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Complex Function ( Limit, Analytic Function)

A function f ...

Question

A function $f$ of the complex variable $z = x + i y$ , is given as $f (x, y) = u (x, y) + i v (x, y), w h e r e u (x, y) = 2 k x y$ and $v (x, y) = x^{2} - y^{2}$ . The value of $k$ , for which the function is analytic, is
-1

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Solution

The correct option is A -1
$u (x, y) = 2 k x y$ and $v (x, y) = x^{2} - y^{2}$
Now, $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} a n d \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$
$\Rightarrow 2 k y = - 2 y$ and $2 k x = - 2 x$
$\Rightarrow k = - 1$ and $k = - 1$

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