Recent questions and answers in Analysis of complex variables

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 ...

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).

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 p...

​​​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 aparabolahyper...

​​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 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 foll...

$F(z)=\frac{1}{1-z}$, 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) } |...

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 c...

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 ...

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

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 $\text{c}$ wit...

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

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 i...

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

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 = $...

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}...

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 th...

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$

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...

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 answerBoth f$_1...