Complex Analysis: A Comprehensive Guide for University Students

Question 1

Under what conditions can the residue theorem be used to evaluate integrals over closed curves in the complex plane? Explain your answer.

Accepted Answer

The residue theorem can only be applied to closed curves that enclose isolated singularities within their interior.

Answer

The residue theorem can be applied to any closed curve in the complex plane.

Question 2

Determine the convergence of the infinite product ∏[n=1,∞) (1 - (z/n))^(n^2) using suitable convergence tests. Explain the implications of the convergence or divergence of the product.

Accepted Answer

The product converges absolutely for |z| < 1.

Answer

The product converges absolutely for all z.

Answer

The product converges conditionally for all z.

Answer

The product diverges for all z.

Question 3

Question:

How can conformal mapping be used to simplify a complex region? Provide an example of using the Schwarz-Christoffel transformation to map a polygon in the complex plane to a rectangle.

Accepted Answer

The Schwarz-Christoffel transformation is a conformal mapping technique that can map a polygon in the complex plane to a rectangle.

Answer

Conformal mapping cannot be used to transform a complex region into a simpler shape.

Answer

Conformal mapping is only applicable to circles and ellipses.

Answer

The concept of conformal mapping has no relevance to engineering applications.

Question 4

Find the zeros and poles of the complex function f(z) = (z^2 - 1)/(z^2 + 1). Draw the pole-zero plot and explain how it affects the behavior of f(z) in the complex plane.

Accepted Answer

Zeros: z = ±i, Poles: z = ±1

Answer

Zeros: z = ±1, Poles: z = ±i

Answer

Zeros: z = 0, Poles: z = ∞

Answer

Zeros: z = ±i, Poles: none

Question 5

What is the modulus of the complex number denoted as z = 3 + 4i?

Accepted Answer

5

Answer

1

Answer

10

Answer

15

Question 6

Which answer choice represents the conjugate of the complex number z = 2 - 3i?

Accepted Answer

2 + 3i

Answer

2 - 3i

Answer

-2 + 3i

Answer

-2 - 3i

Question 7

Which statement accurately defines Cauchy's theorem?

Accepted Answer

If a function is analytic both inside and on a simple closed curve, its integral around that curve is zero.

Answer

If a function is continuous on a closed curve, its integral around that curve is zero.

Answer

If a function is differentiable on a closed curve, its integral around that curve is zero.

Answer

If a function is analytic on a closed curve, its integral around that curve is zero.

Question 8

Determine the complex conjugate of z = 2 - 3i.

Accepted Answer

2 + 3i

Answer

2 - 3i

Answer

-2 + 3i

Answer

-2 - 3i

Question 9

Express the complex number z = 1 + i in polar form.

Accepted Answer

(√2, π/4)

Answer

(1, 1)

Answer

(1, π/4)

Answer

(√2, 3π/4)

Question 10

Evaluate the integral ∫(1 + z)^2 dz around the unit circle in the positive direction.

Accepted Answer

2πi

Answer

0

Answer

4πi

Answer

8πi

Question 11

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

Accepted Answer

i

Answer

1

Answer

-1

Answer

-i

Question 12

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

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 13

Find the image of the circle |z| = 1 under the mapping w = z^2.

Accepted Answer

|w| = 2

Answer

|w| = 1

Answer

Re(w) > 0

Answer

Im(w) > 0

Question 14

Define a complex number.

Accepted Answer

A number expressed as "a + bi", where "a" and "b" are real numbers.

Answer

A number with magnitude and phase.

Answer

A non-real number.

Answer

An imaginary number.

Question 15

Find the conjugate of the complex number: z = 3 + 4i

Accepted Answer

3 - 4i

Answer

4 + 3i

Answer

-3 - 4i

Answer

-4 + 3i

Question 16

Identify the analytic function among the following:

Accepted Answer

f(z) = z^2

Answer

f(z) = 1/z

Answer

f(z) = |z|

Answer

f(z) = z*

Question 17

Find the residue of the function: f(z) = 1/(z - 1) at z = 1

Accepted Answer

1

Answer

0

Answer

i

Answer

-1

Question 18

Which of the following is NOT a complex number?

Accepted Answer

3

Answer

1 + 2i

Answer

4 - 5i

Answer

i

Question 19

Find the modulus of the complex number 2 + 3i.

Accepted Answer

5

Answer

1

Answer

2

Answer

10

Question 20

Which of the following is the conjugate of the complex number 4 - 2i?

Accepted Answer

4 + 2i

Answer

4 - 2i

Answer

-4 + 2i

Answer

-4 - 2i

Question 21

Find the derivative of the complex function f(z) = z^2 + 2z + 3.

