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Recent questions in Complex Analysis
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1
GATE ECE 2021 | Question: 26
Consider the integral $\oint _{c}\frac{sin\left ( x \right )}{x^{2}\left ( x^{2}+4 \right )}dx$ where $C$ is a counter-clockwise oriented circle defined as $\left | x-i \right |=2$. The value of the integral is $-\frac{\pi }{8}\sin\left ( 2i \right )$ $\frac{\pi }{8}\sin\left ( 2i \right )$ $-\frac{\pi }{4}\sin\left ( 2i \right )$ $\frac{\pi }{4}\sin\left ( 2i \right )$
Arjun
asked
in
Complex Analysis
Feb 20, 2021
by
Arjun
6.0k
points
215
views
gateec-2021
complex-analysis
2
votes
0
answers
2
GATE ECE 2019 | Question: 1
Which one of the following functions is analytic over the entire complex plane? $\ln(z)$ $e^{1/z}$ $\frac{1}{1-z}$ $\cos(z)$
Arjun
asked
in
Complex Analysis
Feb 12, 2019
by
Arjun
6.0k
points
270
views
gate2019-ec
complex-analysis
0
votes
0
answers
3
GATE ECE 2016 Set 3 | Question: 2
For $f(z)= \large\frac{\sin(z)}{z^2}$, the residue of the pole at $z = 0$ is _______
Milicevic3306
asked
in
Complex Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
60
views
gate2016-ec-3
numerical-answers
complex-analysis
0
votes
0
answers
4
GATE ECE 2016 Set 3 | Question: 29
The values of the integral $\large\frac{1}{2\pi j}\oint_c\frac{e^z}{(z-2)} \small dz$ along a closed contour $c$ in anti-clockwise direction for the point $z_0=2$ inside the contour $c$, and the point $z_0=2$ outside the contour $c$, respectively,are $(i)2.72, \: (ii) 0$ $(i)7.39, \: (ii) 0$ $(i)0, \: (ii) 2.72$ $(i)0, \: (ii) 7.39$
Milicevic3306
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in
Complex Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
44
views
gate2016-ec-3
complex-analysis
0
votes
0
answers
5
GATE ECE 2016 Set 2 | Question: 2
Consider the complex valued function $f(z)=2z^{3}+b\mid z \mid^{3}$ where $z$ is a complex variable. The value of $b$ for which function $f(z)$ is analytic is _________
Milicevic3306
asked
in
Complex Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
49
views
gate2016-ec-2
numerical-answers
complex-analysis
0
votes
0
answers
6
GATE ECE 2016 Set 2 | Question: 27
Suppose $C$ is the closed curve defined as the circle $x^{2}+y^{2}=1$ with $C$ oreinted anti-clockwise. The value of $\oint$ ( $xy^{2}$ $dx$ + $ x^{2}y$ $dy$ )over the curve $C$ equals _________
Milicevic3306
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in
Complex Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
33
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gate2016-ec-2
numerical-answers
complex-analysis
0
votes
0
answers
7
GATE ECE 2016 Set 1 | Question: 6
Which one of the following is an eigen function of the class of all continuous-time, linear, time-invariant systems ($u(t)$ denotes the unit-step function)? $e^{j\omega_0t}u(t)$ $\cos(\omega_0t)$ $e^{j\omega_0t}$ $\sin(\omega_0t)$
Milicevic3306
asked
in
Complex Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
38
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gate2016-ec-1
complex-analysis
0
votes
0
answers
8
GATE ECE 2016 Set 1 | Question: 28
In the following integral, the contour $C$ encloses the points $2 \pi j$ and $- 2\pi j$ $-\frac{1}{2\pi}\oint_C\frac{\sin z}{(z-2\pi j)^3} \,dz$ The value of the integral is _________
Milicevic3306
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in
Complex Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
53
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gate2016-ec-1
numerical-answers
complex-analysis
0
votes
0
answers
9
GATE ECE 2015 Set 3 | Question: 3
If $C$ is a circle of radius $r$ with centre $z_{0},$ in the complex $z$-plane and if $n$ is a non-zero integer, then $\oint _{C}\frac{dz}{(z-z_{0})^{n+1}}$ equals $2\pi n j$ $0$ $\frac{nj}{2\pi}$ $2\pi n$
Milicevic3306
asked
in
Complex Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
59
views
gate2015-ec-3
vector-analysis
0
votes
0
answers
10
GATE ECE 2015 Set 3 | Question: 51
The complex envelope of the bandpass signal $x(t)=-\sqrt{2}\left(\dfrac{\sin (\pi t/5)}{\pi t/5}\right)\sin (\pi t - \dfrac{\pi}{4}),$ centered about $f=\dfrac{1}{2}\:Hz,$ is $\left(\dfrac{\sin (\pi t/5)}{\pi t/5}\right)e^{j\dfrac{\pi}{4}}$ ... $\sqrt{2} \left(\dfrac{\sin (\pi t/5)}{\pi t/5}\right)e^{-j\dfrac{\pi}{4}}$
Milicevic3306
asked
in
Complex Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
42
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gate2015-ec-3
complex-analysis
0
votes
0
answers
11
GATE ECE 2015 Set 2 | Question: 3
Let $f(z)=\dfrac{az+b}{cz+d}.$ If $f(z_{1})=f(z_{2})$ for all $z_{1}\neq z_{2},a=2,b=4$ and $c=5,$ then $d$ should be equal to ________.
