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- Jan 26, 2012

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**Problem**: Define the operator

\[\frac{\partial}{\partial \overline{z}} = \frac{1}{2}\left(\frac{\partial}{\partial x} + i\frac{\partial}{\partial y}\right).\]

Show that if the first order derivatives of the real and imaginary parts of a function $f(z)=u(x,y) + iv(x,y)$ satisfy the Cauchy-Riemann equations, then $\dfrac{\partial f}{\partial \overline{z}}=0$.

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**Recall**: The Cauchy-Riemann equations for a function $f(z)=u(x,y)+iv(x,y)$ are

\[\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\qquad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.\]

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