Complex Analysis Quizzes: Master Complex Variables and Functions

Question 1

Which of the following is NOT a type of singularity for a complex function?

Accepted Answer

Branch cut

Answer

Pole

Answer

Essential singularity

Answer

Removable singularity

Question 2

Which condition guarantees that a complex function f(z) is analytic within a region R?

Accepted Answer

f(z) satisfies the Cauchy-Riemann equations in R.

Answer

f(z) is continuous and has a continuous first derivative in R.

Answer

f(z) is differentiable in the complex plane.

Question 3

Consider an analytic function f(z) in a domain D. Let γ be a closed contour in D. What is the value of the integral of f(z) along γ?

Accepted Answer

2πi times the sum of the residues of f(z) at its singularities inside γ

Answer

Zero

Answer

The length of the contour γ

Answer

The area enclosed by the contour γ

Question 4

In complex analysis, an analytic function is defined as:

Accepted Answer

A function that satisfies the Cauchy-Riemann equations at each point in its domain

Answer

A function with a continuous derivative at every point in its domain

Answer

A function that is continuous throughout its domain and admits a Taylor series expansion at every point

Answer

A function that accepts complex numbers, outputs complex numbers, and is invertible

Question 5

Which of the following is NOT a necessary condition for a function f(z) to be analytic at a point z0?

Accepted Answer

f''(z0) is defined.

Answer

f(z0) is defined.

Answer

f'(z0) is defined.

Question 6

Define a complex function and specify its domain and codomain.

Accepted Answer

A complex function has a domain and codomain that are both subsets of the complex numbers.

Answer

A complex function has a domain and codomain that are subsets of the real numbers.

Answer

A complex function has a domain that is a subset of the complex numbers and a codomain that is a subset of the real numbers.

Question 7

What is the Riemann sphere, and how does it extend the complex plane?

Accepted Answer

A two-dimensional sphere that is homeomorphic to the complex plane, including a point at infinity.

Answer

A three-dimensional sphere that excludes a point at infinity.

Answer

A two-dimensional plane that includes two points at infinity.

Answer

A three-dimensional plane that excludes two points at infinity.

Question 8

Which function is holomorphic (analytic) for all complex numbers?

Accepted Answer

f(z) = e^z

Answer

f(z) = |z|

Answer

f(z) = z^2 + 1

Answer

f(z) = 1/z

Question 9

Determine the modulus of the complex number (2 + 3i).

Accepted Answer

5

Answer

1

Answer

2

Answer

13

Question 10

Which of the following is the correct definition of a conformal mapping?

Accepted Answer

A mapping that preserves angles

Answer

A mapping that preserves lengths

Answer

A mapping that preserves areas

Answer

A mapping that transforms circles into straight lines

Question 11

What is the residue of a function at a pole?

Accepted Answer

The coefficient of the term in the Laurent expansion of the function at the pole

Answer

The coefficient of the term in the Taylor expansion of the function at the pole

Answer

The constant term in the Laurent expansion of the function at the pole

Answer

The leading coefficient in the Taylor expansion of the function at the pole

Question 12

In complex analysis, which equations are known as the Cauchy-Riemann equations?

Accepted Answer

u_x + v_y = 0, u_y - v_x = 0

Answer

u_x - v_y = 0, u_y + v_x = 0

Answer

u_x*v_y + v_x*u_y = 0

Answer

u_y*v_x + v_y*u_x = 0

Question 13

Which formula famously connects the mathematical constants e, i (the imaginary unit), and π (pi)?

Accepted Answer

Euler's identity

Answer

Cauchy's integral formula

Answer

Riemann's zeta function

Answer

Poincaré's conjecture

Question 14

Which condition is necessary for a function to be meromorphic?

Accepted Answer

It has only poles.

Answer

It has only removable singularities.

Answer

It has no singularities.

Answer

It has only essential singularities.

Question 15

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

Accepted Answer

1/2i

Answer

-1/4

Answer

1/4

Answer

-1/2i

Question 16

What is the order of the zero of the function f(z) = (z-2)^3(z+1) at z = 2?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 17

Which of the following is equivalent to the Cauchy-Riemann equations for a complex function f(z) = u(x, y) + iv(x, y)?

