Complex Analysis - Applications of Harmonic Functions

Since is analytic, the Cauchy-Riemann Equations give us $u_x(x,y) = v_y(x,y) \text{ and } u_y(x,y) = -v_x(x,y) \forall x,y$

Since $\overline{f(z)} = u(x,y) - iv(x,y)$ is analytic, the Cauchy-Riemann Equations give us $u_x(x,y) = -v_y(x,y) \text{ and } u_y(x,y) = v_x(x,y) \forall x,y$ .

Putting these two results together gives us $u_x(x,y) = 0 \text{ and } u_y(x,y) = 0 \forall x,y$

which implies that is constant. Similar reasoning gives is that is constant. This implies that must be constant throughout .

Applications of Harmonic Functions

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