Milicevic3306
asked
in
Complex Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
36
views
gate2015-ec-2
numerical-answers
complex-analysis
0
votes
0
answers
12
GATE ECE 2015 Set 2 | Question: 28
If $C$ denotes the counterclockwise unit circle, the value of the contour integral $\dfrac{1}{2\pi j}\oint_{C} Re\{z\}dz$ is __________.
Milicevic3306
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Complex Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
50
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gate2015-ec-2
numerical-answers
complex-analysis
0
votes
0
answers
13
GATE ECE 2015 Set 1 | Question: 4
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? The residue of $\frac{z}{z^2-1}$ at $z=1$ is $1/2$ $\oint_C z^2 dz=0$ $\frac{1}{2 \pi i} \oint_C \frac{1}{z} dz =1$ $\overline{z}$ (complex conjugate of $z$ is an analytical function
Milicevic3306
asked
in
Complex Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
55
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gate2015-ec-1
complex-analysis
analytic-functions
0
votes
0
answers
14
GATE ECE 2014 Set 2 | Question: 27
The real part of an analytic function $f(z)$ where $z = x + jy$ is given by $e^{-y} \cos(x)$. The imaginary part of $f(z)$ is $e^{y} \cos( x )$ $e^{-y} \sin( x )$ $-e^{y} \sin ( x )$ $-e^{-y} \sin (x )$
Milicevic3306
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in
Complex Analysis
Mar 26, 2018
by
Milicevic3306
15.8k
points
88
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gate2014-ec-2
analytic-functions
complex-analysis
0
votes
0
answers
15
GATE ECE 2014 Set 1 | Question: 3
$C$ is a closed path in the $z$-plane given by $\mid z \mid = 3.$ The value of the integral $\displaystyle{}\oint_{C}\bigg(\dfrac{z^{2}-z+4j}{z+2j}\bigg)dz$ is $-4\pi(1+j2)$ $4\pi(3-j2)$ $-4\pi(3+j2)$ $4\pi(1-j2)$
Milicevic3306
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Complex Analysis
Mar 26, 2018
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Milicevic3306
15.8k
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46
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gate2014-ec-1
complex-analysis
0
votes
0
answers
16
GATE ECE 2014 Set 1 | Question: 27
For a function $g(t),$ it is given that $\int_{- \infty}^{ + \infty} g(t)e^{-j\omega t}\:dt = \omega e^{-2\omega^{2}}$ for any real value $\omega.$ If $y(t) = \int_{- \infty}^{t}\:g(\tau)\:d\tau,$ then $\int_{- \infty}^{ + \infty}y(t)dt$ is $0$ $-j$ $-\frac{j}{2}$ $\frac{j}{2}$
Milicevic3306
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Complex Analysis
Mar 26, 2018
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Milicevic3306
15.8k
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39
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gate2014-ec-1
complex-analysis
0
votes
0
answers
17
GATE ECE 2018 | Question: 51
The contour $C$ given below is on the complex plane $z=x+j y,$ where $j=\sqrt{-1}.$ The value of the integral $\displaystyle{}\dfrac{1}{\pi j}\oint _{C}\dfrac{dz}{z^{2}-1}$ is _______.
gatecse
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in
Complex Analysis
Feb 19, 2018
by
gatecse
1.5k
points
333
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gate2018-ec
numerical-answers
complex-analysis
0
votes
0
answers
18
GATE ECE 2017 Set 2 | Question: 4
The residues of a function $f(z)=\frac1{(z-4)(z+1)^3 }$ are $\frac{-1}{27}$ and $\frac{-1}{125} \\$ $\frac{1}{125}$ and $\frac{-1}{125} \\$ $\frac{-1}{27}$ and $\frac{1}{5} \\$ $\frac{1}{125}$and $\frac{-1}{5}$
admin
asked
in
Complex Analysis
Nov 23, 2017
by
admin
32.7k
points
55
views
gate2017-ec-2
complex-analysis
0
votes
0
answers
19
GATE ECE 2017 Set 1 | Question: 48
Which one of the following options correctly describes the locations of the roots of the equation $s^{4}+s^{2}+1=0$ on the complex plane? Four left half plane(LHP) roots One right half plane(RHP) root,one LHP root and two roots on the imaginary axis Two RHP roots and two LHP roots All four roots are on the imaginary axis
admin
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Complex Analysis
Nov 17, 2017
by
admin
32.7k
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159
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gate2017-ec-1
complex-analysis
0
votes
0
answers
20
GATE ECE 2017 Set 1 | Question: 28
Let $I=\int_{c}\left ( 2zdx+2ydy+2xdx \right )$ where $x,y,z$ are real, and let $C$ be the straight line segment from point $A:(0,2,1)$ to point $B:(4,1,-1)$.The value of $I$ is ____________.
admin
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in
Complex Analysis
Nov 17, 2017
by
admin
32.7k
points
103
views
gate2017-ec-1
complex-analysis
numerical-answers
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