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An analytic function $f(z)$ of complex variable $z=x+iy$ may be written as $f(z)=u(x,y)+iv(x,y)$. Then $u(x,y)$ and $v(x,y)$ must satisfy

- $\frac{\partial u}{ \partial x} = \frac{\partial v}{ \partial y} \text{ and } \frac{\partial u}{ \partial y} = \frac{\partial v}{ \partial x}$
- $\frac{\partial u}{ \partial x} = \frac{\partial v}{ \partial y} \text{ and } \frac{\partial u}{ \partial y} = – \frac{\partial v}{ \partial x}$
- $\frac{\partial u}{ \partial x} = – \frac{\partial v}{ \partial y} \text{ and } \frac{\partial u}{ \partial y} = \frac{\partial v}{ \partial x}$
- $\frac{\partial u}{ \partial x} = – \frac{\partial v}{ \partial y} \text{ and } \frac{\partial u}{ \partial y} = – \frac{\partial v}{ \partial x}$

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