Question

Let $z=x+iy$ be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE?

• A. The residue of $\frac{z}{z^2-1}$ at $z=1$ is $1/2$
• B. ${\oint }_C{z}^{2}dz=0$
• C. $\frac{1}{2\pi i}{\oint }_C\frac{1}{z}dz=1$
• D. $\overline{z}$ (complex conjugate of $z$) is an analytical function

Answer 1

The correct option is D $\overline{z}$ (complex conjugate of $z$) is an analytical function

We will go through options
(a) $f\left(z\right)=\frac{z}{z^2-1}$
$Resf\left(z\right)={lim}_{z\to 1}(z-1)f\left(z\right)={lim}_{z\to 1}\frac{1}{z+1}=\frac{1}{2}$
(b) $f\left(z\right)=z^2$ is analytic every where
By cauchy's theorem
${\oint }_C{z}^{2}dz=0$
(c) By Cauchy integral formula
$\frac{1}{2\pi i}{\oint }_C\frac{f\left(z\right)dz}{z-z_0}=f\left(z_o\right)$
$\Rightarrow \frac{1}{2\pi i}{\oint }_C\left(\frac{1}{z-0}\right)dz=f\left(0\right)=1$
(d) $f\left(z\right)=\overline{z}=x-iy$
$u=x,v=-y$
$u_x=1,v_x=0$
${}_y=0,v_y=-1$
$u_x\ne v_y$
$f\left(z\right)$ does nto satisfy C.R equ
$\Rightarrow f\left(z\right)$ is not analytic