Complex Variable · Engineering Mathematics · GATE EE

Marks 1

Which of the following complex functions is/are analytic on the complex plane?

GATE EE 2024

Consider the complex function $f(z) = \cos z + e^{z^2}$. The coefficient of $z^5$ in the Taylor series expansion of $f(z)$ about the origin is ______ (rounded off to 1 decimal place).

GATE EE 2024

Let $P(z)=z^3+(1+j) z^2+(2+j) z+3$, where $z$ is complex number. Which one of the following is true?

GATE EE 2021

For a complex number $$z,$$
$$\mathop {Lim}\limits_{z \to i} {{{z^2} + 1} \over {{z^3} + 2z - i\left( {{z^2} + 2} \right)}}$$ is

GATE EE 2017 Set 1

Consider the function $$f\left( z \right) = z + {z^ * }$$ where $$z$$ is a complex variable and $${z^ * }$$ denotes its complex conjugate. Which one of the following is TRUE?

GATE EE 2016 Set 2

Given $$f\left( z \right) = g\left( z \right) + h\left( z \right),$$ where $$f,g,h$$ are complex valued functions of a complex variable $$z.$$ Which ONE of the following statements is TRUE?

GATE EE 2015 Set 2

Integration of the complex function $$f\left( z \right) = {{{z^2}} \over {{z^2} - 1}},$$ in the counterclockwise direction, around $$\left| {z - 1} \right| = 1,$$ is

GATE EE 2014 Set 3

All the values of the multi valued complex function $${1^i},$$ where $$i = \sqrt { - 1} $$ are

GATE EE 2014 Set 2

Square roots of $$-i,$$ where $$i = \sqrt { - 1} $$ are

GATE EE 2013

Given $$X(z) = {z \over {{{(z - a)}^2}}}$$ with |z| > a, the residue of $$X(z){z^{n - 1}}$$ at z = a for $$n \ge 0$$ will be

GATE EE 2008

Marks 2

The magnitude of the contour integral

$$ \int_c \frac{(z+1)^2}{(z-i)(z-2)} d z $$

over the contour $C:|z-2-i|=\frac{3}{2}$ is $\_\_\_\_$ . [Round off to two decimal places]

Note : $z$ is a complex variable and $i=\sqrt{-1}$.

GATE EE 2026

Let $C$ be a clockwise oriented closed curve in the complex plane defined by $|\lambda|=1$. Further, let $f(x)=j z$ be a complex function, where $j=\sqrt{-1}$. Then, $\oint_C f(z) d z=$ ___________ .

GATE EE 2025

Let $(-1-j),(3-j),(3+j)$ and $(-1+j)$ be the vertices of rectangle $C$ in the complex plane. Assuming that $C$ is traversed in counter-clockwise direction, the value of contour integral $\oint_C \frac{d z}{z^2(z-4)}$ is

GATE EE 2021

The value of the contour integral in the complex - plane $$\oint {{{{z^3} - 2z + 3} \over {z - 2}}} dz$$ along the contour $$\left| z \right| = 3,$$ taken counter-clockwise is

GATE EE 2017 Set 2

Consider the line integral $${\rm I} = \int\limits_c {\left( {{x^2} + i{y^2}} \right)dz,} $$ where $$z=x+iy.$$ The line $$c$$ is shown in the figure below.