1
GATE ECE 2018
Numerical
+2
-0.67
The contour C given below is on the complex plane $$z = x + jy$$, where $$j = \sqrt { - 1}$$. The value of the integral $${1 \over {\pi j}}\oint\limits_C {{{dz} \over {{z^2} - 1}}}$$ is ________________.
2
GATE ECE 2017 Set 2
+2
-0.6
An integral $${\rm I}$$ over a counter clock wise circle $$C$$ is given by $${\rm I} = \oint\limits_c {{{{z^2} - 1} \over {{z^2} + 1}}} \,\,{e^z}\,dz$$
If $$C$$ is defined as $$\left| z \right| = 3,$$ then the value of $${\rm I}$$ is
A
$$- \pi i\,\,\sin \left( 1 \right)$$
B
$$- 2\pi i\,\,\sin \left( 1 \right)$$
C
$$- 3\pi i\,\,\sin \left( 1 \right)$$
D
$$- 4\pi i\,\,\sin \left( 1 \right)$$
3
GATE ECE 2016 Set 2
Numerical
+2
-0
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 $$f(z)$$ is analytic is __________.
4
GATE ECE 2016 Set 3
+2
-0.6
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 for
(i) the point $${z_0} = 2$$ inside the contour $$c,$$ and
(ii) the point $${z_0} = 2$$ outside the contour $$c,$$ respectively, are
A
(i) $$2.72,$$ (ii) $$0$$
B
(i) $$7.39,$$ (ii) $$0$$
C
(i) $$0,$$ (ii) $$2.72$$
D
(i) $$0,$$ (ii) $$7.39$$
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