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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2013 • Question 7

Question

Math 6210 • Real Analysis Spring 2013 • Question 7

Let \(f\) be a holomorphic function and \(a\) an isolated singularity of \(f\). Prove that the following statements are equivalent:

(a) there exists a disk \(D(a,\epsilon )=\{z\in \mathbb C\mid |z-a|<\epsilon \}\) and a constant \(C>0\) such that \[ |f(z)|\ge \frac {C}{|z-a|^n} \] for all \(z\in D(a,\epsilon )-\{a\}\);

(b) \(f\) has a pole of order \(m\ge n\) at \(a\).

Raw LaTeX

Let \(f\) be a holomorphic function and \(a\) an isolated singularity
of \(f\). Prove that the following statements are equivalent:

(a) there exists a disk
\(D(a,\epsilon)=\{z\in\mathbb C\mid |z-a|<\epsilon\}\) and a constant \(C>0\)
such that
\[
|f(z)|\ge \frac{C}{|z-a|^n}
\]
for all \(z\in D(a,\epsilon)-\{a\}\);

(b) \(f\) has a pole of order \(m\ge n\) at \(a\).

Topic Tags

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