Complex Analysis for Engineering: Master the Fundamentals

Question 1

In fluid mechanics, a complex potential f(z) = z^2 + 2iz describes a potential flow. Calculate the velocity potential and stream function from this complex potential.

Accepted Answer

Velocity potential: 2x + y; Stream function: x^2 - y^2 + 2x

Answer

Velocity potential: x^2 - y^2; Stream function: 2x + y

Answer

Velocity potential: 2x; Stream function: x^2

Question 2

Question: How is conformal mapping applied in electromagnetism? Explain the concept of complex permittivity and its significance in analyzing electromagnetic fields.

Accepted Answer

Conformal mapping helps determine potential distribution in regions with complex permittivity, aiding in understanding electromagnetic field behavior.

Answer

Conformal mapping is not applicable in electromagnetism because it involves real-valued functions.

Answer

Complex permittivity is irrelevant to conformal mapping.

Answer

Conformal mapping is solely used in fluid mechanics to analyze flow patterns.

Question 3

If a complex function f(z) is entire and bounded on the entire complex plane, what can you conclude about its nature?

Accepted Answer

It is a constant function.

Answer

It is continuous everywhere in the complex plane.

Answer

It is differentiable everywhere in the complex plane.

Answer

It need not be entire.

Question 4

Find the derivative of the complex function f(z) = e^(z^2) using the chain rule. Explain your steps and discuss the implications of the result in the context of complex analysis.

Accepted Answer

f'(z) = 2ze^(z^2)

Answer

f'(z) = ze^(z^2)

Answer

f'(z) = 2e^(z^2)

Answer

f'(z) = e^z

Question 5

Determine the differentiability of the function f(z) = z^2 + 1 at z = 0.

Accepted Answer

Differentiable at z = 0

Answer

Not differentiable at z = 0

Answer

Removable singularity at z = 0

Question 6

Identify the singularities of the function f(z) = (z - 1)/(z + 1) and classify the type of singularity at each point.

Accepted Answer

Removable singularity at z = 1, pole of order 1 at z = -1

Answer

Pole of order 1 at z = 1, removable singularity at z = -1

Answer

Essential singularity at z = 1, pole of order 2 at z = -1

Question 7

How can the residue theorem be utilized to evaluate integrals of real-valued functions over real intervals?

Accepted Answer

By expressing the real-valued function as the real or imaginary part of a complex function, the residue theorem can be applied to evaluate the integral.

Answer

The residue theorem can directly evaluate integrals of real-valued functions over real intervals without any modifications.

Answer

The residue theorem is not applicable to evaluating integrals of real-valued functions over real intervals.

Question 8

Question: How can conformal mapping be utilized to solve boundary value problems governed by Laplace's equation? Illustrate the process with an example.

Accepted Answer

Conformal mapping simplifies Laplace's equation by transforming a complex region into a simpler one, enabling the use of separation of variables.

Accepted Answer

The solution obtained in the transformed region can be mapped back to the original region to solve Laplace's equation there.

Answer

Conformal mapping cannot be used to solve Laplace's equation.

Answer

Conformal mapping is limited to solving Laplace's equation in two dimensions.

Question 9

Which property is NOT characteristic of analytic functions?

Accepted Answer

Constant modulus

Answer

Continuous and differentiable throughout their domain

Answer

Satisfy the Cauchy-Riemann equations

Answer

Representable by power series

Question 10

In complex analysis, which statement about Cauchy's Integral Formula is correct?

Accepted Answer

Cauchy's Integral Formula can be used to find the value of a complex function at any point within its region of analyticity.

Answer

The integral of a complex function around a closed contour is always zero.

Answer

The residue theorem can be used to evaluate integrals of complex functions along any contour.

Question 11

Which of the following is NOT a fundamental theorem in complex analysis?

Accepted Answer

Gauss' Theorem

Answer

Cauchy's Integral Theorem

Answer

Taylor's Theorem

Answer

Morera's Theorem

Question 12

The complex number z = 2 + 3i can be represented in polar form as:

Accepted Answer

z = √13(cos 33.7° + i sin 33.7°)

Answer

z = √13(cos 56.3° + i sin 56.3°)

Answer

z = √13(cos 45° + i sin 45°)

Answer

z = √13(cos 90° + i sin 90°)

Question 13

The contour integral of f(z) = 1/z around the unit circle centered at the origin is:

Accepted Answer

2πi

Answer

0

Answer

-2πi

Answer

π

Question 14

Which of the following transformations is NOT a conformal mapping?

Accepted Answer

w = |z| + i arg(z)

Answer

w = z^2

Answer

w = 1/z

Answer

w = exp(z)

Question 15

Which of the following functions is NOT entire?

Accepted Answer

f(z) = 1/(z-1)

Answer

f(z) = e^z

Answer

f(z) = sin(z)

Answer

f(z) = z^2

Question 16

Which of the following is the complex conjugate of (2 + 3i)?

