Complex Variable · Engineering Mathematics · GATE ECE

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Marks 1

GATE ECE 2022
A simple closed path C in the complex plane is shown in the figure. If $$\oint\limits_c {{{{2^z}} \over {{z^2} - 1}}dz = - i\pi A} $$, where $$i = \s...
GATE ECE 2022
Consider the following series: $$\sum\limits_{n = 1}^\infty {{{{n^d}} \over {{c^n}}}} $$ For which of the following combinations of c, d values does ...
GATE ECE 2017 Set 2
The residues of a function $$f\left( z \right) = {1 \over {\left( {z - 4} \right){{\left( {z + 1} \right)}^3}}}$$ are
GATE ECE 2016 Set 3
For $$f\left( z \right) = {{\sin \left( z \right)} \over {{z^2}}},$$ the residue of the pole at $$z=0$$ ________.
GATE ECE 2014 Set 2
The real part of an analytic function $$f(z)$$ where $$z=x+jy$$ is given by $${e^{ - y}}\cos \left( x \right).$$ The imaginary part of $$f(z)$$ is
GATE ECE 2014 Set 1
$$C$$ is a closed path in the $$z-$$plane given by $$\left| z \right| = 3.$$ The value of the integral $$\oint\limits_c {{{{z^2} - z + 4j} \over {z ...
GATE ECE 2011
The value of the integral $$\oint\limits_c {{{ - 3z + 4} \over {{z^2} + 4z + 5}}} \,\,dz,$$ when $$C$$ is the circle $$|z| = 1$$ is given by
GATE ECE 2010
The residues of a complex function $$X\left( z \right) = {{1 - 2z} \over {z\left( {z - 1} \right)\left( {z - 2} \right)}}$$ at it poles
GATE ECE 2009
If $$f\left( z \right) = {C_0} + {C_1}{z^{ - 1}}\,\,$$ then $$\oint\limits_{|z| = 1} {{{1 + f\left( z \right)} \over z}} \,\,dz$$ is given
GATE ECE 2008
The equation sin(z) = 10 has
GATE ECE 2008
The residue of the function $$f(z) = {1 \over {{{\left( {z + 2} \right)}^2}{{\left( {z - 2} \right)}^2}}}$$ at z = 2 is
GATE ECE 2007
The value of $$\oint\limits_C {{1 \over {\left( {1 + {z^2}} \right)}}} dz$$ where C is the contour $$\,\left| {z - {i \over 2}} \right| = 1$$ is
GATE ECE 2006
The value of the counter integral $$$\int\limits_{\left| {z - j} \right| = 2} {{1 \over {{z^2} + 4}}\,} dz\,\,in\,the\,positive\,sense\,is$$$
GATE ECE 2006
For the function of a complex variable w = lnz (where w = u + jv and z = x + jy) the u = constant lines get mapped in the z-plane as

Marks 2

GATE ECE 2018
The contour C given below is on the complex plane $$z = x + jy$$, where $$j = \sqrt { - 1} $$. The value of the integral $${1 \over {\pi j}}\oint\lim...
GATE ECE 2017 Set 2
An integral $${\rm I}$$ over a counter clock wise circle $$C$$ is given by $${\rm I} = \oint\limits_c {{{{z^2} - 1} \over {{z^2} + 1}}} \,\,{e^z}\,dz$...
GATE ECE 2016 Set 2
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...
GATE ECE 2016 Set 3
The value of the integral $${1 \over {2\pi j}}\oint\limits_C {{{{e^z}} \over {z - 2}}dz} $$ along a closed contour $$c$$ in anti-clockwise direction f...
GATE ECE 2016 Set 1
In the following integral, the contour $$C$$ encloses the points $${2\pi j}$$ and $$-{2\pi j}$$. The value of the integral $$ - {1 \over {2\pi }}\oint...
GATE ECE 2015 Set 2
Let $$f\left( z \right) = {{az + b} \over {cz + d}}.$$ If $$f\left( {{z_1}} \right) = f\left( {{z_2}} \right)$$ for all $${z_1} \ne {z_2}.\,\,a = 2,\,...
GATE ECE 2015 Set 2
If $$C$$ denotes the counter clockwise unit circle. The value of the contour integral $${1 \over {2\pi i}}\oint\limits_c {{\mathop{\rm Re}\nolimits} \...
GATE ECE 2015 Set 3
If $$C$$ is a circle of radius $$r$$ with centre $${z_0}$$ in the complex $$z$$-plane and if $$'n'$$ is a non-zero integer, then $$\oint\limits_c {{{d...
GATE ECE 2015 Set 1
Let $$z=x+iy$$ be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction . Which one of t...
GATE ECE 2012
If $$x = \sqrt { - 1} ,\,\,$$ then the value of $${X^x}$$ is
GATE ECE 2012
Given $$f\left( z \right) = {1 \over {z + 1}} - {2 \over {z + 3}}.$$ If $$C$$ is a counterclockwise path in the $$z$$-plane such that $$\left| {z + 1}...
GATE ECE 2007
If the semi-circular contour D of radius 2 is as shown in the figure, then the value of the integral $$\oint\limits_D {{1 \over {{s^2} - 1}}} ds$$ is ...
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