1
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 __________.
2
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) $$7.39$$
3
GATE ECE 2016 Set 1
Numerical
+2
-0
In the following integral, the contour $$C$$ encloses the points $${2\pi j}$$ and $$-{2\pi j}$$. The value of the integral $$- {1 \over {2\pi }}\oint\limits_c {{{\sin z} \over {{{\left( {z - 2\pi j} \right)}^3}}}} dz$$ is ___________.
4
GATE ECE 2015 Set 2
Numerical
+2
-0
Let $$f\left( z \right) = {{az + b} \over {cz + d}}.$$ If $$f\left( {{z_1}} \right) = f\left( {{z_2}} \right)$$ for all $${z_1} \ne {z_2}.\,\,a = 2,\,\,b = 4$$ and $$C=5,$$ then $$d$$ should be equal to