Complex Variable MCQ
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The real part of an analytic function f(z) where z = x + jy is given by e<sup>-y</sup> cos(x). The imaginary part of f(z) is
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For an analytic function, f(x + iy) = u(i, y) + iv(i, y), u is given by u = 3x<sup>2</sup> - 3y. The expression for v, considering K to be a constant is
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If $${\text{x}} = \sqrt { - 1} ,$$ then the value of x<sup>x</sup> is
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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
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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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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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Square roots of -i, where $${\text{i}} = \sqrt { - 1} ,$$ are
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Solutions of Laplace equation having continuous second-order partial derivatives are called
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In the neighborhood of z = 1, the function f(z) has a power series expansion of the form $${\text{f}}\left( {\text{z}} \right) = 1 + \left( {1 - {\text{z}}} \right) + {\left( {1 - {\text{z}}} \right)^2} + \,...$$<br>Then f(z) is
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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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$$\oint {\frac{{{{\text{z}}^2} - 4}}{{{{\text{z}}^2} + 4}}{\text{dz}}} $$ evaluated anticlockwise around the circle |z - i| = 2, where $${\text{i}} = \sqrt { - 1} ,$$ is
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If a complex number ω satisfied the equation ω<sup>3</sup> = 1 then value of $$1 + \omega + \frac{1}{\omega }$$ is
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Given $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{{\text{z}} + 1}} - \frac{2}{{{\text{z}} + 3}}.$$ If C is a counter clockwise path in the z-plane such that |z + 1| = 1, the value of $$\frac{1}{{2\pi {\text{j}}}}\oint_{\text{C}} {{\text{f}}\left( {\text{z}} \right){\text{dz}}} $$ is
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The residue of $${\text{f}}\left( {\text{z}} \right) = \frac{{{{\text{z}}^3}}}{{{{\left( {{\text{z}} - 1} \right)}^4}\left( {{\text{z}} - 2} \right)\left( {{\text{z}} - 3} \right)}}$$ at z = 3 is
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For a complex number z, $$\mathop {\lim }\limits_{{\text{z}} \to {\text{i}}} \frac{{{{\text{z}}^2} + 1}}{{{{\text{z}}^3} + 2{\text{z}} - {\text{i}}\left( {{{\text{z}}^2} + 2} \right)}}$$ is
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The residue of the function $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{{{\left( {{\text{z}} + 2} \right)}^2}{{\left( {{\text{z}} - 2} \right)}^2}}}$$ at z = 2 is
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An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) + iv(x, y), where $${\text{i}} = \sqrt { - 1} .$$ If u(x, y) = x<sup>2</sup> - y<sup>2</sup>, then expression for v(x, y) in terms of x, y and a general constant c would be
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The value of the integral $$\oint {\frac{{2{\text{z}} + 5}}{{\left( {{\text{z}} - \frac{1}{2}} \right)\left( {{{\text{z}}^2} - 4{\text{z}} + 5} \right)}}{\text{dz}}} $$ over the contour |z| = 1, taken in the anti-clockwise direction, would be
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Which one of the following functions is analytic in the region |z| ≤ 1?
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The residues of a complex function $${\text{X}}\left( {\text{z}} \right) = \frac{{1 - 2{\text{z}}}}{{{\text{z}}\left( {{\text{z}} - 1} \right)\left( {{\text{z}} - 2} \right)}}$$ at its poles are
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The complex number $${\left( {\frac{{2 + {\text{i}}}}{{3 - {\text{i}}}}} \right)^2}$$ is
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A complex function f(z) = u(x, y) + iv(x, y) and its complex conjugate, f'(z) = u(x, y) - iv(x, y) are both analytic in the entire complex plane, where z = x + iy and $${\text{i}} = \sqrt { - 1} .$$ The function f is then given by
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The contour integral $$\oint\limits_{\text{C}} {{{\text{e}}^{\frac{1}{{\text{z}}}}}{\text{dz}}} $$ with C as the counter-clockwise unit circle in the z-plane is equal to
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The value of $$\oint {\Gamma \frac{{3{\text{z}} - 5}}{{\left( {{\text{z}} - 1} \right)\left( {{\text{z}} - 2} \right)}}{\text{dz}}} $$ along a closed path $$\Gamma $$ is is equal to (4πi), where z = x + iy and $${\text{i}} = \sqrt { - 1} .$$ The correct path $$\Gamma $$ is
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The value of the contour integral $$\oint\limits_{\left| {{\text{z}} - {\text{i}}} \right| = 2} {\frac{1}{{{{\text{z}}^2} + 4}}{\text{dz}}} $$ in positive sense is
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For a complex number z = 1 - 4i with $${\text{i}} = \sqrt { - 1} ,$$ the value of $$\left| {\frac{{{\text{z}} + 3}}{{{\text{z}} - 1}}} \right|$$ is
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Consider likely applicability of Cauchy's Integral Theorem to evaluate the following integral counter clockwise around the unit circle c.<br>$$I = \oint\limits_{\text{c}} {\sec {\text{z}}} {\text{dz,}}$$ z being a complex variable. The value of $$I$$ will be
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Potential function $$\phi $$ is given as $$\phi $$ = x<sup>2</sup> - y<sup>2</sup>. What will be the stream function $$\psi $$ with the condition $$\psi $$ = 0 at x = y = 0?
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If f(x + iy) = x<sup>3</sup> - 3xy<sup>2</sup> + i$$\phi $$(x, y) where $${\text{i}} = \sqrt { - 1} $$ and f(x + iy) is an analytic function then $$\phi $$(x, y) is
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All the values of the multi-valued complex function 1<sup>i</sup>, where $${\text{i}} = \sqrt { - 1} ,$$ are