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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Fall 2020 • Question 2

Question

Math 6220 • Complex Analysis Fall 2020 • Question 2

Prove that if \(f:\{z:1<|z|<2\}\to \mathbb {C}\) is holomorphic then \[ f(z)=\sum _{j=-\infty }^{\infty } a_j z^j. \] What can you say about the power series \(\sum _{j=0}^{\infty }a_jz^j\) and \(\sum _{j=-\infty }^{0}a_jz^{-j}\)? Prove this.

Raw LaTeX

Prove that if \(f:\{z:1<|z|<2\}\to\mathbb{C}\) is holomorphic then
\[
f(z)=\sum_{j=-\infty}^{\infty} a_j z^j.
\]
What can you say about the power series \(\sum_{j=0}^{\infty}a_jz^j\) and \(\sum_{j=-\infty}^{0}a_jz^{-j}\)? Prove
this.

Topic Tags

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