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Complex Function ( Limit, Analytic Function)

For an analyt...

Question

For an analytic function, $f (x + i y) = u (x, y) + i v (x, y), u$ is given by $u = 3 x^{2} - 3 y^{2}$ . Th expression for $v$ , considering $K$ to be a constant is

3 y^{2} - 3 x^{2} + K

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6 x - 6 y + K

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6 y - 6 x + K

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6 x y + K

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Solution

The correct option is D $6 x y + K$
For an analytic function,
$f (x + i y) = u (x, y) + i v (x, y)$ .
Where, $u = 3 x^{2} - 3 y^{2}$
$u_{x} = 6 x = v_{y}$ (By C.R. Eq's)
$\Rightarrow$ $v = 6 x y + f (x)$ ...(i)
Again, by C.R.eq's
$u_{y} = - v_{x}$
$\Rightarrow - 6 y = - 6 y - f^{'} (x)$
$\Rightarrow f^{'} (x) = 0$
$\Rightarrow f (x) = k$
So by (i), $v = 6 x y + k$

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