GATE ECE
Engineering Mathematics
Complex Variable
Previous Years Questions

## Marks 1

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 $$i = \s... Consider the following series:$$\sum\limits_{n = 1}^\infty {{{{n^d}} \over {{c^n}}}} $$For which of the following combinations of c, d values does ... The residues of a function$$f\left( z \right) = {1 \over {\left( {z - 4} \right){{\left( {z + 1} \right)}^3}}}$$are For$$f\left( z \right) = {{\sin \left( z \right)} \over {{z^2}}},$$the residue of the pole at$$z=0$$________. The real part of an analytic function$$f(z)$$where$$z=x+jy$$is given by$${e^{ - y}}\cos \left( x \right).$$The imaginary part of$$f(z)$$is$$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 ...
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
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
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
The equation sin(z) = 10 has
The residue of the function $$f(z) = {1 \over {{{\left( {z + 2} \right)}^2}{{\left( {z - 2} \right)}^2}}}$$ at z = 2 is
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
The value of the counter integral $$\int\limits_{\left| {z - j} \right| = 2} {{1 \over {{z^2} + 4}}\,} dz\,\,in\,the\,positive\,sense\,is$$$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 The contour C given below is on the complex plane $$z = x + jy$$, where $$j = \sqrt { - 1}$$. The value of the integral $${1 \over {\pi j}}\oint\lim... An integral$${\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$...
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...
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...
In the following integral, the contour $$C$$ encloses the points $${2\pi j}$$ and $$-{2\pi j}$$. The value of the integral $$- {1 \over {2\pi }}\oint... If$$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...
Let $$f\left( z \right) = {{az + b} \over {cz + d}}.$$ If $$f\left( {{z_1}} \right) = f\left( {{z_2}} \right)$$ for all $${z_1} \ne {z_2}.\,\,a = 2,\,... If$$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} \...
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...
If $$x = \sqrt { - 1} ,\,\,$$ then the value of $${X^x}$$ is
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 $$\left| {z + 1}... If the semi-circular contour D of radius 2 is as shown in the figure, then the value of the integral$$\oint\limits_D {{1 \over {{s^2} - 1}}} ds is ...
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