wiki/Nachbin{}s_theorem?oldid=649088516 Contributors: Michael Hardy, CharlesMatthews, Giftlite, Linas, Ruud Koot, Sodin, JoshuaZ, Rschwieb, A. Pichler, CRGreathouse, Karl-H, Classicalecon, Yinweichen, Yobot,Sawomir Biay, Mkelly86, Brad7777, K9re11 and Anonymous: 108.2 Images8.3 Content license Creative Commons Attribution-Share Alike 3.0