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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2015 • Question 6

Question

Math 6210 • Real Analysis Fall 2015 • Question 6

Prove or give a counterexample: If \(\Omega \) is a region, \(f:\Omega \to \mathbb C\) admits unrestricted analytic continuation on a region \(D\supset \Omega \), then there exists \(F:D\to \mathbb C\), holomorphic and with \(F|_\Omega =f\).

Recall: If \(f:\Omega \to \mathbb C\) is an analytic function, we say \(f\) admits unrestricted analytic continuation on \(D\) if for any \(\gamma :[0,1]\to D\), a piecewise smooth path, there exists an analytic continuation of \((f,\Omega )\) along the curve \(\gamma \).

Raw LaTeX

Prove or give a counterexample: If \(\Omega\) is a region,
\(f:\Omega\to\mathbb C\) admits unrestricted analytic continuation on a region
\(D\supset\Omega\), then there exists \(F:D\to\mathbb C\), holomorphic and with
\(F|_\Omega=f\).

Recall: If \(f:\Omega\to\mathbb C\) is an analytic function, we say \(f\) admits
unrestricted analytic continuation on \(D\) if for any
\(\gamma:[0,1]\to D\), a piecewise smooth path, there exists an analytic
continuation of \((f,\Omega)\) along the curve \(\gamma\).

Topic Tags

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