TRB BT Assistant - Complex Numbers and Complex Functions Question Bank with Options

1. The function f(z) = z^2 is analytic everywhere. What are its Cauchy-Riemann equations?

A) 2x = 2y and 2y = -2x

B) 2x = -2y and 2y = 2x

C) x = y and y = -x

D) x = -y and y = x

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

A) The entire complex plane

B) The complex plane excluding z=0

C) The complex plane excluding z=1

D) The complex plane excluding z=-1

3. Which complex function is often used to map the unit disk to itself conformally?

A) Möbius transformation (or fractional linear transformation)

B) Exponential function

C) Logarithmic function

D) Power function

4. What is a branch cut for a multi-valued function?

A) A curve or line segment used to define a single-valued branch of the function

B) A point where the function is undefined

C) The domain of analyticity

D) The region of convergence of a power series

5. What is the branch point of a multi-valued complex function?

A) A point around which the function's values do not return to their original value after a circuit

B) A point where the function is analytic

C) A point where the function has a removable singularity

D) A point where the function is constant

6. What is the fundamental theorem of algebra in the context of complex numbers?

A) Every polynomial of degree n has exactly n roots in the complex numbers (counting multiplicity)

B) Every polynomial of degree n has exactly n real roots

C) Every complex number has a unique square root

D) Every non-constant polynomial has at least one real root

7. The Schwarz-Christoffel transformation is used to map:

A) The upper half-plane onto a polygon

B) A polygon onto the upper half-plane

C) The unit disk onto a polygon

D) A polygon onto the unit disk

8. Which of the following is a fundamental property of conformal mappings?

A) They are always linear

B) They are always constant

C) They are locally equivalent to multiplication by a complex number

D) They are always harmonic

9. What is conformal mapping?

A) A mapping that preserves angles between curves

B) A mapping that preserves lengths of curves

C) A mapping that preserves areas

D) A mapping that maps circles to circles

10. What is the Maximum Modulus Principle?

A) If f(z) is analytic and non-constant in a bounded domain D, then |f(z)| attains its maximum value on the boundary of D

B) If f(z) is analytic and non-constant in a bounded domain D, then |f(z)| attains its minimum value on the boundary of D

C) If f(z) is analytic and non-constant in a bounded domain D, then |f(z)| attains its maximum value inside D

D) If f(z) is analytic and non-constant in a bounded domain D, then |f(z)| attains its minimum value inside D

11. An entire function is a function that is:

A) Analytic everywhere in the complex plane

B) Analytic in a specific region

C) Differentiable only at the origin

D) Continuous everywhere

12. What is the Liouville's Theorem in complex analysis?

A) A bounded entire function must be constant

B) An analytic function is constant if its derivative is zero

C) The integral of an analytic function over a closed loop is zero

D) A function with isolated singularities is constant

13. The part of the Laurent series with negative powers of (z-z0) is called the:

A) Analytic part

B) Principal part

C) Integral part

D) Real part

14. A Laurent series expansion of a function f(z) around an isolated singularity z0 is of the form:

A) sum(a_n * (z-z0)^n)

B) sum(a_n * (z-z0)^-n)

C) sum(a_n * (z-z0)^n) + sum(b_n * (z-z0)^-n)

D) sum(a_n * (z-z0)^-n) + sum(b_n * (z-z0)^n)

15. What is the radius of convergence of a power series?

A) The radius of the disk of convergence

B) The distance to the nearest singularity

C) The sum of the coefficients

D) The exponent of the highest power

16. The region of convergence for a power series centered at z0 is typically:

A) A disk

B) A square

C) An annulus

D) A line

17. What is a power series in the complex plane?

A) A series of the form sum(a_n * z^n)

B) A series of the form sum(a_n / z^n)

C) A series of the form sum(a_n * (z-z0)^n)

D) A series of the form sum(a_n * (z-z0)^-n)

18. The Residue Theorem provides a powerful way to evaluate contour integrals. It states that the integral of f(z) around a simple closed contour C is equal to:

A) The sum of the residues of f(z) at all singularities inside C

B) 2*pi*i times the sum of the residues of f(z) at all singularities inside C

C) The sum of the residues of f(z) at all singularities outside C

D) Zero

19. What is the residue of a complex function f(z) at an isolated singularity z0?

A) The coefficient of the (z-z0)^-1 term in the Laurent series expansion of f(z) around z0

B) The value of the function at z0

C) The principal part of the Laurent series

D) The analytic part of the Laurent series

20. An essential singularity is a singularity where:

A) The function can be made analytic by redefining it

B) The function has a pole of finite order

C) The function's behavior near the singularity is very complex (e.g., Casorati-Weierstrass theorem)

D) The function is bounded

21. A pole is a type of singularity where:

A) The function approaches infinity as z approaches the pole

B) The function remains bounded near the singularity

C) The function can be made analytic by redefining it

D) The function has a finite value

22. A removable singularity is a point z0 where:

A) The function can be defined or redefined at z0 to make it analytic there

B) The function has a pole of order 1

C) The function has an essential singularity

D) The function is undefined and cannot be made analytic

23. What is a singularity of a complex function?

A) A point where the function is analytic

B) A point where the function is not analytic

C) A point where the function is continuous

D) A point where the function is differentiable

24. Cauchy's Integral Formula relates the value of an analytic function at a point inside a contour to the integral of the function on the contour. The formula is:

