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GATE2019-2-4

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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

  1. $\frac{\partial u}{ \partial x} = \frac{\partial v}{ \partial y} \text{ and } \frac{\partial u}{ \partial y} = \frac{\partial v}{ \partial x}$
  2. $\frac{\partial u}{ \partial x} = \frac{\partial v}{ \partial y} \text{ and } \frac{\partial u}{ \partial y} = –  \frac{\partial v}{ \partial x}$
  3. $\frac{\partial u}{ \partial x} = –  \frac{\partial v}{ \partial y} \text{ and } \frac{\partial u}{ \partial y} = \frac{\partial v}{ \partial x}$
  4. $\frac{\partial u}{ \partial x} = –  \frac{\partial v}{ \partial y} \text{ and } \frac{\partial u}{ \partial y} = – \frac{\partial v}{ \partial x}$
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Answer :- $(B)$

Reference :- Cauchy-Riemann Equations

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