Accepted Answer

2 - 3i

Answer

2 + 3i

Answer

-2 + 3i

Answer

-2 - 3i

Question 17

Find the derivative of the complex function f(z) = z^2 + e^z.

Accepted Answer

2z + e^z

Answer

2z - e^z

Answer

2z

Answer

e^z

Question 18

Which of the following is an application of complex analysis in fluid mechanics?

Accepted Answer

Potential flow around an airfoil

Answer

Heat transfer in a pipe

Answer

Stress analysis in a beam

Answer

Vibration analysis of a structure

Question 19

Find the Laurent expansion of the function f(z) = 1/(z - 1) around z = 0.

Accepted Answer

1 + z + z^2 + z^3 + ...

Answer

1 - z + z^2 - z^3 + ...

Answer

z + z^2 + z^3 + ...

Answer

1 + z - z^2 + z^3 + ...

Question 20

Which of the following is NOT a valid complex number?

Accepted Answer

6

Answer

2 + 3i

Answer

4 - 5j

Answer

7i

Question 21

Find the derivative of the complex function f(z) = z^3 + 2z + 1.

Accepted Answer

3z^2 + 2

Answer

z^3 + 2

Answer

3z + 2

Answer

z + 1

Question 22

Which of the following integrals can be evaluated using the Residue Theorem?

Accepted Answer

∫[C] 1/(z^2 + 1) dz, where C is the unit circle

Answer

∫[0,2π] e^iθ dθ

Answer

∫[0,∞] x^2 e^(-x) dx

Answer

∫[C] z^3 e^z dz, where C is the rectangle with vertices at ±1 ± i

Question 23

Which of the following functions is analytic?

Accepted Answer

f(z) = z^2 + 2z + 1

Answer

f(z) = |z|

Answer

f(z) = Re(z)

Answer

f(z) = Im(z)

Question 24

Find the residue of the function f(z) = z/(z^2 + 1) at z = i.

Accepted Answer

1/4

Answer

1/2

Answer

-1/2

Answer

-1/4

Question 25

Which of the following theorems is used to evaluate the integral ∫_γ f(z) dz, where γ is a closed contour?

Accepted Answer

Residue Theorem

Answer

Cauchy's Integral Formula

Answer

Fundamental Theorem of Calculus

Answer

Green's Theorem

Question 26

Compute the integral of $e^{iz}$ around the unit circle centered at $0$.

Accepted Answer

$2\pi i$

Answer

$0$

Answer

$2$

Answer

$-2$

Question 27

Find the modulus of the complex number z = 3 - 4i.

Accepted Answer

7

Answer

1

Answer

5

Answer

9

Question 28

The integral of the complex function f(z) = z around the unit circle is:

Accepted Answer

2πi

Answer

0

Answer

1

Answer

-2πi

Question 29

The residue of the function f(z) = (z+1)/(z^2+1) at z = i is:

Accepted Answer

1/2

Answer

0

Answer

1

Answer

2

Question 30

Which of the following is a branch cut for the function f(z) = sqrt(z)?

Accepted Answer

The negative real axis

Answer

The positive real axis

Answer

The imaginary axis

Answer

The unit circle

Question 31

The conformal mapping w = z^2 maps the unit disk |z| < 1 to:

Accepted Answer

A circle

Answer

A semicircle

Answer

A quarter circle

Answer

A half-plane

Question 32

The residue of the function f(z) = 1/(z-a) at z = a is:

Accepted Answer

1

Answer

0

Answer

a

Answer

1/a

Question 33

The integral of f(z) = exp(z) around the unit circle |z| = 1 is:

Accepted Answer

2πi

Answer

0

Answer

-2πi

Answer

2π

Question 34

Which of the following represents the principal branch of the complex logarithm function, log(z)?

Accepted Answer

{z : |arg(z)| < π}

Answer

{z : |arg(z)| < 2π}

Answer

{z : |arg(z)| ≤ π/2}

Answer

{z : |arg(z)| ≥ π}

Question 35

Which complex function represents a conformal mapping?

Accepted Answer

f(z) = e^z

Answer

f(z) = z²

Answer

f(z) = |z|

Answer

f(z) = z + 1

Question 36

Which of the following is NOT a fundamental concept in complex analysis?

Accepted Answer

Euler's number

Answer

Complex numbers

Answer

Conformal mapping

Answer

Complex functions

Question 37

The complex number represented by the point (2, 3) in the complex plane is:

Accepted Answer

(2 + 3i)

Answer

(-2 - 3i)

Answer

(-2 + 3i)

Answer

(3 + 2i)

Question 38

The integral of the function f(z) = 1/(z - a) around a closed contour that encloses the point a is:

Accepted Answer

2πi

Answer

-2πi

Answer

0

Answer

πi

Question 39

In heat transfer, complex analysis is used to model:

Accepted Answer

Temperature distribution

Answer

Stress-strain analysis

Answer

Structural vibrations

Answer

Flow around airfoils

Question 40

The residue of the function f(z) = (z - 1)/(z^2 - 4) at z = 2 is:

Accepted Answer

1/4

Answer

1

Answer

0

Answer

-1/4

Question 41

Which of the following functions is analytic in the entire complex plane?

