Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2001 • Question 9

Question

Math 6210 • Real Analysis 2001 • Question 9

Suppose \(f\) is holomorphic in \(r<|z|<R\), and suppose that for some \(\rho \), \(r<\rho <R\), \[ \int _{|z|=\rho } f(z)z^n\,dz=0 \] for all negative integers \(n\). Show that \(f\) has a holomorphic extension to the set \(|z|>r\).

Raw LaTeX

Suppose \(f\) is holomorphic in \(r<|z|<R\), and suppose that for some
\(\rho\), \(r<\rho<R\),
\[
\int_{|z|=\rho} f(z)z^n\,dz=0
\]
for all negative integers \(n\). Show that \(f\) has a holomorphic extension to the
set \(|z|>r\).

Topic Tags

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