# Problem of the Week #12 - June 18th, 2012

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#### Chris L T521

##### Well-known member
Staff member
Thanks to those who participated in last week's POTW!! Here's this week's problem.

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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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#### Chris L T521

##### Well-known member
Staff member
This week's problem was correctly answered by hmmm16 and Sudharaka.

Here's Sudharaka's solution:

$u(x,y) = \phi_x\phi_y+\psi_x\psi_y$

$\Rightarrow\frac{\partial}{\partial x}u(x,y)=\phi_{xx}\phi_y+\phi_{yx}\phi_x+\psi_{xx}\psi_y+\psi_{x}\psi_{yx}~~~~~~~~~~(1)$

$v(x,y) = \tfrac{1}{2}(\phi_x^2+\psi_x^2-\phi_y^2 - \psi_y^2)$

$\Rightarrow\frac{\partial}{\partial y}v(x,y) = \phi_{x}\phi_{xy}+\psi_{x}\psi_{xy}-\phi_{y}\phi_{yy}-\psi_{y}\psi_{yy}~~~~~~~~~~~(2)$

We shall assume that $$\phi$$ and $$\psi$$ have commutative second partial derivatives. Then,

$\phi_{xy}=\phi_{yx}\mbox{ and }\psi_{xy}=\psi_{yx}$

By (2),

$\frac{\partial}{\partial y}v(x,y) = \phi_{x}\phi_{yx}+\psi_{x}\psi_{yx}-\phi_{y}\phi_{yy}-\psi_{y}\psi_{yy}~~~~~~~~~~~~~~(3)$

Since $$\phi$$ and $$\psi$$ are harmonic functions,

$\phi_{xx}=-\phi_{yy}\mbox{ and }\psi_{xx}=-\psi_{yy}$

By (3),

$\frac{\partial}{\partial y}v(x,y) = \phi_{x}\phi_{yx}+\psi_{x}\psi_{yx}+\phi_{y}\phi_{xx}+\psi_{y}\psi_{xx}~~~~~~~~~~~~(4)$

By (1) and (4),

$\frac{\partial}{\partial x}u(x,y)=\frac{\partial}{\partial y}v(x,y)$

Similarly,

$\frac{\partial}{\partial y}u(x,y)=\phi_{x}\phi_{yy}+\phi_{xy}\phi_y+\psi_{x}\psi_{yy}+\psi_{xy}\psi_{y}~~~~~~~~~~(5)$

$-\frac{\partial}{\partial x}v(x,y) = -\phi_{x}\phi_{xx}-\psi_{x}\psi_{xx}+\phi_{y}\phi_{yx}+\psi_{y}\psi_{yx}$

By our previous assumption,

$-\frac{\partial}{\partial x}v(x,y) = -\phi_{x}\phi_{xx}-\psi_{x}\psi_{xx}+\phi_{y}\phi_{xy}+\psi_{y}\psi_{xy}~~~~~~~~~~~~~(6)$

Since $$\phi$$ and $$\psi$$ are harmonic functions,

$\phi_{xx}=-\phi_{yy}\mbox{ and }\psi_{xx}=-\psi_{yy}$

By (6),

$-\frac{\partial}{\partial x}v(x,y) = \phi_{x}\phi_{yy}+\psi_{x}\psi_{yy}+\phi_{y}\phi_{xy}+\psi_{y}\psi_{xy}~~~~~~~~~~~(7)$

By (5) and (7),

$\frac{\partial}{\partial y}u(x,y)=-\frac{\partial}{\partial x}v(x,y)$

Q.E.D.

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