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Recent questions tagged complex-analysis
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GATE EC 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 $-\frac{\pi }{8}\sin\left ( 2i \right )$ $\frac{\pi }{8}\sin\left ( 2i \right )$ $-\frac{\pi }{4}\sin\left ( 2i \right )$ $\frac{\pi }{4}\sin\left ( 2i \right )$
asked
Feb 20
in
Complex Analysis
by
Arjun
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4.4k
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68
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gateec-2021
complex-analysis
+2
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0
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2
GATE ECE 2019 | Question: 1
Which one of the following functions is analytic over the entire complex plane? $\ln(z)$ $e^{1/z}$ $\frac{1}{1-z}$ $\cos(z)$
asked
Feb 12, 2019
in
Complex Analysis
by
Arjun
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4.4k
points)
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216
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gate2019-ec
complex-analysis
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0
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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 _______
asked
Mar 28, 2018
in
Complex Analysis
by
Milicevic3306
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15.8k
points)
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20
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gate2016-ec-3
numerical-answers
complex-analysis
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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 $(i)2.72, \: (ii) 0$ $(i)7.39, \: (ii) 0$ $(i)0, \: (ii) 2.72$ $(i)0, \: (ii) 7.39$
asked
Mar 28, 2018
in
Complex Analysis
by
Milicevic3306
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15.8k
points)
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12
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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 _________
asked
Mar 28, 2018
in
Complex Analysis
by
Milicevic3306
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15.8k
points)
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17
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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 _________
asked
Mar 28, 2018
in
Complex Analysis
by
Milicevic3306
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15.8k
points)
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12
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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)? $e^{j\omega_0t}u(t)$ $\cos(\omega_0t)$ $e^{j\omega_0t}$ $\sin(\omega_0t)$
asked
Mar 28, 2018
in
Complex Analysis
by
Milicevic3306
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15.8k
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17
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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 _________
asked
Mar 28, 2018
in
Complex Analysis
by
Milicevic3306
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15.8k
points)
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19
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gate2016-ec-1
numerical-answers
complex-analysis
0
votes
0
answers
9
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 $\left(\dfrac{\sin (\pi t/5)}{\pi t/5}\right)e^{j\dfrac{\pi}{4}}$ ... $\sqrt{2} \left(\dfrac{\sin (\pi t/5)}{\pi t/5}\right)e^{-j\dfrac{\pi}{4}}$
asked
Mar 28, 2018
in
Complex Analysis
by
Milicevic3306
(
15.8k
points)
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11
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gate2015-ec-3
complex-analysis
0
votes
0
answers
10
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 ________.
asked
Mar 28, 2018
in
Complex Analysis
by
Milicevic3306
(
15.8k
points)
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11
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gate2015-ec-2
numerical-answers
complex-analysis
0
votes
0
answers
11
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 __________.
asked
Mar 28, 2018
in
Complex Analysis
by
Milicevic3306
(
15.8k
points)
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16
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gate2015-ec-2
numerical-answers
complex-analysis
0
votes
0
answers
12
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
asked
Mar 28, 2018
in
Complex Analysis
by
Milicevic3306
(
15.8k
points)
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19
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gate2015-ec-1
complex-analysis
analytic-functions
0
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0
answers
13
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 $e^{y} \cos( x )$ $e^{-y} \sin( x )$ $-e^{y} \sin ( x )$ $-e^{-y} \sin (x )$
asked
Mar 26, 2018
in
Complex Analysis
by
Milicevic3306
(
15.8k
points)
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33
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gate2014-ec-2
analytic-functions
complex-analysis
0
votes
0
answers
14
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 $-4\pi(1+j2)$ $4\pi(3-j2)$ $-4\pi(3+j2)$ $4\pi(1-j2)$
asked
Mar 26, 2018
in
Complex Analysis
by
Milicevic3306
(
15.8k
points)
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16
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gate2014-ec-1
complex-analysis
0
votes
0
answers
15
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 $0$ $-j$ $-\frac{j}{2}$ $\frac{j}{2}$
asked
Mar 26, 2018
in
Complex Analysis
by
Milicevic3306
(
15.8k
points)
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15
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gate2014-ec-1
complex-analysis
0
votes
0
answers
16
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 _______.
asked
Feb 19, 2018
in
Complex Analysis
by
gatecse
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1.5k
points)
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155
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gate2018-ec
numerical-answers
complex-analysis
0
votes
0
answers
17
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 $\frac{-1}{5}$
asked
Nov 23, 2017
in
Complex Analysis
by
admin
(
2.8k
points)
|
27
views
gate2017-ec-2
complex-analysis
0
votes
0
answers
18
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
asked
Nov 17, 2017
in
Complex Analysis
by
admin
(
2.8k
points)
|
50
views
gate2017-ec-1
complex-analysis
0
votes
0
answers
19
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 ____________.
asked
Nov 17, 2017
in
Complex Analysis
by
admin
(
2.8k
points)
|
62
views
gate2017-ec-1
complex-analysis
numerical-answers
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