Accepted Answer

∂u/∂x = ∂v/∂y, ∂u/∂y = -∂v/∂x

Answer

∂u/∂x - ∂v/∂y = 0

Answer

∂u/∂x + ∂v/∂y = 0

Answer

∂u/∂y + ∂v/∂x = 0

Question 18

Which of the following is the residue of f(z) = e^z/(z-1) at z = 1?

Accepted Answer

e

Answer

0

Answer

e^2

Answer

1

Question 19

Which of the following is a characteristic property of complex numbers?

Accepted Answer

They can be represented geometrically as points on the complex plane.

Answer

They cannot be multiplied or divided.

Answer

They are always real numbers.

Answer

They have no imaginary parts.

Question 20

Find the derivative of the complex function f(z) = z^3 + 2z^2 - 5z + 1 with respect to z.

Accepted Answer

f'(z) = 3z^2 + 4z - 5

Answer

f'(z) = z^3 - 5z + 1

Answer

f'(z) = z^3 + 4z^2 - 5

Answer

f'(z) = 3z^2 - 5

Question 21

Which of the following transformations is NOT a conformal mapping?

Accepted Answer

w = |z|^2

Answer

w = z^2 + 1

Answer

w = e^z

Answer

w = sin(z)

Question 22

Which of the following equations is **NOT** true for the complex conjugate of a complex number *z*, denoted as *z̄*?

Accepted Answer

z + z̄ = 2Im(z)

Answer

(z1 + z2)̄ = z1̄ + z2̄

Answer

(z1 * z2)̄ = z1̄ * z2̄

Answer

z * z̄ = |z|^2

Question 23

What is the image of the unit circle *|z| = 1* under the mapping *w = 1/z*?

Accepted Answer

The unit circle *|w| = 1*

Answer

The imaginary axis

Answer

The line *Re(w) = 1*

Answer

The real axis

Question 24

Which theorem states that an analytic function within a simply connected domain, continuous on its boundary, attains its maximum modulus on the boundary of the domain?

Accepted Answer

Maximum Modulus Principle

Answer

Cauchy's Integral Theorem

Answer

Liouville's Theorem

Answer

Cauchy's Integral Formula

Question 25

The function *f(z) = 1/z* has a singularity at *z = 0*. What type of singularity is this?

Accepted Answer

Pole

Answer

Essential singularity

Answer

Removable singularity

Question 26

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

Accepted Answer

(z - i)/(z + i)

Answer

z^2

Answer

e^z

Answer

1/z

Question 27

The Laplace equation in two dimensions can be solved using which technique from complex analysis?

Accepted Answer

Conformal mapping and harmonic functions

Answer

Cauchy's integral formula

Answer

Laurent series expansion

Answer

Residue theorem

Question 28

What is the real part of the complex number *z = 3 - 4i*?

Accepted Answer

3

Answer

-4

Answer

-4i

Answer

3i

Question 29

Given the complex function *f(z) = z^2 + 2z*, find its derivative *f'(z)*.

Accepted Answer

2z + 2

Answer

z + 2

Answer

z^2 + 2

Answer

2z

Question 30

Which of the following is a harmonic conjugate of u(x, y) = x^2 - y^2?

Accepted Answer

2xy

Answer

-2xy

Answer

y^2 - x^2

Answer

x^2 + y^2

Question 31

What is the image of the unit circle |z| = 1 under the mapping w = 1/z?

Accepted Answer

The unit circle |w| = 1

Answer

The imaginary axis

Answer

The line Re(w) = 1

Question 32

Which of the following functions has a pole of order 2 at z = 0?

Accepted Answer

f(z) = 1/z^2

Answer

f(z) = z^2

Answer

f(z) = 1/z

Question 33

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

Accepted Answer

1/2

Answer

-i

Answer

i

Answer

1

Question 34

What is the value of the integral of f(z) = 1/z around the unit circle |z| = 1, traversed counterclockwise?

Accepted Answer

2πi

Answer

-2πi

Answer

πi

Answer

0

Question 35

The function f(z) = e^z is:

Accepted Answer

Entire

Answer

Analytic only in the upper half-plane

Answer

Analytic only in the right half-plane

Answer

Nowhere analytic

Question 36

The transformation w = z + 2 - i represents:

Accepted Answer

A translation

Answer

An inversion

Answer

A rotation

Answer

A scaling