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Questions without answers in Complex Analysis

1 votes
0 answers
1
TIFR ECE 2010 | Question: 15
Let $\imath=\sqrt{-1}$. Then $\imath^{\imath}$ could be $\exp (\pi / 2)$ $\exp (\pi / 4)$ Can't determine Takes infinite values Is a complex number
Let $\imath=\sqrt{-1}$. Then $\imath^{\imath}$ could be$\exp (\pi / 2)$$\exp (\pi / 4)$Can't determineTakes infinite valuesIs a complex number
2 votes
0 answers
3
GATE ECE 2019 | Question: 1
Which one of the following functions is analytic over the entire complex plane? $\ln(z)$ $e^{1/z}$ $\frac{1}{1-z}$ $\cos(z)$
Which one of the following functions is analytic over the entire complex plane?$\ln(z)$$e^{1/z}$$\frac{1}{1-z}$$\cos(z)$
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4
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 _______
For $f(z)= \large\frac{\sin(z)}{z^2}$, the residue of the pole at $z = 0$ is _______
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6
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 _________
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 _________
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7
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 _________
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 cu...
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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)? $e^{j\omega_0t}u(t)$ $\cos(\omega_0t)$ $e^{j\omega_0t}$ $\sin(\omega_0t)$
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...
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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 _________
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 integra...
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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 $2\pi n j$ $0$ $\frac{nj}{2\pi}$ $2\pi n$
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...
0 votes
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12
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 ________.
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 ________.
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13
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 __________.
If $C$ denotes the counterclockwise unit circle, the value of the contour integral $$\dfrac{1}{2\pi j}\oint_{C} Re\{z\}dz$$ is __________.
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15
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 $e^{y} \cos( x )$ $e^{-y} \sin( x )$ $-e^{y} \sin ( x )$ $-e^{-y} \sin (x )$
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} \s...
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16
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 $-4\pi(1+j2)$ $4\pi(3-j2)$ $-4\pi(3+j2)$ $4\pi(1-j2)$
$C$ 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...
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18
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 _______.
The contour $C$ given below is on the complex plane $z=x+j y,$ where $j=\sqrt{-1}.$ The value of the integral $\disp...
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19
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 $\frac{-1}{5}$
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}...
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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 ____________.
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...
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