Master Complex Functions: A Comprehensive Guide for University Students

Question 1

Which of the following is **not** a property of complex functions?

Accepted Answer

They are always entire functions.

Answer

They have a real and imaginary part.

Answer

They can be continuous at a point.

Answer

They can be differentiated at a point.

Question 2

Let f(z) = z^2 + 2z. Find f'(z).

Accepted Answer

2z + 2

Answer

2z - 2

Answer

z^2 + 2

Answer

z^2 - 2

Question 3

Determine whether the function f(z) = e^z is continuous at z = 0.

Accepted Answer

Continuous

Answer

Discontinuous

Answer

Not differentiable

Answer

Not integrable

Question 4

Find the integral of f(z) = 1/(z-1) around the circle |z-1| = 2.

Accepted Answer

2πi

Answer

0

Answer

-2πi

Answer

πi

Question 5

Which of the following is a branch of the logarithm function?

Accepted Answer

Log(z) = ln(|z|) + i arg(z)

Answer

Log(z) = ln(|z|) - i arg(z)

Answer

Log(z) = |z| + i arg(z)

Answer

Log(z) = |z| - i arg(z)

Question 6

Find the conformal mapping that transforms the upper half-plane onto the unit disk.

Accepted Answer

w = z + i

Answer

w = z - i

Answer

w = iz

Answer

w = 1/z

Question 7

Which of the following properties is NOT a fundamental property of complex functions?

Accepted Answer

Reversibility

Answer

Continuity

Answer

Differentiability

Answer

Integrability

Question 8

The domain of the complex function f(z) = z^2 + 1 is:

Accepted Answer

The set of all complex numbers

Answer

The set of all real numbers

Answer

The set of all integers

Answer

The set of all negative complex numbers

Question 9

Which of the following conditions is sufficient for a complex function to be continuous at a point z0?

Accepted Answer

lim z->z0 f(z) = f(z0)

Answer

f(z0) exists

Answer

f(z) is differentiable at z0

Answer

f(z) is integrable on the interval containing z0

Question 10

The integral of the complex function f(z) = e^z around the unit circle centered at the origin is:

Accepted Answer

2πi

Answer

0

Answer

-2πi

Answer

π

Question 11

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

Accepted Answer

1

Answer

0

Answer

a

Answer

∞

Question 12

Which of the following is the definition of a complex function?

Accepted Answer

A function that maps complex numbers to complex numbers.

Answer

A function that maps real numbers to real numbers.

Answer

A function that maps complex numbers to real numbers.

Answer

A function that maps real numbers to complex numbers.

Question 13

What is the domain of the complex function f(z) = z^2 + 1?

Accepted Answer

The set of all complex numbers.

Answer

The set of all real numbers.

Answer

The set of all positive real numbers.

Answer

The set of all zero divisors.

Question 14

Which of the following is a necessary condition for a complex function to be continuous at a point z0?

Accepted Answer

The real and imaginary parts of the function must be continuous at z0.

Answer

The function must be differentiable at z0.

Answer

The function must be continuous at all points in an open disc centered at z0.

Answer

The function must be bounded in an open disc centered at z0.

Question 15

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

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 16

How does the residue theorem aid in the calculation of complex integrals?

Accepted Answer

Relates the integral's value to the residues of the integrand

Answer

Deforms the integral's contour

Answer

Provides an alternative method to compute antiderivatives

Answer

Determines the count of a function's zeros within a specified region

Question 17

Determine the domain of the complex function f(z) = log(z-2).

Accepted Answer

C \ {2}

Answer

R, z > 2

Answer

R, z < 2

Answer

C

Question 18

Find the range of the complex function f(z) = e^z.

Accepted Answer

C

Answer

R, z > 0

Answer

C, Re(z) = 0

Answer

C, Im(z) = 0

Question 19

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

Accepted Answer

2z + 2

Answer

2

Answer

z^2 + 2

Question 20

Identify the complex function that is not differentiable at z = 0.

Accepted Answer

f(z) = |z|

Answer

f(z) = sin(z)

Answer

f(z) = e^z

Answer

f(z) = z^2

Question 21

Determine the complex function that is continuous at z = 0.

Accepted Answer

f(z) = (z^2 - 1)/(z - 1)

Answer

f(z) = 1/z

Answer

f(z) = |z|

Answer

f(z) = log(z)

Question 22

Find the integral of the complex function f(z) = z^2.

