MDoubt No 000014 - GATE

GATE [EC, 2015, 1 mark]
Question : Let $z=x+iy$ be a complex variable. Consider that contour integartion is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE?

(a) The residue of $\dfrac{z}{z^2-1}$ at $z=1$ is $1/2$
(b) $\displaystyle\oint\limits_{C}z^2dz=0$
(c) $\displaystyle\dfrac{1}{2\pi i}\oint\limits_{C}\dfrac{1}{z}dz=1$
(d) $\bar{z}$ (complex conjugate of $z$) is analytical function

Solution : $\rightarrow$ Residue of $\dfrac{z}{z^2-1}$ at $\displaystyle z=1=\lim_{z\to 1}(z-1)\dfrac{z}{z^2-1}$
$\displaystyle =\lim_{z\to 0}\dfrac{z}{z+1}$
$=\dfrac{1}{2}$
Option (A) is TRUE

$\rightarrow$ $z^2$ is analytical at every point within and on circle $|z|=1$
by Cauchy's integral theorem, we get
$\displaystyle\oint\limits_{C}z^2dz=0$
Option (B) is TRUE

$\rightarrow\dfrac{1}{z}$ has a pole at $z=0$
$\therefore$ by Cauchy's integral theorem, we get
$\displaystyle\oint\limits_{C}\dfrac{1}{z}dz=2\pi i\left[z\dfrac{1}{z}\right]_{z=0}$
$\displaystyle\dfrac{1}{2\pi i}\oint\limits_{C}\dfrac{1}{z}dz=1$
Option (C) is TRUE

$\rightarrow\bar{z}=x-iy=u+iv$
$\implies u=x$ and $v=-y$
$\implies\begin{array} \\ u_x=1 & v_x=0\\ u_y=0 & v_y=-1 \end{array}$
$u_x\ne v_y$ i.e. one of the Cauchy-Reimann equations is not satisfied.
$\therefore \bar{z}$ is not analytical.
Option (D) is NOT TRUE
CORRECT ANSWER : D