GATE EC Complex Analysis questions

Practise Complex Analysis questions for GATE EC. Explore official PYQs and original topic-wise quizzes, check answers and view available solutions. Free, no sign-in.

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SubjectTopicYearType

The ROC (region of convergence) of the

z

-transform of a discrete-time signal is represented by the shaded region in the

z

-plane. If the signal

x[n]=(2.0)^{\mid n\mid},-\infty<n<+\infty

, then the ROC of its

z

-transform is represented by

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?

Let

f(z)=\dfrac{az+b}{cz+d}.

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 ________.

C

denotes the counterclockwise unit circle, the value of the contour integral

\dfrac{1}{2\pi j}\oint_{C} Re\{z\}dz

is __________.

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

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,

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

Let

x[n] = \bigg( – \dfrac{1}{9}\bigg)^{n}u(n) \:– \bigg( – \dfrac{1}{3}\bigg)^{n}u(-n-1).

The Region of Convergence (ROC) of the

z

-transform of

x[n]

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)

x[n]=(\frac{1}{3})^{|n|}-(\frac{1}{2})^{|n|}u[n]

, then the region of convergence (ROC) of its Z-transform in the Z-plane will be

Given

f(z)=\frac{1}{z+1}-\frac{2}{z+3}

. If

C

is a counterclockwise path in the z-plane such that

\mid z+1 \mid=1

, the value of

\frac{1}{2\pi j}\oint_cf(z)dz

x=\sqrt{-1}

, then the value of

x^x

For the function of a complex variable

\text{W}=\ln \text{Z}\; (

where,

\mathrm{W}=u+j \mathrm{v}

and

\mathrm{Z}=x+j y),

the

u=

constant lines get mapped in

\text{Z}

-plane as

The value of the contour integral

\displaystyle{}\oint_{\mid z-j \mid =2} \;\frac{1}{z^2+4} d z

in positive sense is