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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1997 • Question 6

Question

Math 6210 • Real Analysis Fall 1997 • Question 6

Let \(f(z)\) be an entire function and let \(F_n(z)=\sum _{k=1}^{n}\frac {f^k(z)}{k!}\). Show that for every \(R>0\) there exists \(N\) such that all the zeros of \(F_n\), \(n\ge N\), are outside the circle of radius \(R\).

Raw LaTeX

Let \(f(z)\) be an entire function and let
\(F_n(z)=\sum_{k=1}^{n}\frac{f^k(z)}{k!}\). Show that for every \(R>0\) there
exists \(N\) such that all the zeros of \(F_n\), \(n\ge N\), are outside the circle of
radius \(R\).

Topic Tags

Entire Functions and Liouville-Type Theorems Zeros of Holomorphic Functions and the Identity Theorem