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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2013 • Question 6

Question

Math 6420 • PDE Spring 2013 • Question 6

  1. Formulate and derive the argument principle [which determines the difference between the number of zeros \((N_0)\) and poles \((N_\infty )\) of an analytic function \(f(z)\)].
  2. Applying this principle, determine the number of zeros located inside the first quadrant \(\{z=x+iy:\ x>0,\ y>0\}\) of the function \(f(z)=z^5+1\).

Raw LaTeX

\begin{enumerate}
\item[(a)] Formulate and derive the \textit{argument principle} [which determines the
difference between the number of zeros \((N_0)\) and poles \((N_\infty)\) of an analytic
function \(f(z)\)].

\item[(b)] Applying this principle, determine the number of zeros located inside the first
quadrant \(\{z=x+iy:\ x>0,\ y>0\}\) of the function \(f(z)=z^5+1\).
\end{enumerate}

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Isolated Singularities and Residues Linear Programming and Optimization Meromorphic Functions and the Argument Principle