MATH 406: Additional Problems for PS 5



  1. Show explicitly whether or not the following functions are analytic by checking the Cauchy-Riemann equations:

       a) $\displaystyle \, z^2$   b) $\displaystyle \, x^2y + ixy^2$   c) $\displaystyle \, \cos z
$


  2. State why the function $ f(z) = 6z^3 - z + 29i -ze^z + e^{-z}$ is entire.


  3. Is the function $ h(z) = \bar{z} / \vert z\vert^2$ analytic? Show why or why not.


  4. Show explicitly that $ \vert\exp(z^2)\vert \le \exp(\vert z^2\vert)$ .


  5. Let $ f(z) = u(x,y) + iv(x,y)$ be analytic in $ D \subset \mathbb{C}$ . Show that

    $\displaystyle \psi(x,y) = e^{u(x,y)}\cos v(x,y) \qquad\qquad
\phi(x,y) = e^{u(x,y)}\sin v(x,y)$

    are harmonic in $ D$ , and that $ \phi$ is a harmonic conjugate of $ \psi$ .


  6. For each given function, find a harmonic conjugate: a) $ u = y^3 - 3x^2y$ ; b) $ u = x-y$ .




Andrew L. Belmonte 2005-02-07