Accepted Answer

2z + 2

Answer

3z

Answer

2z

Answer

1

Question 22

Which of the following is the residue of the function 1/(z - 2) at z = 2?

Accepted Answer

1

Answer

0

Answer

2

Answer

3

Question 23

Which of the following is the integral of the function e^z around a unit circle centered at the origin?

Accepted Answer

2πi

Answer

0

Answer

-2πi

Answer

1

Question 24

Which of the following is the correct definition of the complex exponential function e^z?

Accepted Answer

e^(x+iy) = cosh(x) + i sinh(y)

Answer

cos(x) + i sin(y)

Answer

tanh(x) + i coth(y)

Answer

sinh(x) + i cosh(y)

Question 25

What is the value of the integral of the function f(z) = 1/z along the positively oriented unit circle centered at the origin?

Accepted Answer

2πi

Answer

0

Answer

π

Answer

2π

Question 26

Which of the following functions has a removable singularity at z = 0?

Accepted Answer

sin(z)/z

Answer

1/z

Answer

z^2

Answer

e^z

Question 27

Which of the following is the correct form of the Cauchy-Riemann equations?

Accepted Answer

u_x = v_y, u_y = -v_x

Answer

u_x = v_x, u_y = v_y

Answer

u_x = -v_y, u_y = v_x

Answer

u_x = -v_x, u_y = -v_y

Question 28

Consider the infinite series ∑[n=1,∞) (z^n)/(n^2 + 1) for all (z ∈ ℂ). Using the Ratio Test, determine whether this series converges or diverges.

Accepted Answer

The series converges when ( |z| < 1 ), and diverges when ( |z| > 1).

Answer

The series converges for all (z ∈ ℂ).

Answer

The series diverges for all ( z ∈ ℂ ).

Answer

The series converges when ( |z| > 1 ) and diverges when ( |z| < 1).

Question 29

In complex analysis, which of these is not a theorem?

Accepted Answer

Green's theorem

Answer

Cauchy's theorem

Answer

Morera's theorem

Answer

Argument principle

Question 30

Find the value of the integral ∮_C (z^2 + 1)/(z-i) dz, where C is the circle with radius 2 centered at z=i.

Accepted Answer

2πi

Answer

0

Answer

4πi

Answer

-2πi

Question 31

Calculate the modulus of the complex number (3 + 4i).

Accepted Answer

7

Answer

1

Answer

5

Answer

9

Question 32

Which of the following is NOT a type of complex function commonly encountered in engineering applications?

Accepted Answer

Trigonometric function

Answer

Analytic function

Answer

Meromorphic function

Answer

Entire function

Question 33

State the Cauchy-Riemann condition for a complex function to be differentiable.

Accepted Answer

The partial derivative of the function with respect to x is equal to the complex conjugate of the partial derivative with respect to y.

Answer

The partial derivatives of the function with respect to x and y are equal.

Answer

The real and imaginary parts of the function are both harmonic.

Answer

The function is continuous and differentiable.

Question 34

Which of the following is NOT a conformal mapping?

Accepted Answer

log(z)

Answer

z^2

Answer

e^z

Answer

sin(z)

Question 35

Find the complex conjugate of z = 3 + 4i.

Accepted Answer

3 - 4i

Answer

3 + 4i

Answer

-3 + 4i

Answer

-3 - 4i

Question 36

Calculate the modulus of the complex number z = (1 + i)(2 + 3i).

Accepted Answer

5

Answer

1

Answer

2

Answer

10

Question 37

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

2

Answer

4

Question 38

What is the order of the pole of the function  f(z) = e^z at z = 0?

Accepted Answer

Order 1 (The function has a simple pole at z = 0)

Answer

Order 0 (The function is continuous at z = 0)

Answer

Order 1 (The function has a removable singularity at z = 0)

Answer

Order 2 (The function has a double pole at z = 0)

Question 39

What is the primary application of the residue theorem?

Accepted Answer

Evaluating integrals of complex functions along closed contours

Answer

Finding the zeros of complex functions

Answer

Solving differential equations

Question 40

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

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 41

Determine the contour integral of f(z) = 1/(z - 2) around the unit circle centered at the origin.

Accepted Answer

2πi

Answer

0

Answer

-2πi

Answer

4πi

Question 42

Define the residue theorem.

Accepted Answer

A theorem connecting a contour integral's value to function residues within the contour

Answer

A theorem stating that the integral of a function along a closed contour is zero

Answer

A theorem providing a formula for the sum of function residues at its poles

Answer

A theorem for finding polynomial roots

Question 43

Explain the relationship between a function's residue at a pole and its Laurent series expansion around the pole.

