GATE Questions & Answers of Complex Analysis Electronics and Communication Engg

In the following integral, the contour C encloses the points 2πj and -2πj

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

The value of the integral is ________

Consider the complex valued function fz=2z3+bz3 where $z$ is a complex variable. The value of $b$ for which the function fz is analytic is ________

For fz=sinzz2, 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 fz=az+bcz+d.If fz1=fz2 for all z1z2,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 fzwhere z=x+jy is given by e-ycosx The imaginary part of fz


$\Large{f\left(z\right)=\frac1{z+1}-\frac2{z+3}}.$ 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=-1, then the value of xx is

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

The residues of a complex function xz=1-2zzz-1z-2 at its poles are

If fz=c0+c1z-1, then ${\Large{\underset{circle}{\underset{unit}{\oint}}\frac{1+f\left(z\right)}zdz}}$ is given by

The Taylor series expansion of sinxx-π at x=π 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 fz=1z+22z-2 at z = 2 is

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

For |x|<< 1, coth(x) can be approximated as

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