A) f(a) = (1 / 2*pi*i) * integral(f(z) / (z-a)) dz

B) f(a) = (1 / pi) * integral(f(z) / (z-a)) dz

C) f(a) = integral(f(z) / (z-a)) dz

D) f(a) = (1 / 2) * integral(f(z) / (z-a)) dz

25. Cauchy's Integral Theorem states that if a function f(z) is analytic inside and on a simple closed contour C, then the integral of f(z) around C is:

A) Equal to 2*pi*i

B) Equal to pi

C) Zero

D) Infinite

26. What is the integral of a complex function f(z) along a curve C called?

A) Line integral

B) Contour integral

C) Surface integral

D) Volume integral

27. If f(z) = u + iv is analytic, then u and v are harmonic conjugates if:

A) u and v satisfy the Cauchy-Riemann equations

B) u and v are linearly related

C) u and v are constants

D) u and v are periodic

28. What is the Laplacian operator, often used in relation to harmonic functions?

A) del^2 = d^2/dx^2 + d^2/dy^2

B) del = d/dx + d/dy

C) del^2 = d^2/dx^2 - d^2/dy^2

D) del = d/dx - d/dy

29. What property do the real and imaginary parts of an analytic function satisfy?

A) They are independent

B) They are harmonic functions

C) They are constant

D) They are linear

30. A function that is analytic in a region is also called:

A) Harmonic

B) Holomorphic

C) Polynomial

D) Rational

31. If f(z) = u(x, y) + iv(x, y) is a complex function, what are the Cauchy-Riemann equations in Cartesian coordinates?

A) du/dx = dv/dy and du/dy = -dv/dx

B) du/dx = -dv/dy and du/dy = dv/dx

C) du/dy = dv/dx and du/dx = dv/dy

D) du/dx = dv/dx and du/dy = dv/dy

32. What are the Cauchy-Riemann equations?

A) Conditions for a complex function to be analytic

B) Formulas for complex number multiplication

C) Equations for finding the modulus of a complex number

D) The definition of the imaginary unit

33. A function f(z) where z is a complex variable is called a complex function. What is the condition for a complex function to be analytic?

A) It must be continuous

B) Its derivative must exist at every point in a region

C) It must be differentiable at a point

D) It must be defined for all complex numbers

34. What is the geometric representation of complex numbers called?

A) Number line

B) Argand diagram

C) Cartesian plane

D) Complex plane

35. What is the principal value of the argument of a complex number z = a + bi?

A) Any angle theta such that z = |z|(cos(theta) + i sin(theta))

B) The unique angle theta such that z = |z|(cos(theta) + i sin(theta)) and -pi < theta <= pi

C) The angle theta such that z = |z|(cos(theta) + i sin(theta)) and 0 <= theta < 2*pi

D) The angle theta such that z = |z|(cos(theta) + i sin(theta)) and 0 < theta <= pi

36. According to De Moivre's theorem, [r(cos(theta) + i sin(theta))]^n is equal to:

A) r^n(cos(n*theta) + i sin(n*theta))

B) r(cos(n*theta) + i sin(n*theta))

C) r^n(cos(theta) + i sin(theta))

D) r(cos^n(theta) + i sin^n(theta))

37. De Moivre's theorem is used to compute:

A) The sum of complex numbers

B) The roots of complex numbers

C) The powers and roots of complex numbers

D) The product of complex numbers

38. In the polar form r(cos(theta) + i sin(theta)), 'theta' represents:

A) The modulus of the complex number

B) The imaginary part of the complex number

C) The argument of the complex number

D) The real part of the complex number

39. In the polar form r(cos(theta) + i sin(theta)), 'r' represents:

A) The argument of the complex number

B) The imaginary part of the complex number

C) The real part of the complex number

D) The modulus of the complex number

40. What is the polar form of a complex number z = a + bi?

A) r(cos(theta) + i sin(theta))

B) a(cos(phi) + i sin(phi))

C) r(sin(theta) + i cos(theta))

D) b(cos(theta) + i sin(theta))

41. Euler's formula states that e^(ix) is equal to:

A) cos(x) - i sin(x)

B) sin(x) + i cos(x)

C) cos(x) + i sin(x)

D) sin(x) - i cos(x)

42. What is the argument of a complex number z = a + bi?

A) The angle it makes with the positive real axis in the complex plane

B) The distance from the origin in the complex plane

C) The real part of the number

D) The imaginary part of the number

43. The modulus of a complex number z = a + bi is denoted by |z| and is calculated as:

A) sqrt(a^2 - b^2)

B) a^2 + b^2

C) sqrt(a^2 + b^2)

D) sqrt(b^2 - a^2)

44. What is the complex conjugate of a + bi?

A) a - bi

B) -a + bi

C) -a - bi

D) b + ai

45. What is the product of (a + bi) and (c + di)?

A) (a+c) + (b+d)i

B) (ac - bd) + (ad + bc)i

C) (ac + bd) + (ad - bc)i

D) (a-c) + (b-d)i

46. What is the result of adding two complex numbers (a + bi) and (c + di)?

A) (ac - bd) + (ad + bc)i

B) (ac + bd) + (ad - bc)i

C) (a+c) + (b+d)i

D) (a-c) + (b-d)i

47. In the complex number a + bi, what is 'b' called?

A) Real part

B) Imaginary part

C) Argument

D) Conjugate

48. A complex number is generally represented in the form a + bi, where 'a' and 'b' are real numbers. What is 'a' called?

A) Imaginary part

B) Real part

C) Modulus

D) Argument

49. What is the imaginary unit, denoted by 'i'?

A) The square root of 1

B) The square root of -1

C) The square root of 0

D) The square root of infinity

Complex Numbers and Complex Functions - Question Bank