Recent activity in Complex Analysis

Consider a sequence of non-negative numbers $\left\{x_{n}: n=1,2, \ldots\right\}$. Which of the following statements cannot be true?$\sum_{n=1}^{\infty} x_{n}=\infty$ but...

Consider the two positive integer sequences, defined for a fixed positive integer $c \geq 2$\[f(n)=\frac{1}{n}\left\lfloor\frac{n}{c}\right\rfloor, \quad g(n)=n\left\lflo...

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle $C$ in the complex plane is/are TRUE?$\oint_{C} e^{z} \: d ...

An ant in the plane travels in a spiral such that its position $(x(t), y(t))$ at time $t \geq 0$ is $\left(e^{t} \cos t, e^{t} \sin t\right)$. At time $t=1$, find the rea...

The value of the contour integral, $\oint_C\left(\frac{z+2}{z^2+2 z+2}\right) d z$, where the contour $\mathrm{C}$ is $\left\{z:\left|z+1-\frac{3}{2} j\right|=1\right\}$,...

A simple closed path $C$ in the complex plane is shown in the figure. If$$\oint_{c} \frac{2^{z}}{z^{2} – 1}dz = – i \pi A,$$where $i = \sqrt{-1},$ then the value of $A$ i...

Consider the integral$$\oint _{c}\frac{sin\left ( x \right )}{x^{2}\left ( x^{2}+4 \right )}dx$$where $C$ is a counter-clockwise oriented circle defined as $\left | x-i \...

The value of the integral $\oint_c \frac{-3 z+4}{\left(z^2+4 z+5\right)} d z$ where $c$ is the circle $|z|=1$ is given by$0$$1 / 10$$4 / 5$$1$

The residues of a complex function $X(z)=\dfrac{1-2 z}{z(z-1)(z-2)}$ at its poles are$\frac{1}{2},-\frac{1}{2}$ and $1$$\frac{1}{2}, \frac{1}{2}$ and $-1$$\frac{1}{2}, 1$...

If $f(z) = c_{0} + c_{1} z^{-1},$ then $\displaystyle{} \oint_{\text{unit circle}} \frac{1+f(z)}{z} dz$ is given by$2 \pi c_{1}$$2 \pi\left(1+c_{0}\right)$$2 \pi j c_{1}$...

The equation $\sin (z)=10$ hasno real or complex solutionexactly two distinct complex solutionsa unique solutionan infinite number of complex solutions

The residue of the function $f(z)=\dfrac{1}{(z+2)^{2}(z-2)^{2}}$ at $z=2$ is$-\;\frac{1}{32}$$-\;\frac{1}{16}$$\frac{1}{16}$$\frac{1}{32}$

If the semi-circular contour $D$ of radius $2$ is as shown in the figure, then the value of the integral $\displaystyle{}\oint_{D} \dfrac{1}{\left(s^{2}-1\right)} d s$ is...

The value of the contour integral $\displaystyle{}\oint_{\mid z-j \mid =2} \;\frac{1}{z^2+4} d z$ in positive sense is$\frac{j \pi}{2}$$-\frac{\pi}{2}$$-\frac{j \pi}{2}$$...

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...

The radius of convergence of the power series$\sum_{0}^{\infty} \frac{(3 m) !}{(m !)^{3}} x^{3m}$ is ___________.

If $x=\sqrt{-1}$, then the value of $x^x$ is$e^{\frac{-\pi}{2}}$$e^{\frac{\pi}{2}}$$x$$1$

​​​​​Let $z$ be a complex variable. If $f(z)=\frac{\sin (\pi z)}{z^{2}(z-2)}$ and $C$ is the circle in the complex plane with $|z|=3$ then $\oint_{C} f(z) d z$ is $\_\_\_...

Let $\imath=\sqrt{-1}$. Then $\imath^{\imath}$ could be$\exp (\pi / 2)$$\exp (\pi / 4)$Can't determineTakes infinite valuesIs a complex number

Let $I=\int_{c}\left ( 2zdx+2ydy+2xdx \right )$ where $x,y,z$ are real, and let $C$ be the straight line segment from point $A:(0,2,1)$ to point $B:(4,1,-1)$.The value of...

The contour $C$ given below is on the complex plane $z=x+j y,$ where $j=\sqrt{-1}.$ The value of the integral $\disp...

The residues of a function $$f(z)=\frac1{(z-4)(z+1)^3 }$$are$\frac{-1}{27}$ and $\frac{-1}{125} \\$$\frac{1}{125}$ and $\frac{-1}{125} \\$$\frac{-1}{27}$ and $\frac{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?Four left half plane(LHP) rootsOn...

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 forthe point $z_0=2$ inside t...

For $f(z)= \large\frac{\sin(z)}{z^2}$, the residue of the pole at $z = 0$ is _______

Suppose $C$ is the closed curve defined as the circle $x^{2}+y^{2}=1$ with $C$ oreinted anti-clockwise. The value of $\oint$ ( $xy^{2}$ $dx$ + $ x^{2}y$ $dy$ )over the cu...

Consider the complex valued function $f(z)=2z^{3}+b\mid z \mid^{3}$ where $z$ is a complex variable. The value of $b$ for which function $f(z)$ is analytic is _________

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)?$e^{j\omega_0t...