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  3. consider z and as independent variables then a show that...

Question: consider z and as independent variables then a show that...

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Consider z and \overline{z} as independent variables. Then x=\frac{z+\bar{z}}{2}, y=\frac{z-\bar{z}}{2i}

a) Show that \frac{\partial f}{\partial z}=\frac{1}{2}(\frac{\partial }{\partial x}-i\frac{\partial }{\partial y})f, \frac{\partial f}{\partial z}=\frac{1}{2}(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y})f.

b) Prove that the Cauchy-Riemann conditions are equivalent to the equation \frac{\partial f}{\partial z}=0.

c) Show that the Laplace equation is equivalent to \frac{\partial^{2} u}{\partial z\partial \bar{z}}=0.

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