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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Summer 2021 • Question 4

Question

Math 6220 • Complex Analysis Summer 2021 • Question 4

Suppose that \(f_n\) are holomorphic on a domain \(U\subset \mathbb {C}\) and \(f_n(U)\subset \mathbb {C}\setminus \{w\}\) for all \(n\) and some fixed \(w\in \mathbb {C}\). Show that if \(f_n\to f\) locally uniformly in \(U\) then either \(f(z)\equiv w\) or \(f(U)\subset \mathbb {C}\setminus \{w\}\).

Raw LaTeX

Suppose that \(f_n\) are holomorphic on a domain \(U\subset\mathbb{C}\) and \(f_n(U)\subset\mathbb{C}\setminus\{w\}\)
for all \(n\) and some fixed \(w\in\mathbb{C}\). Show that if \(f_n\to f\) locally uniformly in
\(U\) then either \(f(z)\equiv w\) or \(f(U)\subset\mathbb{C}\setminus\{w\}\).

Topic Tags

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