Accepted Answer

The residue equals the coefficient of the (z - a)^-1 term in the Laurent series

Answer

The residue equals the sum of coefficients for (z - a)^-n terms, where n is the pole order

Answer

The residue is the integral of the function around a small circle centered at the pole

Answer

The residue equals the function's value at the pole

Question 44

Which of the following functions is holomorphic everywhere in the complex plane?

Accepted Answer

f(z) = z^2

Answer

f(z) = |z|

Answer

f(z) = arg(z)

Answer

f(z) = 1/z

Question 45

Convert the complex number z = 1 + i to its polar form.

Accepted Answer

(√2, π/4)

Answer

(√2, 0)

Answer

(1, 0)

Question 46

Find the derivative of the complex function f(z) = z^2 + 2z - 1.

Accepted Answer

2z + 2

Answer

z^2

Answer

1

Answer

2z - 1

Question 47

Compute the integral of f(z) = e^z, where z is a complex variable.

Accepted Answer

e^z + C

Answer

e^(z^2)

Answer

z^2*e^z

Question 48

Determine the Laurent series expansion of f(z) = 1/(z - 1) for the region |z - 1| > 1.

Accepted Answer

∑(n=0 to ∞) (z - 1)^-n-1

Answer

∑(n=1 to ∞) (z - 1)^n-1

Answer

∑(n=0 to ∞) n(z - 1)^-n-1

Question 49

Which application of complex analysis is employed in fluid mechanics?

Accepted Answer

Solving Laplace's equation

Answer

Computing integrals

Answer

Solving differential equations

Question 50

Which of the following represents the real part of the complex number 3 + 4i?

Accepted Answer

3

Answer

4

Answer

3i

Answer

4i

Question 51

Determine the modulus of the complex number -1 + √3i.

Accepted Answer

2

Answer

1

Answer

√2

Answer

√3

Question 52

Which function possesses the property of being analytic throughout the complex plane?

Accepted Answer

e^z

Answer

1/z

Answer

z^2

Answer

sin(z)

Question 53

Identify the branch cut of the function √(z^2 - 4).

Accepted Answer

[-2, 2]

Answer

[0, 2]

Answer

[2, 4]

Answer

[-4, 4]

Question 54

Find the image of the unit circle under the mapping w = z^3.

Accepted Answer

Cardioid

Answer

Ellipse

Answer

Circle

Answer

Parabola

Question 55

Calculate the value of the integral  ∫(z^2 + 1) / (z^2 - 1) dz, where the contour is a circle centered at the origin with radius 2, traversed counterclockwise.

Accepted Answer

2πi

Answer

-2πi

Answer

0

Answer

πi

Question 56

Identify the function that is analytic throughout the entire complex plane.

Accepted Answer

e^z

Answer

log(z)

Answer

1/z

Answer

z^2

Question 57

Which statement accurately describes Cauchy's Integral Formula?

Accepted Answer

It establishes a relationship between the value of an analytic function at a point and its integral along a closed contour.

Answer

It provides a method for locating the poles of an analytic function.

Answer

It states that the integral of an analytic function over any closed contour is always zero.

Question 58

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

Accepted Answer

1/z - 1 + z + z^2 + ...

Answer

-1/z - 1 + z + z^2 + ...

Answer

1/z + 1 + z + z^2 + ...

Question 59

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

Accepted Answer

w = (z - i) / (z + i)

Answer

w = e^z

Answer

w = z^2

Answer

w = 1/z

Question 60

Employing the residue theorem, what is the value of the integral ∫(e^z) / (z^2 + 1) dz, where the contour is the unit circle centered at the origin, traversed counterclockwise?

Accepted Answer

πi*e^i

Answer

2πi*e^i

Answer

0

Answer

-πi*e^i

Question 61

The principle of the argument states that the change in the argument of a function f(z) as z traverses a closed contour is equivalent to:

Accepted Answer

2πi times the difference between the number of zeros and the number of poles of f(z) enclosed within the contour.

Answer

The residue of f(z) at the origin.

Answer

The value of the integral of f(z) along the contour.

Question 62

What is the significance of the Cauchy-Riemann equations in the context of complex analysis?

Accepted Answer

They provide necessary and sufficient conditions for a complex function to be differentiable (analytic) at a point.

Answer

They define the residue of a function at a pole.

Question 63

Calculate the value of (3 + 4i) multiplied by its complex conjugate.

Accepted Answer

25

Answer

9 - 16i

Answer

-7

Answer

1

Question 64

Express the complex number z = 4(cos(2π/3) + i sin(2π/3)) in rectangular form.

Accepted Answer

-2 + 2√3i

Answer

-2 - 2√3i

Answer

2 + 2√3i

Answer

2 - 2√3i