Analytic Functions: Master Complex Functions for Electrical Engineering

Question 1

If f(z) = u(x, y) + iv(x, y) is analytic, what is the relationship between u and v?

Accepted Answer

u and v satisfy the Laplace equation.

Answer

u and v satisfy the heat equation.

Answer

u and v satisfy the wave equation.

Answer

u and v satisfy the Poisson equation.

Question 2

If f(z) is entire and bounded, what can be concluded about f(z)?

Accepted Answer

f(z) is constant.

Answer

f(z) has a finite number of zeros.

Answer

f(z) has an infinite number of zeros.

Answer

f(z) is not differentiable at any point.

Question 3

What is the radius of convergence of the power series ∑[n=0,∞] (z-1)^n / n?

Accepted Answer

1

Answer

0

Answer

2

Answer

∞

Question 4

What is the complex potential function for a uniform flow of velocity V in the x-direction?

Accepted Answer

Vz

Answer

V

Answer

V/z

Answer

Vz^2

Question 5

Which of the following functions is entire?

Accepted Answer

f(z) = e^z

Answer

f(z) = z^2

Answer

f(z) = 1/z

Answer

f(z) = sin(z)

Question 6

Determine the residue of the function f(z) = 1/(z-a) at z = a.

Accepted Answer

1

Answer

0

Answer

a

Answer

infinity

Question 7

Identify the contour that should be used to evaluate the integral of f(z) = 1/z around the unit circle.

Accepted Answer

The unit circle itself

Answer

A circle of radius 2 centered at the origin

Answer

A square with vertices at ±1 ±i

Answer

A semicircle in the upper half-plane

Question 8

In complex analysis, how do the Cauchy-Riemann equations characterize analytic functions?

Accepted Answer

Necessary and sufficient conditions

Answer

Necessary conditions only

Answer

Sufficient but not necessary conditions

Answer

Not related

Question 9

Which of the following is a consequence of the maximum modulus principle?

Accepted Answer

Analytic functions cannot have isolated singularities inside their domain of convergence

Answer

Analytic functions are continuous on their domain of convergence

Answer

Analytic functions are differentiable on their domain of convergence

Question 10

Which of the following functions is NOT analytic?

Accepted Answer

f(z) = |z|^2

Answer

f(z) = z^2

Answer

f(z) = e^z

Answer

f(z) = sin(z)

Question 11

What is the residue of the function f(z) = (z-1)/(z^2+1) at z = i?

Accepted Answer

1/2

Answer

-1/2

Answer

1

Answer

0

Question 12

What is the order of the pole of the function f(z) = 1/(z-a)^n at z = a?

Accepted Answer

n

Answer

0

Answer

n-1

Answer

n+1

Question 13

Which function is a branch of an analytic function with a branch cut along the negative real axis?

Accepted Answer

log(z)

Answer

sin(z)

Answer

exp(z)

Question 14

Evaluate the integral ∮_C dz/(z-a).

Accepted Answer

2πi

Answer

∮_C dz/(z^2-a^2)

Answer

∮_C dz/(z^3-a^3)

Answer

∮_C dz/z

Question 15

What is the order of the pole of f(z) = 1/(z-1)^3 at z = 1?

Accepted Answer

3

Answer

1

Answer

4

Answer

2

Question 16

Which of the following sets of equations represent the Cauchy-Riemann equations?

Accepted Answer

∂u/∂x = ∂v/∂y, ∂u/∂y = -∂v/∂x

Answer

∂u/∂y = ∂v/∂y, ∂u/∂x = -∂v/∂x

Answer

∂u/∂x = -∂v/∂y, ∂u/∂y = ∂v/∂x

Answer

∂u/∂x = ∂v/∂x, ∂u/∂y = ∂v/∂y

Question 17

A function f(z) = u(x, y) + iv(x, y) is analytic if and only if:

Accepted Answer

u and v satisfy the Cauchy-Riemann equations and f'(z) exists at every point in the domain.

Answer

u and v satisfy the Cauchy-Riemann equations but f'(z) does not exist at some points.

Answer

u and v do not satisfy the Cauchy-Riemann equations but f'(z) exists at every point in the domain.

