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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2014 • Question 7

Question

Math 6420 • PDE Fall 2014 • Question 7

  1. Show that any analytic function (not identically equal to zero) can have only isolated zeros inside its analyticity domain.
  2. Prove the Uniqueness Theorem: If two functions are analytic in a domain \(D\) and equal on some set of points that has a limiting point inside \(D\), then these functions are identically equal in \(D\).

Raw LaTeX

\begin{enumerate}
\item[(a)] Show that any analytic function (not identically equal to zero) can have only
isolated zeros inside its analyticity domain.

\item[(b)] Prove the Uniqueness Theorem: If two functions are analytic in a domain \(D\) and
equal on some set of points that has a limiting point inside \(D\), then these functions are
identically equal in \(D\).
\end{enumerate}

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Isolated Singularities and Residues Linear Programming and Optimization