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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Fall 2021 • Question 6

Question

Math 6220 • Complex Analysis Fall 2021 • Question 6

Let \(D\subset \mathbb {C}\) be a connected open set and \(f_1,\ldots :D\to \mathbb {C}\) be a sequence of holomorphic functions that converges uniformly on compact sets to a function \(g\). Show that \(g\) is holomorphic and \(g'(z)=\lim _{j\to \infty } f_j'(z)\) for all \(z\in D\).

Raw LaTeX

Let \(D\subset\mathbb{C}\) be a connected open set and \(f_1,\ldots:D\to\mathbb{C}\) be a sequence of
holomorphic functions that converges uniformly on compact sets to a function
\(g\). Show that \(g\) is holomorphic and \(g'(z)=\lim_{j\to\infty} f_j'(z)\) for all \(z\in D\).

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Riemann Mapping Theorem and Simply Connected Domains Zeros of Holomorphic Functions and the Identity Theorem