Complex Variable MCQs Question and Answer in Engineering Maths

31

Let f₁(z) = z² and f₂(z) = $$\overline {\text{z}} $$ be two complex variable function. Here $$\overline {\text{z}} $$ is the complex conjugate of z. Choose the correct answer.

A) Both f₁(z) and f₂(z) are analytic B) Only f₁(z) is analytic C) Only f₂(z) is analytic D) Both f₁(z) and f₂(z) are not analytic

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Answer & Solution

Answer: Option B

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32

If $${\text{W}} = \phi + {\text{i}}\psi $$ represents the complex potential for an electric field. Given $$\psi = {{\text{x}}^2} - {{\text{y}}^2} + \frac{{\text{x}}}{{{{\text{x}}^2} + {{\text{y}}^2}}},$$ then the function $$\phi $$ is

A) $$ - 2{\text{xy}} + \frac{{\text{y}}}{{{{\text{x}}^2} + {{\text{y}}^2}}} + {\text{C}}$$ B) $$2{\text{xy}} + \frac{{\text{y}}}{{{{\text{x}}^2} + {{\text{y}}^2}}} + {\text{C}}$$ C) $$ - 2{\text{xy}} + \frac{{\text{x}}}{{{{\text{x}}^2} + {{\text{y}}^2}}} + {\text{C}}$$ D) $$2{\text{xy}} + \frac{{\text{x}}}{{{{\text{x}}^2} + {{\text{y}}^2}}} + {\text{C}}$$

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Answer: Option A

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33

An integral $$I$$ over a counter-clockwise circle C is given by $$I = \oint_{\text{C}} {\frac{{{{\text{z}}^2} - 1}}{{{{\text{z}}^2} + 1}}{{\text{e}}^{\text{z}}}{\text{dz}}{\text{.}}} $$
If C is defined as |z| = 3, then the value of $$I$$ is

A) -πi sin(1) B) -2πi sin(1) C) -3πi sin(1) D) -4πi sin(1)

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Answer & Solution

Answer: Option D

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34

The function $${\text{f}}\left( {\text{z}} \right) = \frac{{{{\text{z}}^2} + 1}}{{{{\text{z}}^2} + 4}}$$ is singular at

A) z = ±2 B) z = ±1 C) z = ±i D) z = ±2i

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Answer & Solution

Answer: Option D

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35

If f(z) has a pole of order n at z = a, then Residue of function f(z) at a is

A) $${\text{Res f}}\left( {\text{a}} \right) = \frac{1}{{\left( {\text{n}} \right)!}}{\left\{ {\frac{{{{\text{d}}^{{\text{n}} - 1}}}}{{{\text{d}}{{\text{z}}^{{\text{n}} - 1}}}}\left( {{{\left( {{\text{z}} - {\text{a}}} \right)}^{{\text{n}} - 1}}{\text{f}}\left( {\text{z}} \right)} \right)} \right\}_{{\text{z}} = {\text{a}}}}$$ B) $${\text{Res f}}\left( {\text{a}} \right) = \frac{1}{{\left( {{\text{n}} - 1} \right)!}}{\left\{ {\frac{{{{\text{d}}^{{\text{n}} - 1}}}}{{{\text{d}}{{\text{z}}^{{\text{n}} - 1}}}}\left( {{{\left( {{\text{z}} - {\text{a}}} \right)}^{{\text{n}} - 1}}{\text{f}}\left( {\text{z}} \right)} \right)} \right\}_{{\text{z}} = {\text{a}}}}$$ C) $${\text{Res f}}\left( {\text{a}} \right) = \frac{1}{{\left( {\text{n}} \right)!}}{\left\{ {\frac{{{{\text{d}}^{{\text{n}} - 1}}}}{{{\text{d}}{{\text{z}}^{{\text{n}} - 1}}}}\left( {{{\left( {{\text{z}} - {\text{a}}} \right)}^{\text{n}}}{\text{f}}\left( {\text{z}} \right)} \right)} \right\}_{{\text{z}} = {\text{a}}}}$$ D) $${\text{Res f}}\left( {\text{a}} \right) = \frac{1}{{\left( {{\text{n}} - 1} \right)!}}{\left\{ {\frac{{{{\text{d}}^{{\text{n}} - 1}}}}{{{\text{d}}{{\text{z}}^{{\text{n}} - 1}}}}\left( {{{\left( {{\text{z}} - {\text{a}}} \right)}^{\text{n}}}{\text{f}}\left( {\text{z}} \right)} \right)} \right\}_{{\text{z}} = {\text{a}}}}$$

