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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2011 • Question 8

Question

Math 6210 • Real Analysis Spring 2011 • Question 8

Let \(f\) be a holomorphic function on \(U\subset \mathbb C\). Let \(a\in U\) and let \(k\) be a positive integer. Prove that the following statements are equivalent:

(i) \(a\) is a zero of order \(\ge k\) of \(f\);

(ii) there exist \(C,\epsilon >0\) such that \[ |f(z)|\le C|z-a|^k \] for all \(z\) such that \(|z-a|<\epsilon \).

Raw LaTeX

Let \(f\) be a holomorphic function on \(U\subset\mathbb C\). Let
\(a\in U\) and let \(k\) be a positive integer. Prove that the following
statements are equivalent:

(i) \(a\) is a zero of order \(\ge k\) of \(f\);

(ii) there exist \(C,\epsilon>0\) such that
\[
|f(z)|\le C|z-a|^k
\]
for all \(z\) such that \(|z-a|<\epsilon\).

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Zeros of Holomorphic Functions and the Identity Theorem