Math — Complex Analysis
Cauchy-Riemann equations and contour integration.
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u_x = v_y, \quad u_y = -v_x
\oint_C f(z)\,dz = 0 \quad (f \text{ analytic})
f(z_0) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z - z_0}\,dz
\oint_C f(z)\,dz = 2\pi i \sum_k \operatorname{Res}(f, z_k)