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

Consider the two series,

S_{A}

S_{B}

\begin{array}{l} S_{A}=\sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}} \\ S_{B}=1+\frac{1}{2}+\frac{1}{8}+\frac{1}{16}+\frac{1}{64}+\frac{1}{128}+\frac{1}{512}+\cdots \end{array}

Which of the following statements is correct for the two given series?

Consider the following series:

\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}

\sum_{n=1}^{\infty} \frac{1}{n(n+1)}

\sum_{n=1}^{\infty} \frac{1}{n!}

Choose the correct option.

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle

C

in the complex plane is/are TRUE?

z

be a complex variable. If

f(z)=\frac{\sin (\pi z)}{z^{2}(z-2)}

C

is the circle in the complex plane with

|z|=3

\oint_{C} f(z) d z

\_\_\_\_\_\_

w^4=16 j

. Which of the following cannot be a value of

w

The value of the contour integral,

\oint_C\left(\frac{z+2}{z^2+2 z+2}\right) d z

, where the contour

\mathrm{C}

\left\{z:\left|z+1-\frac{3}{2} j\right|=1\right\}

, taken in the counter clockwise direction, is

A simple closed path

C

in the complex plane is shown in the figure. If

\oint_{c} \frac{2^{z}}{z^{2} – 1}dz = – i \pi A,

i = \sqrt{-1},

then the value of

A

is _________ (rounded off to two decimal places).

Consider the following series:

\sum_{n=1}^{\infty} \frac{n^{d}}{c^{n}}

For which of the following combinations of

c, d

values does this series converge?

Consider the integral

\oint _{c}\frac{sin\left ( x \right )}{x^{2}\left ( x^{2}+4 \right )}dx

C

is a counter-clockwise oriented circle defined as

\left | x-i \right |=2

. The value of the integral is

Which one of the following functions is analytic over the entire complex plane?

The value of the contour integral

\frac{1}{2\pi j} \oint\left(z+\frac{1}{z}\right)^{2}dz

evaluated over the unit circle

\mid z \mid=1

C

given below is on the complex plane

z=x+j y,

j=\sqrt{-1}.

The value of the integral

\displaystyle{}\dfrac{1}{\pi j}\oint _{C}\dfrac{dz}{z^{2}-1}

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?

The residues of a function

f(z)=\frac1{(z-4)(z+1)^3 }

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)?

Consider the sequence

x[n] = a^nu[n] + b^nu[n]

u[n]

denotes the unit-step sequence and

0<\mid a \mid < \mid b \mid<1

. The region of convergence (ROC) of the

Z

x[n]

In the following integral, the contour

C

encloses the points

2 \pi j

- 2\pi j

-\frac{1}{2\pi}\oint_C\frac{\sin z}{(z-2\pi j)^3} \,dz

The value of the integral is _________

Consider the complex valued function

f(z)=2z^{3}+b\mid z \mid^{3}

z

is a complex variable. The value of

b

for which function

f(z)

is analytic is _________

f(z)= \large\frac{\sin(z)}{z^2}

, the residue of the pole at

z = 0

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