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

   
   

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

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