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Answer & Solution

Answer: Option D

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36

Which one of the following functions is analytic over the entire complex plane?

A) cos(z) B) $${{\text{e}}^{\frac{1}{{\text{z}}}}}$$ C) $$l$$n(z) D) $$\frac{1}{{1 - {\text{z}}}}$$

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Answer & Solution

Answer: Option A

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37

Let z = x + iy be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE?

A) The residue of $$\frac{{\text{z}}}{{{{\text{z}}^2} - 1}}$$ at z = 1 is $$\frac{1}{2}$$ B) $$\oint_{\text{c}} {{{\text{z}}^2}{\text{dz}} = 0} $$ C) $$\frac{1}{{2\pi {\text{i}}}}\oint_{\text{c}} {\frac{1}{{\text{z}}}\,{\text{dz}} = 1} $$ D) $$\overline {\text{z}} $$ (complex conjugate of z) is analytical function

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Answer & Solution

Answer: Option D

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38

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) = 2xy, then v(x, y) must be

A) x² + y² + constant B) x² - y² + constant C) -x² + y² + constant D) -x² - y² + constant

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Answer & Solution

Answer: Option C

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39

Polar form of the Cauchy-Reimann equations is

A) $$\frac{{\partial {\text{u}}}}{{\partial {\text{r}}}} = {\text{r}}\frac{{\partial {\text{v}}}}{{\partial \theta }}{\text{ and }}\frac{{\partial {\text{v}}}}{{\partial {\text{r}}}} = - {\text{r}}\frac{{\partial {\text{u}}}}{{\partial \theta }}$$ B) $$\frac{{\partial {\text{u}}}}{{\partial {\text{r}}}} = \frac{1}{{\text{r}}}\frac{{\partial {\text{v}}}}{{\partial \theta }}{\text{ and }}\frac{{\partial {\text{v}}}}{{\partial {\text{r}}}} = - \frac{1}{{\text{r}}}\frac{{\partial {\text{u}}}}{{\partial \theta }}$$ C) $$\frac{{\partial {\text{u}}}}{{\partial {\text{r}}}} = \frac{1}{{\text{r}}}\frac{{\partial {\text{v}}}}{{\partial \theta }}{\text{ and }}\frac{{\partial {\text{v}}}}{{\partial {\text{r}}}} = - {\text{r}}\frac{{\partial {\text{u}}}}{{\partial \theta }}$$ D) $$\frac{{\partial {\text{u}}}}{{\partial {\text{r}}}} = {\text{r}}\frac{{\partial {\text{v}}}}{{\partial \theta }}{\text{ and }}\frac{{\partial {\text{v}}}}{{\partial {\text{r}}}} = - \frac{1}{{\text{r}}}\frac{{\partial {\text{u}}}}{{\partial \theta }}$$

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Answer & Solution

Answer: Option B

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40

Let S be the set of points in the complex plane corresponding to the unit circle. (That is, S = {z : |z| = 1}. Consider the function f(z) = zz^∗ where z^∗ denotes the complex conjugate of z. The f(z) maps S to which one of the following in the complex plane

A) unit circle B) horizontal axis line segment from origin to (1, 0) C) the point (1, 0) D) the entire horizontal axis

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Answer & Solution

Answer: Option C

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