GATE EE
Engineering Mathematics
Complex Variable
Previous Years Questions

## Marks 1

For a complex number $$z,$$ $$\mathop {Lim}\limits_{z \to i} {{{z^2} + 1} \over {{z^3} + 2z - i\left( {{z^2} + 2} \right)}}$$ is
Consider the function $$f\left( z \right) = z + {z^ * }$$ where $$z$$ is a complex variable and $${z^ * }$$ denotes its complex conjugate. Which one ...
Given $$f\left( z \right) = g\left( z \right) + h\left( z \right),$$ where $$f,g,h$$ are complex valued functions of a complex variable $$z.$$ Which O...
Integration of the complex function $$f\left( z \right) = {{{z^2}} \over {{z^2} - 1}},$$ in the counterclockwise direction, around $$\left| {z - 1} \r... All the values of the multi valued complex function$${1^i},$$where$$i = \sqrt { - 1} $$are Square roots of$$-i,$$where$$i = \sqrt { - 1} $$are Given$$X(z) = {z \over {{{(z - a)}^2}}}$$with |z| > a, the residue of$$X(z){z^{n - 1}}$$at z = a for$$n \ge 0$$will be ## Marks 2 The value of the contour integral in the complex - plane$$\oint {{{{z^3} - 2z + 3} \over {z - 2}}} dz$$along the contour$$\left| z \right| = 3,$$t... Consider the line integral$${\rm I} = \int\limits_c {\left( {{x^2} + i{y^2}} \right)dz,} $$where$$z=x+iy.$$The line$$c$$is shown in the figure b... The value of the integral$$\oint\limits_c {{{2z + 5} \over {\left( {z - {1 \over 2}} \right)\left( {{z^2} - 4z + 5} \right)}}} dz$$over the contour ... Let$$S$$be the set of points in the complex plane corresponding to the unit circle.$$\left( {i.e.,\,\,S = \left\{ {z:\left| z \right| = 1} \right\}...
$$\oint {{{{z^2} - 4} \over {{z^2} + 4}}} dz\,\,$$ evaluated anticlockwise around the circular $$\left| {z - i} \right| = 2,$$ where $$i = \sqrt { - 1... If$$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}...
A point $$z$$ has been plotted in the complex plane as shown in the figure below The plot of the complex number $$w = 1/z$$ ...
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