New dimensional approach to the measurement of growths of complex valued functions

Abstract

A function f de...ned in the open complex plane C is said to be analytic newlineat at a point z 0 if there exists a neighbourhood of z 0 at all points of which newlinef 0 (z) exists. If f is not analytic at z 0 then the point z 0 is called a singular newlinepoint or the singularity of f . Now if f be a single valued analytic function newlineon an annulus D : r 2 lt jz newlinej lt r 1 then at each point z 2 D, f can be newline1 newline1 newlineX newlineX newlinen newlinerepresented by a series of the form f (z) = newlinea n (z newline) + newlineb n (z newline) n ; newlinen=0 newlinen=1 newlineR newlineR newlinef (z) newline1 newlinewhere a n = 2 1 i C (z f (z) newlinedz newlineand newlineb newline= newlinedz: newlinen newline2 i C (z newline) n+1 newline) n+1 newlineThe above series is called the Laurent s series of f about the point z = : newlineA function f de...ned in the open complex plane C is said to be mero- newlinemorphic is it is analytic except at its poles. A function f is said to be newlinean entire or an integral function if it is analytic everywhere in the ...nite newlinecomplex plane. The Taylor series expansion of f about z = 0 is given by newlinef = a 0 + a 1 z + a 2 z 2 + ::: ::: ::: + a n z n + :::::: , which can be expressed as newlinean extension of a polynomial. The rate of growth of of a polynomial is es- newlinetimated by the degree of the polynomial, which is equal to the number of newlinezeros, as independent variable moves without bound. newlineThe maximum modulus function of an entire function f on jzj = r is newlinede...ned as M (r; f ) = max jf (z)j which is especially used to characterise the newlinejzj=r newlinegrowth of an entire function and the distribution of its zeros. Also M (r; f ) newlineis unbounded for any non-constant entire function and by maximum mod- newlineulus theorem M (r; f ) increases monotonically as r increases. The function newlinelog M (r; f ) is a continuous, convex and increasing function of log r: newline