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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Fall 2025 • Question 5

Question

Math 6220 • Complex Analysis Fall 2025 • Question 5

Let \(f_1,f_2,\ldots :\mathbb {C}\to \mathbb {C}\) be a sequence of non-vanishing entire functions that converge uniformly on compact sets to an entire function \(f_\infty \). Show that \(f_\infty \) is either identically 0 or non-vanishing.

Raw LaTeX

Let \(f_1,f_2,\ldots:\mathbb{C}\to\mathbb{C}\) be a sequence of non-vanishing entire functions
that converge uniformly on compact sets to an entire function \(f_\infty\). Show
that \(f_\infty\) is either identically 0 or non-vanishing.

Topic Tags

Entire Functions and Liouville-Type Theorems