Recent questions and answers in Complex variables

Consider the following complex numbers\[\begin{array}{l}z_{1}=r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right) \\z_{2}=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\rig...

​​​​Consider two complex numbers $z_{1}=1-i$ and $z_{2}=i$. The argument of $z_{1} z_{2}$ is$0$$\frac{\pi}{4}$$\frac{\pi}{2}$$\pi$

If $z_{1}=-1+i$ and $z_{2}=2 i$, where $i=\sqrt{-1}$, then $\operatorname{Arg}\left(z_{1} / z_{2}\right)$ is$\frac{3 \pi}{4}$$\frac{\pi}{4}$$\frac{\pi}{2}$$\frac{\pi}{3}$

The value of $(1 + i)^{12}$, where $i = \sqrt{-1}$, is$-64i$$64i$$64$$-64$

Consider the hyperbolic functions in $\textbf{Group-1}$ and their definitions in $\textbf{Group-2}$.$\begin{array}{llll} & \textbf{Group-1} & & \textbf{Group-2} \\ \textb...

The value of the complex number $i^{-1/2}$ (where $i=\sqrt{-1}$) is $\frac{1}{\sqrt{2}}\left ( 1-i \right )$$-\frac{1}{\sqrt{2}}i$$\frac{1}{\sqrt{2}}i$$\frac{1}{\sqrt{2}}...

The real part of $6e^{i\pi /3}$ is _______________.

What are the modulus $\left ( r \right )$ and argument $\left ( \theta \right )$ of the complex number $3+4i$ ?$r=\sqrt{7}, \theta =tan^{-1}\left ( \frac{4}{3} \right )$$...

A complex-valued function, $f(z)$, given below is analytic in domain $D$:$$f\left ( z \right )=u\left ( x,y\right )+i\nu \left ( x,y\right )\:\:\:\:\:\:\:z=x+iy$$Which of...

For complex variable $z$, the value of the contour integral $\frac{1}{2\pi i} \underset{C}{\int }\frac{e^{-2z}}{z(z-3)}dz$ along the clockwise contour $C:\left | z \right...

For the functionf(z) = $\frac{1}{(2-z)(z+2)}$the residue at z = 2 is $\_\_\_\_\_$

If $i = \sqrt{-1}$, the value of the integral $$\oint_{c}\dfrac{7z + i}{z(z^2 + 1)}dz \quad\quad\quad\quad|z|<2$$,u...

The decay ratio for a system having complex conjugate poles as $\left ( -\frac{1}{10}+j\frac{2}{15} \right )$ and $\left ( -\frac{1}{10}-j\frac{2}{15} \right )$ is $7\ti...