Complex Variables

Question 1
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 1 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 2
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 2 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 3
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 3 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 4
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 4 Explanation: 
f(z) = \cos z is analytic every where.
Question 5
The contour C given below is on the complex plane z=x+jy , where j=\sqrt{-1} .

The value of the integral \frac{1}{\pi j}\oint_{C}^{}\frac{dz}{z^{2}-1} is _______.
A
1
B
2
C
3
D
4
GATE EC 2018   Engineering Mathematics
Question 5 Explanation: 


\begin{array}{l} \frac{1}{\pi \dot{i}} \oint \frac{d z}{z^{2}-1}=2\left[\frac{1}{2 \pi i} \oint_{C_{1}} \frac{d z}{(z+1)(z-1)}+\frac{1}{2 \pi i} \oint_{C_{2}} \frac{d z}{(z+1)(z-1)}\right] \\ =2\left[-\left.\left(\frac{1}{z-1}\right)\right|_{z=-1}+\left.\left(\frac{1}{z+1}\right)\right|_{z=1}\right] \\ =2\left[-\left(-\frac{1}{2}\right)+\left(\frac{1}{2}\right)\right]=2 \end{array}
Question 6
An integral I over a counter clock wise circle C is given by I=\oint_{0}^{c}\frac{z^{2}-1}{z^{2}+1}e^{z}dz.
If C is defined as |z| = 3, then the value of I is
A
-\pi i sin(1)
B
-2\pi i sin(1)
C
-3\pi i sin(1)
D
-4\pi i sin(1)
GATE EC 2017-SET-2   Engineering Mathematics
Question 6 Explanation: 
Poles are z^{2}+1=0
z=\pm i
z=i lies inside |z|=3
z=-i lies inside |z|=3
Residue at z=i is
\begin{aligned} &=\lim _{z \rightarrow i}(z-i) \frac{z^{2}-1}{(z-i)(z+i)} e^{z} \\ &=\frac{-1-1}{2 i} e^{i}=i e^{i} \end{aligned}
Residue at z=-i is
\begin{aligned} &=\lim _{z \rightarrow-i}(z+i) \frac{z^{2}-1}{(z-i)(z+i)} e^{z}\\ &=\frac{-1-1}{-2 i} e^{-i}=\frac{1}{i} e^{-i}=-i e^{-i} \end{aligned}
By Residues theorem
\begin{aligned} I &=2 \pi i\left(i e^{i}-i e^{-i}\right) \\ &=-2 \pi\left(e^{i}-e^{-i}\right) \\ &=-2 \pi(\cos 1+i \sin 1-\cos 1+i \sin 1) \\ &=-2 \pi(2 i \sin 1)=-4 \pi i \sin (1) \end{aligned}
Question 7
The residues of a function f(z)=\frac{1}{(z-4)(z+1)} are
A
\frac{-1}{27} and \frac{-1}{125}
B
\frac{1}{125} and \frac{-1}{125}
C
\frac{-1}{27} and \frac{1}{5}
D
\frac{1}{125} and \frac{-1}{5}
GATE EC 2017-SET-2   Engineering Mathematics
Question 7 Explanation: 
Residue at, z=4 is
=\lim _{x \rightarrow 1}(z-4) \frac{1}{(z-4)(z+1)^{3}}=\frac{1}{(4+1)^{3}}=\frac{1}{125}
Residue at z=-1 is
=\lim _{z \rightarrow-1} \frac{1}{2 !} \frac{d^{2}}{d z^{2}}\left((z-1)^{3} \frac{1}{(z-4)(z+1)^{3}}\right)
=\lim _{z \rightarrow-1} \frac{1}{2 !}\left(\frac{2}{(z-4)^{3}}\right)=\frac{1}{(-1-4)^{3}}=\frac{-1}{125}
Question 8
The values of the integral \frac{1}{2\pi j}\oint_c \frac{e^{z}}{z-2}dz along a closed contour c in anti-clockwise direction for

(i) the point z_{0} = 2 inside the contour c, and
(ii) the point z_{0} = 2 outside the contour c,

respectively, are
A
(i) 2.72, (ii) 0
B
(i) 7.39, (ii) 0
C
(i) 0, (ii) 2.72
D
(i) 0, (ii) 7.39
GATE EC 2016-SET-3   Engineering Mathematics
Question 8 Explanation: 
\begin{aligned} (i) \quad\quad Z_{0}&=2 -lies inside C \\ \text { So, } \text{Res}(z)&=\lim _{z \rightarrow 2}(z-2) \cdot \frac{e^{z}}{z-2} \\ &=e^{2}=7.39 \\ \frac{1}{2 \pi i} \int \frac{e^{z}}{z-2} d z &=2 \pi i \cdot \frac{1}{2 \pi i}(7.39)=7.39 \\ (ii)\quad \quad Z_{0}&=-2\text{ lies out side C then} \\ \text{Res}(z)&=0 \\ \text{So, }\int_{c} \frac{e^{z}}{z-2} d z&=2 \pi i \frac{1}{2 \pi i}(0)=0 \end{aligned}
Question 9
For f(z)=\frac{sin(z)}{z^{2}}, the residue of the pole at z=0 is __________
A
0.5
B
1
C
2
D
3
GATE EC 2016-SET-3   Engineering Mathematics
Question 9 Explanation: 
Residue of \frac{\sin z}{z^{2}}
= coefficient of \frac{1}{2} in \left\{\frac{z-\frac{z^{3}}{3 !}+\frac{z^{5}}{5 !}---}{z^{2}}\right\}
= coefficient of \frac{1}{z} in \left\{\frac{1}{z}-\frac{z}{3 !}+\frac{z^{3}}{5 !}---\right\}
=1
Question 10
Consider the complex valued function f(z)=2z^{3}+b|z|^{3} where z is a complex variable. The value of b for which the function f(z) is analytic is ________
A
1
B
2
C
3
D
0
GATE EC 2016-SET-2   Engineering Mathematics
Question 10 Explanation: 
f(z)=2 z^{3}+b_{1}|z|^{3}
Given that f(z) is analytic.
which is possible only when b=0
since \left|z^{3}\right| is differentiable at the origin but not analytic.
2 z^{3} is analytic everywhere
\begin{aligned} \therefore \quad f(z)&=2 z^{3}+b\left|z^{3}\right| \text{ is analytic}\\ \text{only when }\quad b&=0 \end{aligned}
There are 10 questions to complete.