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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1994 • Question 7

Question

Math 6210 • Real Analysis Fall 1994 • Question 7

Let \(P(z)=a_nz^n+\cdots +a_0\) and suppose that \(P\) has zeroes at \(z_1,\ldots ,z_r\) each with multiplicity \(m_1,\ldots ,m_r\) respectively. Let \(\Omega \) be some domain not containing \(z_1,\ldots ,z_r\), and \(\gamma \) a cycle in \(\Omega \). Suppose further that there exists \(g\) holomorphic on \(\Omega \) such that \(g^p=P\). Show that there exists \(k\in \mathbb Z\) such that \[ m_1\operatorname {Ind}(\gamma ,z_1)+\cdots +m_r\operatorname {Ind}(\gamma ,z_r)=kp. \]

Raw LaTeX

Let \(P(z)=a_nz^n+\cdots+a_0\) and suppose that \(P\) has zeroes at
\(z_1,\ldots,z_r\) each with multiplicity \(m_1,\ldots,m_r\) respectively. Let
\(\Omega\) be some domain not containing \(z_1,\ldots,z_r\), and \(\gamma\) a cycle
in \(\Omega\). Suppose further that there exists \(g\) holomorphic on \(\Omega\) such
that \(g^p=P\). Show that there exists \(k\in\mathbb Z\) such that
\[
m_1\operatorname{Ind}(\gamma,z_1)+\cdots+m_r\operatorname{Ind}(\gamma,z_r)=kp.
\]

Topic Tags

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