Master Complex Analysis: Online Quizzes and Practice Questions

Question 1

Which of the following accurately defines a complex function?

Accepted Answer

A function that maps complex numbers to complex numbers

Answer

A function whose derivative is complex-valued

Answer

A function that can only be represented graphically in the complex plane

Answer

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

Question 2

According to Cauchy's theorem, when can the integral of a complex function around a closed contour be zero?

Accepted Answer

When the function is analytic inside the contour

Answer

When the function is differentiable on the contour

Answer

When the contour is oriented clockwise

Answer

When the function has a pole inside the contour

Question 3

Which of the following represents the principal branch of the complex logarithm function?

Accepted Answer

Log(z)

Answer

Log(z) + 2πi

Answer

Log(z) - 2πi

Answer

Exp(z)

Question 4

The Cauchy integral formula can be used to evaluate complex integrals with integrands of the form:

Accepted Answer

f(z)dz / (z - a)

Answer

f(z^2)dz / z^2

Answer

f(sin(z))dz / cos(z)

Answer

f(exp(z))dz / z^2

Question 5

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

Accepted Answer

∑(n=1 to infinity) 1/z^n

Answer

∑(n=0 to infinity) z^n

Answer

∑(n=0 to infinity) n z^n

Answer

∑(n=1 to infinity) n/z^n

Question 6

Describe a characteristic of an entire function.

Accepted Answer

It is analytic in the entire complex plane

Answer

It has a finite number of zeros

Answer

It has a pole at every point in the complex plane

Answer

It has a singularity at every point in the complex plane

Question 7

How does the presence of a branch cut impact the continuity of the complex logarithm function, ln(z)? Describe the resulting discontinuity concisely.

Accepted Answer

The branch cut makes ln(z) continuous along the cut, except at endpoints where it may exhibit a jump discontinuity.

Answer

Branch cuts make ln(z) differentiable everywhere in the complex plane.

Answer

ln(z) becomes undefined at points on the branch cut.

Answer

Branch cuts have no effect on the continuity of ln(z).

Question 8

To evaluate the integral of f(z) = (z^2 + 1)/(z-1) around the unit circle centered at the origin using the residue theorem, what is the first step?

Accepted Answer

Decompose f(z) into partial fractions and find the residue at z = 1.

Answer

Apply the residue theorem directly to f(z) without decomposition.

Answer

Use the Cauchy integral formula to evaluate the integral.

Answer

The residue theorem is not applicable to this integral.

Question 9

Which of the following is a necessary and sufficient condition for a complex function f(z) to be analytic at a point z0?

Accepted Answer

f(z0) is continuous at z0 and the Cauchy-Riemann equations are satisfied at z0

Answer

f'(z0) exists

Answer

f''(z0) exists

Answer

The contour integral of f(z) around any closed contour containing z0 is zero

Question 10

What is the period of the complex exponential function e^iz?

Accepted Answer

2π

Answer

π

Answer

4π

Answer

∞

Question 11

Which of the following is a conformal mapping?

Accepted Answer

w = 1/z

Answer

w = z^2

Answer

w = z + 1

Answer

w = e^z

Question 12

What is the region of convergence of the power series ∑(n=0 to ∞) (z - 2)^n?

Accepted Answer

|z - 2| < 1

Answer

|z - 2| > 1

Answer

|z - 2| ≤ 1

Answer

|z - 2| ≥ 1

Question 13

Determine the geometric transformation represented by the Mobius transformation w = (z - i) / (z + i).

Accepted Answer

A rotation by 90 degrees counterclockwise around the origin.

Answer

A translation to the left by 1 unit and a scaling by a factor of i.

Answer

A reflection across the real axis.

Question 14

State the Fundamental Theorem of Algebra.

Accepted Answer

Every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Answer

Every continuous function on a closed interval has a maximum and minimum value.

Question 15

Define the Cauchy-Riemann equation.

Accepted Answer

A set of partial differential equations that characterize the differentiability of a complex function.

Answer

A formula for calculating the integral of a complex function.

Question 16

State the conformal mapping of the unit disk onto the upper half-plane.

Accepted Answer

w = z + (1-z^2)/(2z)

Answer

w = z^2

Answer

w = 1/z

Answer

w = e^z

Question 17

Explain the Riemann Mapping Theorem.

Accepted Answer

Every simply connected open set in the complex plane can be conformally mapped onto the unit disk.

Answer

Every open set in the complex plane can be conformally mapped onto the unit disk.

Answer

Every closed set in the complex plane can be conformally mapped onto the unit disk.

Question 18

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

Accepted Answer

2πi

Answer

πi

Answer

0

Answer

4πi

Question 19

Which of the following is true about the function f(z) = e^z?

Accepted Answer

It is entire.

Answer

It is analytic but not entire.

Answer

It is meromorphic.

Answer

It has an essential singularity at z=0.

Question 20

Evaluate the integral ∫e^z/(z-1) dz along the contour C consisting of the line segment [0, 1] and the semicircle of radius 1 centered at the origin, lying in the upper half-plane.

Accepted Answer

2πi

Answer

πi

Answer

0

Answer

4πi

Question 21

Which of the following functions is conformal?

Accepted Answer

f(z) = z^2

Answer

f(z) = |z|^2

Answer

f(z) = z*

Answer

f(z) = 1/z

Question 22

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

Accepted Answer

3

Answer

2

Answer

1

Question 23

Which of the following is a necessary condition for the existence of a Taylor series expansion of a complex function f(z) at z=a?

Accepted Answer

f(z) must be analytic in a neighborhood of z=a.

Answer

f(z) must be continuous at z=a.

Answer

f(z) must be differentiable at z=a.

Answer

f(z) must be bounded in a neighborhood of z=a.

Question 24

Find the number of zeros of the function f(z) = z^4+1 inside the unit circle.

Accepted Answer

0

Answer

2

Answer

1

Answer

4

Question 25

Which of the following is a branch of the multi-valued function f(z) = √(z-1)?

Accepted Answer

f(z) = √(z-1) for z in the right half-plane.

Answer

f(z) = √(z-1) for z in the left half-plane.

Answer

f(z) = -√(z-1) for z in the left half-plane.

Answer

f(z) = -√(z-1) for z in the right half-plane.