GATE Overflow for GATE IN - Recent activity in Analysis of complex variables

Edited: GATE IN 2026 | Question: 3

Sat, 21 Mar 2026 07:20:00 +0000

Consider a function $f(z)=z^{2}+z+1$ where $z \in \mathbb{C}$ is a complex variable. A simple closed contour $\gamma$ in $z$-plane encloses the point $z=1+0 j$.

The value of integral $\oint_{\gamma} \dfrac{f(z)}{z-1} d z=$ $\_\_\_\_\_\_$.

$6 \pi j$
$3 \pi j$
$12 \pi j$
$\pi j$

Edited: GATE IN 2024 | Question: 39

Sat, 29 Nov 2025 13:20:47 +0000

The complex functions $f(z)=u(x, y)+i ~v(x, y)$ and $\overline{f(z)}=u(x, y)-i~ v(x, y)$ are both analytic in a given domain. Choose the correct option(s) from the following.

$\dfrac{\partial u}{\partial x}=\dfrac{\partial v}{\partial y}=0$
$\dfrac{\partial u}{\partial y}=-\dfrac{\partial v}{\partial x} \neq 0$
$\dfrac{d f(z)}{d z}=0$
$\dfrac{d f(z)}{d z} \neq 0$

Edited: GATE IN 2024 | Question: 36

Sat, 29 Nov 2025 13:19:07 +0000

Let $C$ be the closed curve in the $x y$-plane, traversed in the counterclockwise direction along the boundary of the rectangle with vertices at $(0,0),(2,0),(2,1),(0,1)$. The value of the line integral

$$ \oint_{C}\left(-e^{y} d x+e^{x} d y\right)$$

$e^{2}+2 e-3$
$e^{2}-2 e-3$
$e^{2}+e-1$
$e^{2}+e+1$

Edited: GATE IN 2024 | Question: 1

Fri, 15 Aug 2025 19:48:01 +0000

Let $\boldsymbol{z}=x+i y$ be a complex variable and $\overline{\boldsymbol{z}}$ be its complex conjugate. The equation $\bar{z}^{2}+z^{2}=2$ represents a

parabola
hyperbola
ellipse
circle

Edited: GATE IN 2025 | Question: 39

Mon, 19 May 2025 08:31:36 +0000

Choose the correct statement(s) from the following options, regarding Cauchy's theorem on complex integration $\oint_{C} f(z) \mathrm{d} z$ where $C$ is a simple closed path in a simply connected domain $D$.

Cauchy's theorem cannot be directly applied to conclude that $\oint_{C} f(z) \mathrm{d} z=0$ when $f(z)=\frac{1}{z^{2}}$, and $C$ is the unit circle
If $f(z)$ is analytic in $D$, then it can be concluded that $\oint_{C} f(z) \mathrm{d} z=0$ for any simple closed path $C$ in $D$
The function $f(z)$ must be analytic in $D$ to conclude $\oint_{C} f(z) \mathrm{d} z=0$ for any simple closed path $C$ in $D$
$\oint_{C} f(z) \mathrm{d} z \neq 0$ when $f(z)=\frac{1}{z^{2}}$, since the function is not analytic at $z=0$

Edited: GATE IN 2025 | Question: 23

Mon, 19 May 2025 07:36:59 +0000

Consider the function $f(z)=\frac{2 z+1}{z^{2}-z}$, where $z$ is a complex variable. The sum of the residues at singular points of $f(z)$ is ______________ (in integer).

Recategorized: GATE IN 2013 | Question: 4

Mon, 21 Apr 2025 15:08:34 +0000

The complex function $\tan h(s)$ is analytic over a region of the imaginary axis of the complex s-plane if the following is $\text{TRUE}$ everywhere in the region for all integers $n$

$Re(s)=0$
$Im(s)\neq n\pi$
$Im(s)\neq \frac{n\pi}{3}$
$Im(s)\neq\frac{(2n+1)\pi}{2}$

Recategorized: GATE IN 2022 | Question: 40

Mon, 21 Apr 2025 11:28:45 +0000

For the complex number $z= \frac{a+jb}{a-jb},$ where $a>0$ and $b>0.$

Which of the following statement(s) is/are true?

The phase is $2 \; \tan^{-1}\frac{b}{a}$
The phase is $ \; \tan^{-1}\frac{2b}{a}$
The magnitude is $1$
The magnitude is $\sqrt{\frac{a^{2}+b^{2}}{a^{2}-b^{2}}}$

Recategorized: GATE IN 2022 | Question: 51

Mon, 21 Apr 2025 11:28:21 +0000

Consider the function $f(z) = \dfrac{1}{(z+1)(z+2)(z+3)}.$ The residue of $f(z)$ at $z = -1,$ is ___________

Recategorized: GATE IN 2023 | Question: 9

Mon, 21 Apr 2025 11:18:49 +0000

$F(z)=\frac{1}{1-z}$, when expanded as a power series around $z=2$, would result in $F(z)=\sum_{k=0}^{\infty} a_k(z-2)^k$, with the region of convergence $\text{(ROC) } |z-2|<1$. The coefficients $a_k, k \geq 0$, are given by the expression _____________.

