Question

An analytic function of a complex variabel $z=x+iy$ is expressed as $f\left(x\right)=u(x,y)+iv(x,y)$ where $i=\sqrt{-1}$. If $u=xy,$ the expression for $v$ should be

• A. $\frac{(x+y{)}^2}{2}+k$
• B. $\frac{x^2-y^2}{2}+k$
• C. $\frac{y^2-x^2}{2}+k$
• D. $\frac{(x-y{)}^2}{2}+k$

Answer 1

The correct option is C $\frac{y^2-x^2}{2}+k$

Given function is
$f\left(z\right)=u(x,y)+iv(x,y)$
$u(x,y)=xy$
By Cauchy - Riemann equations
$\frac{\partial v}{\partial y}=\frac{\partial u}{\partial x}=y$
$\Rightarrow v=\frac{y^2}{2}+f\left(x\right)$ ... (i)
Again, $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$
$\Rightarrow x=-f^{\prime }\left(x\right)$
$\Rightarrow f\left(x\right)=-\frac{x^2}{2}+k$
So by (i), $v=\frac{y^2-x^2}{2}+k$