scholarly journals Hole probability for entire functions represented by Gaussian Taylor series

2012 ◽  
Vol 118 (2) ◽  
pp. 493-507 ◽  
Author(s):  
Alon Nishry
Keyword(s):  
Taylor Series ◽  
Entire Functions ◽  
Hole Probability

scholarly journals On a Result of Hayman Concerning the Maximum Modulus Set

2021 ◽  
Author(s):  
Vasiliki Evdoridou ◽  
Leticia Pardo-Simón ◽  
David J. Sixsmith
Keyword(s):  
Entire Function ◽  
Taylor Series ◽  
Upper Bound ◽  
Entire Functions ◽  
Maximum Modulus ◽  
Algebraic Condition ◽  
Exact Number ◽  
Small Set ◽  
Set Of Points ◽  
Analytic Curves

AbstractThe set of points where an entire function achieves its maximum modulus is known as the maximum modulus set. In 1951, Hayman studied the structure of this set near the origin. Following work of Blumenthal, he showed that, near zero, the maximum modulus set consists of a collection of disjoint analytic curves, and provided an upper bound for the number of these curves. In this paper, we establish the exact number of these curves for all entire functions, except for a “small” set whose Taylor series coefficients satisfy a certain simple, algebraic condition. Moreover, we give new results concerning the structure of this set near the origin, and make an interesting conjecture regarding the most general case. We prove this conjecture for polynomials of degree less than four.

On the Growth of Entire Function Defined by Multiple Taylor

2014 ◽  
Vol 905 ◽  
pp. 786-789
Author(s):  
Wan Chun Lu ◽  
You Hua Peng
Keyword(s):  
Entire Function ◽  
Taylor Series ◽  
Necessary Conditions ◽  
Entire Functions ◽  
Polar Coordinates ◽  
Growth Of Entire Functions ◽  
Sufficient And Necessary Conditions

It is studied that the growth of entire functions defined by multiple Taylor series by means of Polar coordinates, and established two sufficient and necessary conditions.

scholarly journals Hole probability for zeroes of Gaussian Taylor series with finite radii of convergence

2017 ◽  
Vol 171 (1-2) ◽  
pp. 377-430 ◽  
Author(s):  
Jeremiah Buckley ◽  
Alon Nishry ◽  
Ron Peled ◽  
Mikhail Sodin
Keyword(s):  
Taylor Series ◽  
Hole Probability
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