Complex Variable MCQs Question and Answer in Engineering Maths

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81.
$$\oint {\frac{{{{\text{z}}^2} - 4}}{{{{\text{z}}^2} + 4}}{\text{dz}}} $$   evaluated anticlockwise around the circle |z - i| = 2, where $${\text{i}} = \sqrt { - 1} ,$$   is

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82.
An analytic function f(z) of complex variable z = x + iy may be written as f(z) = u(x, y) + iv(x, y). Then, u(x, y) and v(x, y) must satisfy,

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83.
In the neighborhood of z = 1, the function f(z) has a power series expansion of the form $${\text{f}}\left( {\text{z}} \right) = 1 + \left( {1 - {\text{z}}} \right) + {\left( {1 - {\text{z}}} \right)^2} + \,...$$
Then f(z) is

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85.
Square roots of -i, where $${\text{i}} = \sqrt { - 1} ,$$   are

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86.
Given two complex numbers $${{\text{z}}_1} = 5 + \left( {5\sqrt 3 } \right){\text{i}}$$    and $${{\text{z}}_2} = \frac{2}{{\sqrt 3 }} + 2{\text{i}}$$    the argument $$\frac{{{{\text{z}}_1}}}{{{{\text{z}}_2}}}$$ in degree is

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87.
The value of the contour integral in the complex plane $$\oint {\frac{{{{\text{z}}^3} - 2{\text{z}} + 3}}{{{\text{z}} - 2}}{\text{dz}}} $$    along the contour |z| = 3, taken counter-clockwise is

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88.
In the Laurent expansion of $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{\left( {{\text{z}} - 1} \right)\left( {{\text{z}} - 2} \right)}}$$    valid in the region 1 < | z | < 2, the coefficient of $$\frac{1}{{{{\text{z}}^2}}}$$ is

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90.
For an analytic function, f(x + iy) = u(i, y) + iv(i, y), u is given by u = 3x2 - 3y. The expression for v, considering K to be a constant is

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