Recent questions tagged complex-variables

The magnitude of the contour integral\[\oint_{C}\left(\frac{(z+1)^{2}}{(z-i)(z-2)}\right) d z\]over the contour $C:|z-2-i|=3 / 2$ is $\_\_\_\_$(Round off to two decimal p...

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

​​​​Which of the following complex functions is/are analytic on the complex plane?$f(z)=j \operatorname{Re}(z)$$f(z)=\operatorname{Im}(z)$$f(z)=e^{|z|}$$f(z)=z^{2}-z$

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 ...

Let $p\left ( z\right )=z^{3}+\left ( 1+j \right )z^{2}+\left ( 2+j \right )z+3$, where $z$ is a complex number.Which one of the following is true?$\text{conjugate}\:\lef...

The value of the following complex integral, with $\text{C}$ representing the unit circle centered at origin in the counterclockwise sense, is:$$\int _{C}\frac{z^{2}+1}{z...

Which one of the following functions is analytic in the region $\mid z \mid \leq 1$ ?$\frac{z^{2}-1}{z} \\ $$\frac{z^{2}-1}{z+2} \\ $$\frac{z^{2}-1}{z-0.5} \\ $$\frac{z^...

The closed-loop line integral $$\underset{\mid z \mid = 5}{\oint} \frac{z^3 + z^2 + 8}{z+2}dz$$evaluated Counter-clockwise, is $+8 j \pi$$-8 j \pi$$-4 j \pi$$+4 j \pi$

Given $f(z) = \dfrac{1}{z+1} – \dfrac{2}{z+3}$. If $C$ is a counterclockwise path in the $z$-plane such that $\mid z+1 \mid =1$, the value of $\dfrac{1}{2 \pi \: j} \oint...

If $C$ is a circle $\mid z \mid=4$ and $f(z)=\frac{z^2}{(z^2-3z+2)^2}$, then $\underset{C}{\oint} f(z) dz$ is$1$$0$$-1$$-2$

The number of roots of the polynomial, $s^7+s^6+7s^5+14s^4+31s^3+73s^2+25s+200$, in the open left half of the complex plane is$3$$4$$5$$6$

The value of the integral $\oint _c \frac{z+1}{z^2-4} dz$ in counter clockwise direction around a circle $C$ of radius $1$ with center at the point $z=-2$ is$\frac{\pi i}...

$\displaystyle{}\int \frac{z^2-4}{z^2+4}\: dz$ evaluated anticlockwise around the circle $\mid z-i \mid=2$ , where $i=\sqrt{-1}$, is$-4\pi$$0$$2+\pi$$2+2i$

Square roots of $-i$,where $i=\sqrt{-1}$, are$i,-i \\$$\cos(-\dfrac{\pi }{4} )+i\sin(-\dfrac{\pi }{4})+\cos(\dfrac{3\pi }{4})+i\sin(\dfrac{3\pi }{4}) \\$$\cos(\dfrac{\pi ...

Integration of the complex function $f(z)=\dfrac{z^2}{z^2-1}$ , in the counterclockwise direction, around $\mid z-1 \mid = 1$, is$-\pi i$$0$$\pi i$$2 \pi i$

All the values of the multi-valued complex function $1^i$,where $i=\sqrt{-1}$ arepurely imaginary.real and non-negativeon the unit circle.equal in real and imaginary part...

Let $S$ be the set of points in the complex plane corresponding to the unit circle. $(\text{That is}, S = \{z :\:\: \mid z \mid =1\}).$ Consider the function $f(z)=zz^{\a...

Given $f(z) = g(z) + h(z)$, where $f, g, h$ are complex valued functions of a complex variable $z$. Which one of the following statements is TRUE?If $f(z)$ is differentia...

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