Complex Analysis (Quick Review)

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• If any pair of points $z_1, z_2 \in S$ can be connected by a polygonal line that consists of a finite number of line segments joined end to end that lies entirely in the set, then $S$ is connected.
• A function $v(x,y)$ is harmonic conjugate of $u(x,y)$ if $f(z)=u(x,y)+iv(x,y)$ is analytics.
• Theorem (Cauchy): If $f$ is analytic in simply connected domain $D$, then for every simple closed contour $C$ in $D$, $\int_C f(z) dz =0$.
• A number $z_0$ is zero of $f(z)$ if $f(z_0)=0$.

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