An analytic function of a complex variable $z = x + i y$ is expressed as $f (z) = u(x, y) + i v(x, y)$ , where $i=\sqrt{-1}$  . If $u(x, y) = x^2 − y^2$ , then expression for $v(x, y)$ in terms of $x$, $y$ and a general constant $c$ would be

1. $xy + c \\$
2. $\dfrac{x^2+y^2}{2}+c \\$
3. $2xy+c \\$
4. $\dfrac{(x-y)^2}{2}+c$