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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2006 • Question 9

Question

Math 6210 • Real Analysis 2006 • Question 9

Suppose \(f\) is holomorphic in an open set \(\Omega \) and \(\overline {D_R}(z_0)\subset \Omega \). If \(f\) has a zero of order \(k\) at \(z_0\) and no other zeroes in \(\overline {D_R}(z_0)\), then prove that there is an \(\epsilon >0\) such that, if \(|\lambda |<\epsilon \), then \(f\) takes on \(\lambda \) as a value \(k\) times (counting multiplicity) in the disc \(D_R(z_0)\).

Raw LaTeX

Suppose \(f\) is holomorphic in an open set \(\Omega\) and
\(\overline{D_R}(z_0)\subset \Omega\). If \(f\) has a zero of order \(k\) at \(z_0\) and
no other zeroes in \(\overline{D_R}(z_0)\), then prove that there is an
\(\epsilon>0\) such that, if \(|\lambda|<\epsilon\), then \(f\) takes on \(\lambda\) as a
value \(k\) times (counting multiplicity) in the disc \(D_R(z_0)\).

Topic Tags

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