# GATE Questions & Answers of Complex Analysis Electronics and Communication Engg

#### Complex Analysis 24 Question(s)

The residues of a function $f\left(z\right)=\frac{1}{\left(z-4\right)\left(z+1{\right)}^{3}}$ are

An integral I over a counterclockwise circle C is given by

If C is defined as |z| = 3, then the value of I is

In the following integral, the contour $C$ encloses the points $2\pi j$ and $-2\pi \mathrm{j}$

$-\frac1{2\mathrm\pi}\oint_c\frac{\sin\;z}{\left(z-2\mathrm\pi j\right)^3}dz$

The value of the integral is ________

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

For $f\left(z\right)=\frac{\mathrm{sin}\left(z\right)}{{z}^{2}}$, the residue of the pole at z = 0 is __________

The values of the integral $\frac1{2\pi j}\oint\limits_c\frac{e^z}{z-2}dz$ along a closed contour c in anti-clockwise direction for

(i) the point z0 = 2 inside the contour c, and
(ii) the point z0 = 2 outside the contour c,
respectively, are

Let $z=x+iy$ be a complex variable. Consider that contour  integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE?

Let $f\left(z\right)=\frac{az+b}{cz+d}$.If $f\left({z}_{1}\right)=f\left({z}_{2}\right)$ for all ${z}_{1}\ne {z}_{2},a=2,b=4,c=5$, then d should be equal to __________.

If C denotes the counterclockwise unit circle, the value of the contour integral

$\frac1{2\mathrm{πj}}\oint_c\;Re\left\{z\right\}dz$

is ________.

If C is a circle of radius r with centre z0, in the complex z-plane and if n is a non-zero integer, then $\oint_c\;\frac{dz}{\left(z-z_0\right)^{n+1}}$ equals

The Taylor series expansion of  3 sin x +  2 cos x is

The real part of an analytic function $f\left(z\right)$where $z=x+jy$ is given by ${e}^{-y}\mathrm{cos}\left(x\right)$ The imaginary part of $f\left(z\right)$

Given

$\Large{f\left(z\right)=\frac1{z+1}-\frac2{z+3}}.$ If C is a counterclockwise path in the z-plane such that $\mid z+1\mid=1$, the value of $\frac1{2\pi j}\oint_Cf\left(z\right)dz$

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

The value of the integral $\underset c{\oint\;}\frac{-3z+4}{\left(z^2+4z+5\right)}dz$ where c is the circle $\left|z\right|=1$ is given by

The residues of a complex function $x\left(z\right)=\frac{1-2z}{z\left(z-1\right)\left(z-2\right)}$ at its poles are

If $f\left(z\right)={c}_{0}+{c}_{1}{z}^{-1}$, then ${\Large{\underset{circle}{\underset{unit}{\oint}}\frac{1+f\left(z\right)}zdz}}$ is given by

The Taylor series expansion of $\frac{\mathrm{sin}x}{x-\mathrm{\pi }}$ at $x=\pi$ is given by

The equation sin (z) = 10 has

Which of the following functions would have only odd powers of x in its Taylor series expansion about the point x=0?

The residue of the function  at z = 2 is

In the Taylor series expansion of exp(x) + sin(x) about the point x=$\pi$, the coefficient of (x-$\pi$)2 is

If the semi-circular contour D of radius 2 is as shown in the figure, then the value of the integral $\oint\limits_D\frac1{\left(s^2-1\right)}ds$ is