Question

The most general complex analytical function$f\left(z\right)=u(x,y)+iv(x,y)foru=x^2-y^2$ is

• A. z
• B. $z^2$
• C. $z^3$
• D. $\frac{1}{z^2}$

Answer 1

The correct option is B $z^2$

(b)

From CR equations,

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}and\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}$

$Givenu=x^2-y^2$
$\frac{\partial u}{\partial x}=2x$
$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$
$2x=\frac{\partial v}{\partial y}$
v = 2xy
$\frac{\partial v}{\partial x}=2y$

$f^{\prime }\left(z\right)=\frac{df}{dz}=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}$

$=2x+i2y=2(x+iy)=2z$

Intergrating the above , we get

$f\left(z\right)=\int 2zdz=2\frac{z^2}{2}=z^2$