Question 18

Evaluate the integral ∮(z^2 + 2z + 1) dz around the unit circle centered at the origin.

Accepted Answer

0

Answer

8πi

Answer

4πi

Answer

2πi

Question 19

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

Accepted Answer

e

Answer

1

Answer

-e

Answer

0

Question 20

Which of the following is a consequence of the Cauchy Integral Formula?

Accepted Answer

Analytic functions have derivatives of all orders in their domain.

Answer

Analytic functions have a finite number of zeros in their domain.

Answer

Analytic functions are continuous but not necessarily differentiable.

Question 21

The Laurent series expansion of f(z) = 1/(z^2 + 1) in the region |z| > 1 is:

Accepted Answer

∑ (-1)^n z^(-2n-2)

Answer

∑ n z^(-n-2)

Answer

∑ z^(-2n-2)

Answer

∑ (-1)^n n z^(-n-2)

Question 22

The function f(z) = log(z^2) has a branch cut along the:

Accepted Answer

Negative real axis

Answer

Imaginary axis

Answer

Positive real axis

Answer

Unit circle

Question 23

The order of the pole of the function f(z) = sin(z)/(z-π/2) at z = π/2 is:

Accepted Answer

1

Answer

2

Answer

0

Question 24

The integral ∫[0,∞) e^(-x^2) dx is:

Accepted Answer

Improper integral of Type I

Answer

Cauchy principal value integral

Answer

Improper integral of Type II

Answer

Not an improper integral

Question 25

The residue theorem can be used to evaluate integrals of the form:

Accepted Answer

∮(f(z)/g(z)) dz where f(z) and g(z) are analytic and g(z) has a zero inside the contour.

Answer

∫(f(z)) dz where f(z) is analytic and has a singularity inside the contour.

Answer

∫(f(x)/g(x)) dx where f(x) and g(x) are real-valued and g(x) has a zero in the interval.

Question 26

Which of the following Cauchy-Riemann equations is responsible for the condition that the real and imaginary parts of an analytic function have equal partial derivatives?

Accepted Answer

∂u/∂x = ∂v/∂y

Answer

∂u/∂y = -∂v/∂x

Answer

∂u/∂x = -∂v/∂y

Answer

∂u/∂y = -∂v/∂x

Question 27

Which of the following is NOT a consequence of the Cauchy-Riemann equations?

Accepted Answer

An analytic function must have a continuous derivative.

Answer

An analytic function is harmonic.

Answer

An analytic function is holomorphic.

Question 28

Consider the function f(z) = z^2. Is f(z) analytic?

Accepted Answer

Yes

Answer

Analytic only for imaginary z

Answer

Analytic only for real z

Answer

No

Question 29

Which of the following is the integral of dz/(z^2 + 1)?

Accepted Answer

arctan(z) + C

Answer

1/(z^2 + 1) + C

Answer

-arctan(z) + C

Answer

ln(z^2 + 1) + C

Question 30

What is the residue of the function f(z) = 1/(z-2) at z=2?

Accepted Answer

1

Answer

0

Answer

-1

Question 31

Which of the following is the Laurent expansion of f(z) = 1/(z-1) around z=0?

Accepted Answer

1 + z + z^2 + ...

Answer

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

Answer

1/z + 1/(z^2) + 1/(z^3) + ...

Answer

1 - z + z^2 - ...

Question 32

What is the order of the pole of the function f(z) = 1/(z^2-4) at z=2?

Accepted Answer

1

Answer

3

Answer

2

Answer

0

Question 33

Which of the following integrals represents the contour integral of f(z) = z^2 around the unit circle centered at the origin?

Accepted Answer

∫[0,2π] e^(2iθ) dθ

Answer

∫[0,2π] z^2 e^(iθ) dθ

Answer

∫[0,2π] e^(2θ) dθ

Answer

∫[0,2π] z^2 dθ

Question 34

What is the value of the improper integral ∫[0,∞) e^(-z^2) dz?

Accepted Answer

√(π/2)

Answer

1

Answer

π

Question 35

Which of the following functions has a branch cut along the negative real axis?

Accepted Answer

f(z) = √(z)

Answer

f(z) = e^z

Answer

f(z) = sin(z)

Answer

f(z) = z^2