### CSIR JUNE 2011 PART B QUESTION 33 SOLUTION

Let $f: \Bbb C$ to $\Bbb C$ be a complex-valued function defined by $f(z) = u(x,y)+iv(x,y)$. Suppose that $v(x,y) = 3xy^2$ then
1) $f$ cannot be holomorphic on $\Bbb C$ for any choice of $u$,
2) $f$ is holomorphic on $\Bbb C$ for a suitable choice of $u$,
3) $f$ isholomorphic on $\Bbb C$ for any choice of $u$,
4) $v$ is  not differentiable as a function of $x$ and $y$.
Solution:
We note that $v_{xx}+v_{yy} = 6x$ which is not zero, so $v$ is not a harmonic function. Hence it cannot be the imaginary part of any homomorphic functions. option (1) is correct.
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