# Complex Variables

 Question 1
Let $w^{4}=16 j$. Which of the following cannot be a value of $w$ ?
 A $2 e^{\frac{j 2 \pi}{8}}$ B $2 e^{\frac{j \pi}{8}}$ C $2 e^{\frac{j 5 \pi}{8}}$ D $2 e^{\frac{j 9 \pi}{8}}$
GATE EC 2023   Engineering Mathematics
Question 1 Explanation:
$w=(2) j^{1 / 4}$
$w=2(0+j)^{1 / 4}$
$w=2\left[e^{j(2 n+1) \pi / 2}\right]^{1 / 4}$ $=2\left[e^{j(2 n+1) \pi / 8}\right]$

For $n=0$, $w=e^{j \pi / 8}$
For $n=2$, $w=2 e^{5 \pi j / 8}$
For $n=4$, $w=2 e^{9 \pi j / 8}$
 Question 2
Consider the following series:
$\sum_{n=1}^{\infty }\frac{n^d}{c^n}$
For which of the following combinations of $c,d$ values does this series converge?
 A $c=1,d=-1$ B $c=2,d=1$ C $c=0.5,d=-10$ D $c=1,d=-2$
GATE EC 2022   Engineering Mathematics
Question 2 Explanation:
\begin{aligned} \Sigma u_n&=\Sigma \frac{n}{2^n}\\ &\text{Ratio test; }\\ \lim_{n \to \infty }\frac{u_{n+1}}{u_n}&=\lim_{n \to \infty }\frac{n+1}{2^{n+1}} \times \frac{2^n}{n}=\frac{1}{2}\\ \frac{1}{2}& \lt 2 \end{aligned}
$\therefore$ By ratio test, $\Sigma u_n$ is convergent.
(A) c=1, d=-1
$\Sigma u_n=\Sigma \frac{1}{n}$ is divergent by P-test
(B) c=0.5, d=-10
\begin{aligned} \Sigma u_n&=\Sigma \frac{n^{-10}}{(0.5)^n}\\ &\text{Ratio test; }\\ \lim_{n \to \infty }\frac{u_{n+1}}{u_n}&=\lim_{n \to \infty }\frac{(n+1)^{-10}}{(0.5)^{n+1}} \times \frac{(0.5)^n}{n^{10}}\\ &=\frac{1}{0.5}=2\\ 2& \gt 1 \end{aligned}
$\therefore$ By ratio test, $\Sigma u_n$ is divergent.
(D) c=1,d=-2 $\Sigma u_n=\Sigma \frac{n^{-2}}{(1)^n}=\Sigma \frac{1}{n^2}$
$\Sigma u_n$ is convergent by P-test.

 Question 3
A simple closed path $C$ in the complex plane is shown in the figure. If
$\oint C\frac{2^z}{z^2-1}dz=-i \pi A$
where $i=\sqrt{-1}$, then the value of $A$ is ______ (rounded off to two decimal places)

 A 0.2 B 0.4 C 0.5 D 0.6
GATE EC 2022   Engineering Mathematics
Question 3 Explanation:
\begin{aligned} Roots\;\; (z-1)(z+1)&=0\\ z&=\pm 1\\ \because \; z&=-1 \text{ is in contour} \end{aligned}
\begin{aligned} \Rightarrow \oint _c \frac{2^z}{z^2-1}&=\left [ \lim_{z \to -1} \frac{(z+1)2^z}{(z+1)(z-1)} \right ] \times 2 \pi i\\ &=\frac{2^{-1}}{(-1-1)} \times 2 \pi i\\ &=-\frac{1}{2} \pi i\\ \Rightarrow \;\; A&=1/2=0.5 \end{aligned}
 Question 4
The value of the contour integral

$\frac{1}{2 \pi j}\oint \left (z+\frac{1}{z} \right )^2 dz$

evaluated over the unit circle |z|=1 is ______
 A 0 B 0.0001 C 0.0005 D 0.0008
GATE EC 2019   Engineering Mathematics
Question 4 Explanation:

$\frac{1}{2 \pi j} \oint\left(z+\frac{1}{z}\right)^{2} d z \text { where } C \text { is }|z|=1$
$I=\frac{1}{2 \pi j} \int_{c}^{\left(z^{2}+1\right)^{2}} d z$
z=0 lies inside the circle,
\begin{aligned} I &=\frac{1}{2 \pi j}\left[\frac{2 \pi j}{1 !} \frac{d}{d z}\left(z^{2}+1\right)^{2}\right]_{2=0} \\ &=\left[\frac{d}{d z}\left(z^{2}+1\right)^{2}\right]_{z=0} \\ &=\left[2\left(z^{2}+1\right) \times 2 z\right]_{z=0}=0 \end{aligned}
 Question 5
Which one of the following functions is analytic over the entire complex plane?
 A $ln(z)$ B $e^{1/z}$ C $\frac{1}{1-z}$ D $cos(z)$
GATE EC 2019   Engineering Mathematics
Question 5 Explanation:
$f(z) = \cos z$ is analytic every where.

There are 5 questions to complete.