Accepted Answer

z^3/3 + C

Answer

2z + C

Answer

z^3 + C

Question 23

Which complex function has a branch point at z = 0?

Accepted Answer

f(z) = sqrt{z}

Answer

f(z) = z^2

Answer

f(z) = e^z

Answer

f(z) = sin(z)

Question 24

Calculate the residue of the complex function f(z) = (z^2 - 1)/(z - 1) at z = 1.

Accepted Answer

2

Answer

0

Answer

-1

Answer

1

Question 25

Which complex function is analytic in the entire complex plane?

Accepted Answer

f(z) = e^z

Answer

f(z) = 1/z

Answer

f(z) = |z|

Answer

f(z) = sqrt{z}

Question 26

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

Accepted Answer

2

Answer

1

Answer

3

Answer

0

Question 27

Which of the following is the most precise definition of a complex function?

Accepted Answer

A mathematical function that takes a complex number as input and produces a complex number as output.

Answer

A function that operates on vectors and matrices.

Answer

A transformation that maps real numbers to real numbers.

Question 28

What is the domain of the complex function f(z) = 1 / z?

Accepted Answer

The set of all complex numbers except for zero.

Answer

The set of all integers.

Answer

The set of all imaginary numbers.

Answer

The set of all real numbers.

Question 29

What is the derivative of the complex function f(z) = e^z?

Accepted Answer

f'(z) = e^z

Answer

f'(z) = 1 / e^z

Answer

f'(z) = 0

Answer

f'(z) = ze^z

Question 30

Which of the following functions is an example of a holomorphic function?

Accepted Answer

f(z) = e^z

Answer

f(z) = |z|

Answer

f(z) = Arg(z)

Answer

f(z) = 1 / z

Question 31

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

Accepted Answer

3

Answer

1

Answer

4

Answer

2

Question 32

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

Accepted Answer

1

Answer

-1

Answer

0

Answer

2

Question 33

Determine the domain of definition for the complex function: f(z) = log(z - 2).

Accepted Answer

{z | z ∈ ℂ, z ≠ 2}

Answer

{z | z ∈ ℂ, z < 2}

Answer

{z | z ∈ ℂ}

Answer

{z | z ∈ ℝ, z > 2}

Question 34

Calculate the integral of the function f(z) = 1/z around the unit circle centered at the origin.

Accepted Answer

2πi

Answer

0

Answer

-2πi

Answer

π

Question 35

What is the domain of the complex function f(z) = (z+1)/(z-i)?

Accepted Answer

The set of all complex numbers except z = i

Answer

The set of all non-zero complex numbers

Answer

The set of all positive real numbers

Answer

The set of all real numbers

Question 36

Which of the following conformal mappings maps the unit disk to the upper half-plane?

Accepted Answer

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

Answer

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

Answer

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

Question 37

For a complex function to be differentiable at a point, which of the following is a necessary condition?

Accepted Answer

The function must be continuous at that point.

Answer

The function must have a derivative at that point (this is the definition of differentiability).

Answer

The function must be real-valued at that point.

Answer

The function must be analytic at that point.

Question 38

What is the domain of the function  f(z) = log(z - 2)?

Accepted Answer

z ∈ ℂ, z ≠ 2

Answer

z ∈ ℂ, z > 0

Answer

z ∈ ℝ, z < 2

Answer

z ∈ ℝ, z > 2

Question 39

What is the range of the function f(z) = e^z?

Accepted Answer

The entire complex plane.

Answer

The positive real axis.

Answer

A half-plane.

Answer

The negative real axis.

Question 40

Which of the following complex functions is NOT analytic?

Accepted Answer

f(z) = |z|

Answer

f(z) = sin(z)

Answer

f(z) = e^z

Answer

f(z) = z^2

Question 41

What is the derivative of the function f(z) = 1/z?

Accepted Answer

-1/z^2

Answer

-z

Answer

1/z^2

Answer

z

Question 42

If a complex function f(z) has a removable singularity at z0, what can be said about f(z0)?

Accepted Answer

It can be defined by a finite value.

Answer

It must be zero.

Answer

It must be infinite.

Answer

It is undefined.

Question 43

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

Accepted Answer

3

Answer

2

Answer

1

Answer

The function does not have a pole.

Question 44

What is a consequence of the Cauchy-Riemann equations?

Accepted Answer

A complex function is analytic if and only if it satisfies the Cauchy-Riemann equations.

Answer

Analytic functions can only have isolated singularities.

Answer

Complex functions are always differentiable.