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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Summer 2022 • Question 4

Question

Math 6220 • Complex Analysis Summer 2022 • Question 4

Suppose \(f:\mathbb {D}\setminus \{0\}\to \mathbb {C}\) is holomorphic and has an essential singularity at 0. Show that for all \(\epsilon > 0\) there is \(z\) near 0 so that \[ \left |f(z)-\frac {1}{z^4}\right |\le \epsilon . \]

Raw LaTeX

Suppose \(f:\mathbb{D}\setminus\{0\}\to\mathbb{C}\) is holomorphic and has an essential singularity at
0. Show that for all \(\epsilon > 0\) there is \(z\) near 0 so that
\[
\left|f(z)-\frac{1}{z^4}\right|\le\epsilon.
\]

Topic Tags

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