# complex analysis: integrals

• Dec 3rd 2007, 10:03 PM
hanahou
complex analysis: integrals
http://i3.photobucket.com/albums/y57...ty/prob3-1.jpg

how can i do this, applying Rouche Theorem and Cauchy Argument Principle?
• Dec 4th 2007, 08:38 AM
ThePerfectHacker
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 $z_1 \mbox{ and }z_2$.