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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Summer 2017 • Question 10

Question

Math 6210 • Real Analysis Summer 2017 • Question 10

Let \(f\) be a holomorphic function on \(D\) such that \(f^{(n)}(0)\ne 0\) for an infinite number of positive integers \(n\). Show that there exists a point \(z_0\in D\) such that \(f^{(n)}(z_0)\ne 0\) for all \(n>0\).

Raw LaTeX

Let \(f\) be a holomorphic function on \(D\) such that
\(f^{(n)}(0)\ne0\) for an infinite number of positive integers \(n\). Show that
there exists a point \(z_0\in D\) such that \(f^{(n)}(z_0)\ne0\) for all \(n>0\).

Topic Tags

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