Question

Conisder the complex valued funciton $f\left(z\right)=2z^3+b|z{|}^3$ where $z$ is a complex variable. The value of $b$ for which the function $f\left(z\right)$ is analytic is

• A. 0

Answer 1

The correct option is A 0

Let $z=x+jy$
$\Rightarrow f\left(z\right)=2(x+jy{)}^2+b|(x+jy){|}^3$
$\Rightarrow f\left(z\right)=2[x^3+3x^{2}jy-3x{y}^2-j{y}^3]+b(x^2+y^2{)}^{3/2}$
$\Rightarrow f\left(z\right)=2(x^3-3x{y}^2)+b(x^2+y^2{)}^{3/2}+j[6x^{2}y-2y^3]$
Let $f\left(z\right)=u+jv$
$u=2(x^3-3x{y}^2)+b(x^2+y^2{)}^{3/2}$
$\Rightarrow \frac{\partial v}{\partial y}=6x^2-6y^2$
Now $f\left(z\right)$ to be analytic
$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$
and this is true only when,
$b=0$