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- Jan 26, 2012

- 995

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**Problem**: We say that a function $f(x,y)$ is

*harmonic*if it satisfies the Laplace equation $\dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} = 0$. Suppose that $\phi(x,y)$ and $\psi(x,y)$ are harmonic functions. Let $u$ and $v$ be functions defined as follows:

\[u(x,y) = \phi_x\phi_y+\psi_x\psi_y\quad\text{and}\quad v(x,y) = \tfrac{1}{2}(\phi_x^2+\psi_x^2-\phi_y^2 - \psi_y^2).\]

Show that $u(x,y)$ and $v(x,y)$ satisfy the Cauchy-Riemann equations

\[\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\quad \text{and}\quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.\]

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