Complex Analysis Quizzes: Master the Fundamentals of Complex Functions

Question 1

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

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 2

State the residue theorem.

Accepted Answer

The integral of a function around a closed contour equals the sum of the residues of the function at its singularities within the contour.

Answer

The sum of the residues of a function at its singularities equals zero.

Answer

The integral of a function along a branch cut equals the sum of the residues of the function at its singularities on the branch cut.

Answer

The integral of a function around a pole equals 2πi times the residue of the function at the pole.

Question 3

Identify an example of an entire function.

Accepted Answer

e<sup>z</sup>

Answer

sin(z)

Answer

cos(z)

Answer

1/(z - 1)

Question 4

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

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 5

Describe the geometric interpretation of taking the inverse of a complex number.

Accepted Answer

Reflection about the real axis

Answer

Reflection about the imaginary axis

Answer

Rotation by 180 degrees around the origin

Answer

Translation by 1 unit in the negative direction

Question 6

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

Accepted Answer

f(z) = z^2 + 1

Answer

f(z) = |z|

Answer

f(z) = 1/z

Answer

f(z) = Re(z)

Question 7

What is the conformal mapping that transforms the upper half-plane to the unit disk?

Accepted Answer

w = 1/z

Answer

w = z

Answer

w = z^2

Answer

w = z + 1

Question 8

What is the geometric interpretation of the argument of a complex number?

Accepted Answer

The angle between the positive real axis and the line connecting the origin to the number

Answer

The magnitude of the number

Answer

The real part of the number

Answer

The imaginary part of the number

Question 9

Graphically, what does the complex number z = 3 + 4i represent?

Accepted Answer

A point at (3, 4) on the complex plane

Answer

A vector extending 3 units along the real axis and 4 units along the imaginary axis

Answer

A circle with radius 5 centered at the origin

Answer

A degree 2 polynomial with complex coefficients

Question 10

Which of the following about complex functions is accurate?

Accepted Answer

The Cauchy-Riemann equations are necessary and sufficient for their differentiability.

Answer

They can be differentiated with respect to a real variable.

Answer

Their derivatives are always complex.

Answer

They are all continuous on the complex plane.

Question 11

In complex analysis, which branch deals with functions on Riemann surfaces?

Accepted Answer

Geometric function theory

Answer

Algebraic geometry

Answer

Analytic number theory

Answer

Numerical analysis

Question 12

Which of the following is NOT a correct statement about the Cauchy-Riemann equations?

Accepted Answer

The imaginary part of a holomorphic function is harmonic, while the real part is not.

Answer

They imply that the function is differentiable at each point.

Answer

They are necessary and sufficient conditions for a function to be holomorphic.

Question 13

Which of the following integrals computes the residue of the function f(z) = (z-1)/(z^2+1) at z = i?

Accepted Answer

∮[C] f(z) dz / 2πi, where C is a circle centered at z = i with any positive radius r.

Answer

∫[C] f(z) dz / 2πi, where C is a circle centered at z = 0 with any positive radius r.

Answer

∮[C] f(z) dz, where C is a circle centered at z = i with any positive radius r.

Question 14

Which of the following series converges uniformly on the closed unit disk?

Accepted Answer

∑[n=1 to ∞] (z^n)/n^2

Answer

∑[n=1 to ∞] (1/n)z^n

Answer

∑[n=1 to ∞] z^n

Answer

∑[n=1 to ∞] (z^n)/n

Question 15

What is the principal branch of the complex logarithm function?

Accepted Answer

Log(z) = ln(|z|) + i Arg(z) for z not equal to 0, where -π < Arg(z) ≤ π

Answer

Log(z) = ln(|z|) + i Arg(z) for z not equal to 0, where 0 ≤ Arg(z) < 2π

Answer

Log(z) = ln(|z|) - i Arg(z) for z not equal to 0, where -π < Arg(z) ≤ π

Question 16

Which of the following is a necessary and sufficient condition for the existence of a meromorphic function?

Accepted Answer

The function has isolated singularities and is holomorphic on the complex plane除去these singularities.

Answer

The function is differentiable at all points except for isolated points.

Answer

The function has no zeros.

Answer

The function is continuous on the entire complex plane.

Question 17

Evaluate the contour integral ∮C (z^2 + 1)/(z - 2) dz, where C is the circle |z| = 3 traversed counterclockwise.

Accepted Answer

10πi

Answer

4πi

Answer

2πi

Answer

0

Question 18

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

Accepted Answer

e^z

Answer

1/z

Answer

√z

Answer

log(z)

Question 19

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

Accepted Answer

-i/2

Answer

1/2

Answer

-1/2

Answer

i/2

Question 20

Identify the branch points of the multi-valued function f(z) = √(z^2 - 1).

Accepted Answer

z = ±1

Answer

z = -i

Answer

z = i

Answer

z = 0

Question 21

The argument principle states that for a function f(z) and a closed contour C, the difference between the number of zeros and poles of f(z) inside C is equal to:

Accepted Answer

1/2πi times the change in the argument of f(z) as we traverse C.

Answer

The number of zeros of f(z) inside C minus the number of poles inside C.

Answer

The number of zeros of f(z) inside C.

Answer

The number of poles of f(z) inside C.

Question 22

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

Accepted Answer

$2z + 2$

Answer

$4z$

Answer

$2$

Answer

$z^2 + 2$

Question 23

Find the residue of the function $f(z) = rac{1}{z(z-1)}$ at $z=0$.

Accepted Answer

$1$

Answer

$0$

Answer

$-1$

Question 24

Which theorem guarantees the existence of a Taylor series expansion for a complex function?

Accepted Answer

Cauchy's Integral Formula

Answer

Cauchy's Residue Theorem

Answer

Liouville's Theorem

Answer

Morera's Theorem

Question 25

Which of the following is an application of complex analysis in physics?

Accepted Answer

Fluid dynamics

Answer

Number theory

Answer

Computer science