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Recent questions tagged complex-variables

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GATE Mechanical 2021 Set 2 | Question: 19
Value of $\left ( 1+i \right )^{8}$, where $i=\sqrt{-1}$, is equal to $4$ $16$ $4i$ $16i$

go_editor asked in Calculus Mar 1, 2021

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GATE Mechanical 2021 Set 1 | Question: 27
Let $\text{C}$ represent the unit circle centered at origin in the complex plane, and complex variable, $z=x+iy$. The value of the contour integral $\oint _{C}\dfrac{\cosh \:3z}{2z}\:dz$ (where integration is taken counter clockwise) is $0$ $2$ $\pi i$ $2 \pi i$

gatecse asked in Calculus Feb 22, 2021

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GATE2020-ME-2: 27
The function $f(z)$ of complex variable $z=x+iy$, where $i=\sqrt{-1}$, is given as $f(z)=(x^3-3xy^2)+i \: v(x,y)$. For this function to be analytic, $v(x,y)$ should be $(3xy^2-y^3) +$ constant $(3x^2y^2-y^3) +$ constant $(x^3-3x^2 y) +$ constant $(3x^2y-y^3) +$ constant

go_editor asked in Calculus Sep 18, 2020

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GATE2020-ME-1: 4
Which of the following function $f(z)$, of the complex variable $z,$ is NOT analytic at all the points of the complex plane? $f\left ( z \right )=z^{2}$ $f\left ( z \right )=e^{z}$ $f\left ( z \right )=\sin z$ $f\left ( z \right )=\log z$

go_editor asked in Calculus Feb 19, 2020

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GATE2020-ME-1: 36
An analytic function of a complex variable $z=x + iy \left ( i=\sqrt{-1} \right )$ is defined as $f\left ( z \right )=x^{2}-y^{2}+i\psi \left ( x,y \right ),$ where $\psi \left ( x,y \right )$ is a real function. The value of the imaginary part of $f(z)$ at $z=\left ( 1+i \right )$ is __________ (round off to $2$ decimal places).

go_editor asked in Calculus Feb 19, 2020

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GATE2019 ME-2: 4
An analytic function $f(z)$ of complex variable $z=x+iy$ may be written as $f(z)=u(x,y)+iv(x,y)$. Then $u(x,y)$ and $v(x,y)$ ...

Arjun asked in Calculus Feb 9, 2019

by Arjun

27.5k points

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GATE2018-2-26
Let $z$ be a complex variable. For a counter-clockwise integration around a unit circle $C$, centered at origin, $\oint_C \frac{1}{5z-4} dz=A \pi i$, the value of $A$ is $2/5$ $1/2$ $2$ $4/5$

Arjun asked in Calculus Feb 17, 2018

by Arjun

27.5k points

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GATE2018-1-4
$F(z)$ is a function of the complex variable $z=x+iy$ given by $F(z)+ i \: z + k \: Re(z) + i \: Im(z)$. For what value of $k$ will $F(z)$ satisfy the Cauchy-Riemann equations? $0$ $1$ $-1$ $y$

Arjun asked in Calculus Feb 17, 2018

by Arjun

27.5k points

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GATE2017 ME-2: 29
If $f(z)=(x^{2}+ay^{2})+i bxy$ is a complex analytic function of $z=x+iy$, where $i=\sqrt{-1}$, then $a=-1, b=-1$ $a=-1, b=2$ $a=1, b= 2$ $a=2, b=2$

Arjun asked in Calculus Feb 27, 2017

by Arjun

27.5k points

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GATE2016-2-4
A function of the complex variable $z= x+iy$, is given as $f(x,y) =u(x,y) +iv(x,y)$ , where $u(x,y) = 2kxy$ and $ v(x,y) =x^2 −y^2$. The value of $k$, for which the function is analytic, is _____

Arjun asked in Calculus Feb 24, 2017

by Arjun

27.5k points

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GATE2016-1-3
$f(z)=u(x,y)+iv(x,y)$ is an analytic function of complex variable $z=x+iy$ where $i=\sqrt{-1}$. If $u(x,y)$=$2xy$ , then $v(x,y)$ may be expressed as $-x^2 + y^2 + $ constant $x^2 – y^2 +$ constant $x^2 + y^2 +$ constant $-(x^2 + y^2) +$ constant

Arjun asked in Calculus Feb 24, 2017

by Arjun

27.5k points

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GATE2015-1-5
Given two complex numbers $z_1=5+(5\sqrt{3})i$ and $z_2=\dfrac{2}{\sqrt{3}}+2i$ , the argument of $\dfrac{z_1}{z_2}$ in degrees is $0$ $30$ $60$ $90$

Arjun asked in Calculus Feb 24, 2017

by Arjun

27.5k points

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GATE Mechanical 2014 Set 4 | Question: 26
If $z$ is a complex variable, the value of $\displaystyle{} \int_{5}^{3i}\dfrac{dz}{z}$ is $− 0.511−1.57i$ $− 0.511+1.57i$ $0.511− 1.57i$ $0.511+1.57i$

Arjun asked in Calculus Feb 19, 2017

by Arjun

27.5k points

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GATE Mechanical 2014 Set 3 | Question: 26
An analytic function of a complex variable $z = x + i y$ is expressed as $f (z) = u(x, y) + i v(x, y)$ , where $i=\sqrt{-1}$ . If $u(x, y) = x^2 − y^2$ , then expression for $v(x, y)$ in terms of $x$, $y$ and a general constant $c$ would be $xy + c \\$ $\dfrac{x^2+y^2}{2}+c \\$ $2xy+c \\$ $\dfrac{(x-y)^2}{2}+c$

Arjun asked in Calculus Feb 19, 2017

by Arjun

27.5k points

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GATE Mechanical 2014 Set 2 | Question: 26
An analytic function of a complex variable $z=x+iy$ is expressed as $f(z)=u(x,y)+iv(x,y)$ , where $i=\sqrt{-1}$ If $u(x,y)=2xy$, then $v(x,y)$ must be $x^2$+$y^2$+constant $x^2$-$y^2$+constant -$x^2$+$y^2$+constant -$x^2$-$y^2$+constant

Arjun asked in Calculus Feb 19, 2017

by Arjun

27.5k points

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GATE Mechanical 2014 Set 1 | Question: 3
The argument of the complex number $\dfrac{1+\imath }{1-\imath }$ where $\imath =\sqrt{-1}$ ,is $-\pi \\$ $\dfrac{-\pi }{2} \\$ $\dfrac{\pi }{2} \\$ $\pi$

Arjun asked in Calculus Feb 19, 2017

by Arjun

27.5k points

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