complex analysis: integrals

Use Cauchy's Argument Principle.

1) $\frac{1}{2\pi i}\oint_{\Gamma}\frac{f'(z)}{f(z)}dz = \mathbb{Z} - \mathbb{P}$ where $\mathbb{P}$ is the number of poles and $\mathbb{Z}$ is the number of zeros (both counting multiplicity). Thus, this equation tells us that $\mathbb{Z} - \mathbb{P} = 2$ but $\mathbb{P} = 0$ because $f(z)$ ia analytic on $\Omega$ . Thus, the function $f(z)$ has two zeros $z_1$ and $z_2$ within the unit disk (and I let $z_1 = z_2$ so if it is the same zero of multiplicity two).

2) $\frac{1}{2\pi i}\oint_{\Gamma}z\frac{f'(z)}{f(z)}dz = z_1 + z_2$ . Thus, $z_1+z_2 = 0$ .

3) $\frac{1}{2\pi i}\oint_{\Gamma}z^2 \frac{f'(z)}{f(z)}dz = z_1^2+z_2^2 = \frac{1}{2}$ .

Now solve for $z_1 \mbox{ and }z_2$ .