Accepted Answer

e^z

Answer

log(z)

Answer

1/z

Answer

sin(z)

Question 42

Find the derivative of the complex function f(z) = z^2 + 2z - 1.

Accepted Answer

2z + 2

Answer

1

Answer

2z - 1

Answer

z^2 + 2

Question 43

Determine the order of the pole of the function f(z) = 1 / (z - 2)^3 at z = 2.

Accepted Answer

3

Answer

2

Answer

1

Answer

4

Question 44

Convert the complex number -3 + 4i to its polar form, expressed as (r∠θ).

Accepted Answer

5∠(π - arctan(4/3))

Answer

√7∠(π - arctan(4/3))

Answer

5∠arctan(4/3)

Answer

√7∠arctan(4/3)

Question 45

Determine the residue of the function ( f(z) = rac{1}{z^2 + 1} ) at ( z = i ).

Accepted Answer

1/(2i)

Answer

0

Answer

i

Answer

1

Question 46

A Laurent series expansion of a complex function ( f(z) ) is primarily used to:

Accepted Answer

Represent ( f(z) ) in an annular region of the complex plane.

Answer

Find the roots of ( f(z) ).

Answer

Determine the derivative of ( f(z) ).

Answer

Calculate the integral of ( f(z) ) along any path.

Question 47

The complex conjugate of the complex number ( z = a + bi ) is given by:

Accepted Answer

a - bi

Answer

a + bi

Answer

|z|

Answer

bi - a

Question 48

Calculate the modulus of the complex number ( z = 2 + 3i ).

Accepted Answer

√13

Answer

13

Answer

3

Question 49

Find the derivative of the complex function f(z) =  z^3 - 2z + i.

Accepted Answer

3z^2 - 2

Answer

z^3 - 2

Answer

z^2 - 2

Answer

3z - 2

Question 50

Which of the following represents the Cauchy-Riemann equations in terms of partial derivatives for an analytic function?

Accepted Answer

u_x = v_y and u_y = -v_x

Answer

v_x = u_y and v_y = -u_x

Answer

u_x = -v_y and u_y = v_x

Answer

u_x = v_x and u_y = v_y

Question 51

Under the conformal mapping w = z^2, the image of the unit circle centered at the origin is:

Accepted Answer

The entire upper half-plane

Answer

The left half-plane

Answer

The right half-plane

Answer

The entire lower half-plane

Question 52

Identify a valid engineering application of complex analysis:

Accepted Answer

Fluid mechanics

Answer

Solid mechanics

Answer

Control theory

Answer

Thermodynamics

Question 53

Find the residue of the function f(z) = (z-1)^2 / (z^2 + 1) at the pole z = i.

Accepted Answer

1

Answer

-1

Answer

-i

Answer

i

Question 54

Which of the following integrals can be evaluated using the residue theorem?

Accepted Answer

∮[z³ / (z-1)(z-2)] dz

Answer

∫[cos(z) / z^2] dz

Answer

∫[ln(z) / z] dz

Answer

∫[e^z / (z-1)] dz

Question 55

Find the Laurent series expansion of the function f(z) = 1 / (z-1)(z-2) around the point z = 1.

Accepted Answer

1 + z + z^2 + ...

Answer

1 - z + z^2 + ...

Answer

1 / (z-1) + 1 / (z-2) + ...

Answer

z + z^2 + z^3 + ...

Question 56

What is the value of the complex number (2 + 3i) * (1 - 2i)?

Accepted Answer

8 - i

Answer

2 - 6i

Answer

-4 + i

Answer

1 + 4i

Question 57

Which of the following is the polar form of the complex number -1 + i?

Accepted Answer

√2 * cis(135°)

Answer

√2 * cis(-45°)

Answer

√2 * cis(45°)

Answer

√2 * cis(-135°)

Question 58

Find the derivative of the complex function f(z) = z^2 + 2z - 3.

Accepted Answer

2z + 2

Answer

2z - 3

Answer

z^2 + 2

Answer

z^2 + 2z

Question 59

The Cauchy-Riemann equations are necessary conditions for a complex function to be:

Accepted Answer

Analytic

Answer

Differentiable

Answer

Integrable

Answer

Continuous

Question 60

The residue of the function f(z) = 1/(z^2 + 1) at z = i is:

Accepted Answer

-i/2

Answer

-1/2

Answer

1/2

Answer

i/2

Question 61

The integral of the function f(z) = 1/z around the unit circle, traversed once counterclockwise, is equal to:

Accepted Answer

2πi

Answer

πi

Answer

-2πi

Answer

0

Question 62

A complex function that maps a region of the complex plane onto another region, preserving angles and orientation, is called:

Accepted Answer

Conformal Mapping

Answer

Homomorphic Mapping

Answer

Bijective Mapping

Answer

Isomorphic Mapping