Question

Consider the function $f\left(z\right)=z+z^{\ast }$ where $z$ is a complex variable and $z^{\ast }$ denotes its complex conjugate.Which oneof the following is TRUE?

• A. $f\left(z\right)$ is both continuous and analytic
• B. $f\left(z\right)$ is a continuous but not analytic
• C. $f\left(z\right)$ is not continuous but is analytic
• D. $f\left(z\right)$ is neither continuous nor analytic

Answer 1

The correct option is B $f\left(z\right)$ is a continuous but not analytic

$z=x+iy$ & $z^{\ast }=x-iy$
Then $f\left(z\right)=z+z^{\ast }=2x$
$\because f\left(z\right)$ is in polynomial form hence continuous
Again $f\left(z\right)=2x=u+iv$
$\Rightarrow u=2x$ & $v=0$
$\because CR$ equation are not satisfied throughout the domain of $f\left(z\right)$ hence not analytic.