$(-1)^k$
$(-1)^{k+1}$
$\left(\frac{1}{2}\right)^k$
$\left(\frac{-1}{2}\right)^{k+1}$

Recategorized: GATE IN 2023 | Question: 29

Mon, 21 Apr 2025 11:17:54 +0000

Let $f(z)=j \frac{1-z}{1+z}$, where $z$ denotes a complex number and $j$ denotes $\sqrt{-1}$. The inverse function $f^{-1}(z)$ maps the real axis to the___________.

unit circle with centre at the origin
unit circle with centre not at the origin
imaginary axis
real axis

Recategorized: GATE IN 2021 | Question: 24

Sun, 11 Apr 2021 13:16:42 +0000

Let $f\left ( z \right )=\dfrac{1}{z^{2}+6z+9}$ defined in the complex plane. The integral $\oint _{c}\:f\left ( z \right )dz$ over the contour of a circle $\text{c}$ with center at the origin and unit radius is _______________.

Recategorized: GATE IN 2021 | Question: 26

Sun, 11 Apr 2021 13:16:42 +0000

$f\left ( Z \right )=\left ( Z-1 \right )^{-1}-1+\left ( Z-1 \right )-\left ( Z-1 \right )^{2}+ \cdots$ is the series expansion of

$\frac{-1}{Z\left ( Z-1 \right )}$ for $\left | Z-1 \right |< 1$
$\frac{1}{Z\left ( Z-1 \right )}$ for $\left | Z-1 \right |< 1$
$\frac{1}{\left ( Z-1 \right )^{2}}$ for $\left | Z-1 \right |< 1$
$\frac{-1}{\left ( Z-1 \right )}$ for $\left | Z-1 \right |< 1$

Recategorized: GATE2020: 15

Sat, 20 Mar 2021 18:07:22 +0000

Let $f(z)=\frac{1}{z+a},a>0.$ the value of the integral $\oint f(z)dz$ over a circle $C$ with center $(-a,0)$ and radius $R>0$ evaluated in the anti-clockwise direction is ____________

$0$
$2\pi i$
$-2\pi i$
$4\pi i$

Recategorized: GATE2019 IN: 29

Sat, 20 Mar 2021 17:18:07 +0000

A complex function f(z) = u(x,y) + i v(x,y) and its complex conjugate f*(z) = u(x,y) – i v(x,y) are both analytic in the entire complex plane, where z = x + i y and i = $\sqrt{-1}$. The function f is then given by

f(z) = x + i y
f(z) = x$^{2}$ – y$^{2}$ + i 2xy
f(z) = constant
f(z) = x$^{2}$ + y$^{2}$

Recategorized: GATE2018IN: 2

Sat, 20 Mar 2021 16:59:33 +0000

Let f$_1$(Z) =Z$^2$ and f$_2$(Z) = $\overline{z}$ be two complex variable functions. Here $\overline{z}$ is the complex conjugate of z. Choose the correct answer

Both f$_1$(Z) and f$_2$(Z) are analytic
Only f$_1$(Z) is analytic
Only f$_2$(Z) is analytic
Both f$_1$(Z) and f$_2$(Z) are not analytic

Recategorized: GATE IN 2017 | Question: 3

Sat, 20 Mar 2021 13:10:16 +0000

Let $z=x+jy$ where $j=\sqrt{-1}$. Then $\overline{\cos z}$ =

$\cos z$
$cos\overline{z}$
$\sin z$
$\sin\overline{z}$

Recategorized: GATE2016-30

Sat, 20 Mar 2021 12:53:28 +0000

The value of the integral $\displaystyle{}\frac{1}{2\pi j}\int_c \frac{Z^2+1}{Z^2-1}dz$ where $z$ is a complex number and $C$ is a unit circle with center at $1+0j$ in the complex plane is $\_\_\_\_\_\_\_\_.$

Retagged: GATE2016-5

Sat, 20 Mar 2021 12:52:27 +0000

In the neighborhood of $z=1$, the function $f(z)$ has a power series expansion of the form $f(z)$ = $1$ + $(1-z)$ + $(1-z)^2+ \ldots$

Then $f(z)$ is

$\frac{1}{z}$
$\frac{-1}{z-2}$
$\frac{z-1}{z+}$
$\frac{1}{2z-1}$

Recategorized: GATE2015-12

Sat, 20 Mar 2021 12:12:41 +0000

The value of $\oint \frac{1}{Z^2} dZ,$ where the contour is the unit circle traversed clockwise, is

$-2\pi i$
$0$
$2\pi i$
$4\pi i$