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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2016 • Question 7

Question

Math 6420 • PDE Fall 2016 • Question 7

  1. Prove: If a function is analytic then its real and imaginary parts are harmonic.
  2. Prove: If a function \(u(x,y)\) is harmonic in a domain \(D\), then \(u(x,y)\) cannot attain a strict local maximum in \(D\). [Hint: If \(f(z)=u+iv\), then function \(\phi (z)=e^{f(z)}\) has absolute value \(\lvert \phi (z)\rvert =e^u\).]

Raw LaTeX

\begin{enumerate}
\item[(a)] Prove: If a function is analytic then its real and imaginary parts are harmonic.

\item[(b)] Prove: If a function \(u(x,y)\) is harmonic in a domain \(D\), then \(u(x,y)\) cannot
attain a strict local maximum in \(D\). [Hint: If \(f(z)=u+iv\), then function
\(\phi(z)=e^{f(z)}\) has absolute value \(\lvert\phi(z)\rvert=e^u\).]
\end{enumerate}

Topic Tags

Absolute Stability and A-Stability Harmonic Functions and Poisson Integral Formula Laplace Equation Linear Programming and Optimization