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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2017 • Question 6

Question

Math 6210 • Real Analysis Fall 2017 • Question 6

Let \(f\) be a function holomorphic for \(z\) in the annulus \(r<z<R\). Show that if for some \(\rho \) with \(r<\rho <R\), \[ \int _{|z|=\rho } z^n f(z)\,dz=0 \] for all nonnegative \(n\), then \(f\) extends to a holomorphic function on the disk \(|z|<R\).

Raw LaTeX

Let \(f\) be a function holomorphic for \(z\) in the annulus
\(r<z<R\). Show that if for some \(\rho\) with \(r<\rho<R\),
\[
\int_{|z|=\rho} z^n f(z)\,dz=0
\]
for all nonnegative \(n\), then \(f\) extends to a holomorphic function on the
disk \(|z|<R\).

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Laurent Series and Annuli Lebesgue Integration