Question

If $f\left(z\right)=(x^2+a{y}^2)+ibxy$ is a complex analytic function of $z=x+iy$, where $i=\sqrt{-1}$, then

• A. $a=-1,b=-1$
• B. $a=-1,b=2$
• C. $a=1,b=2$
• D. $a=2,b=2$

Answer 1

The correct option is B $a=-1,b=2$

$f\left(z\right)=(x^2+a{y}^2)+ibxy=u(x,y)+iv(x,y)$
$u(x,y)=x^2+a{y}^2$
$v(x,y)=bxy$
$\frac{\partial u}{\partial x}=2x,\frac{\partial u}{\partial y}=2ay,\frac{\partial v}{\partial x}=by,\frac{\partial v}{\partial y}=bx$
Using Cauchy Reimann equations:
$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\Rightarrow 2x=bx\Rightarrow b=2$
and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x},a=\frac{-b}{2}=-1$
Hence, $a=-1,b=2$