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 $y$.

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.

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**.**Share to your groups:**

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