Index of a point with respect to a closed curve, local properties of analytic functions, removable singularities, Taylor's theorem - Question Bank

1. According to Taylor's theorem, if f(z) is analytic in a disk |z - z₀| < R, then f(z) = ∑_{n=0}^∞ aₙ(z - z₀)ⁿ for |z - z₀| < R. What is the relationship between aₙ and f⁽ⁿ⁾(z₀)?
A) aₙ = f⁽ⁿ⁾(z₀) / n!
B) aₙ = f⁽ⁿ⁾(z₀) * n!
C) aₙ = f⁽ⁿ⁾(0) / n!
D) aₙ = f(z₀) / n!
2. If a function f(z) has a removable singularity at z₀, then f(z) can be made analytic at z₀ by:
A) Redefining f(z₀) to be lim (z→z₀) f(z)
B) Setting f(z₀) = 0
C) Removing z₀ from the domain
D) Replacing f(z) with its derivative
3. The index of a point z₀ with respect to a curve γ is constant for all z₀ in a given connected component of the complement of γ. This component is:
A) The unbounded component
B) The bounded component
C) Any component
D) The component containing infinity
4. If f(z) is analytic in a domain D and has a zero of order m at z₀, then in a neighborhood of z₀, f(z) behaves like:
A) c(z - z₀)ᵐ for some non-zero constant c
B) c(z - z₀) for some non-zero constant c
C) c for some non-zero constant c
D) 0
5. The coefficients of the Taylor series expansion of f(z) around z₀ are given by f⁽ⁿ⁾(z₀)/n!. This formula relies on which theorem?
A) Cauchy's Integral Formula for derivatives
B) Cauchy's Integral Theorem
C) Liouville's Theorem
D) Morera's Theorem
6. Consider the function f(z) = z⁻¹ sin(z). What is the nature of the singularity at z=0?
A) Removable singularity
B) Pole
C) Essential singularity
D) Branch point
7. If f(z) = 1/z², what is the nature of the singularity at z=0?
A) Pole of order 2
B) Removable singularity
C) Essential singularity
D) Simple pole
8. What is the condition for a function f(z) to be analytic at a point z₀?
A) f(z) is differentiable at z₀ and in a neighborhood around z₀.
B) f(z) is continuous at z₀.
C) f(z) has a finite derivative at z₀.
D) f(z) is bounded in a neighborhood of z₀.
9. The index of a point z₀ with respect to a curve γ can be calculated using the integral:
A) 1/(2πi) ∫[γ] dz / (z - z₀)
B) ∫[γ] (z - z₀) dz
C) 2πi ∫[γ] dz / (z - z₀)
D) 1/2π ∫[γ] |dz| / |z - z₀|
10. If f(z) has a pole of order m at z₀, then lim (z→z₀) (z - z₀)ᵐ f(z) is:
A) A finite non-zero complex number
B) Zero
C) Infinity
D) Undefined
11. The Taylor series expansion of f(z) around z₀ converges in the largest open disk centered at z₀ within which:
A) f(z) is analytic
B) f(z) is continuous
C) f(z) is defined
D) f(z) is bounded
12. If f(z) = z² sin(1/z), what is the nature of the singularity at z=0?
A) Removable singularity
B) Pole
C) Essential singularity
D) Branch point
13. A point z₀ where a function f(z) fails to be analytic is called a(n):
A) Singularity
B) Analytic point
C) Critical point
D) Limit point
14. If f(z) is analytic and non-constant in a domain D, then f(D) is:
A) An open set
B) A closed set
C) A compact set
D) A discrete set
15. The index of a point z₀ with respect to a curve γ is zero if z₀ is:
A) Outside the region enclosed by γ
B) Inside the region enclosed by γ
C) On the curve γ
D) At infinity
16. What does the Cauchy's Integral Formula for derivatives state?
A) f⁽ⁿ⁾(z₀) = n! / (2πi) ∫[γ] f(z) / (z - z₀)ⁿ⁺¹ dz
B) f(z₀) = 1 / (2πi) ∫[γ] f(z) / (z - z₀) dz
C) f'(z₀) = 1 / (2πi) ∫[γ] f(z) / (z - z₀)² dz
D) f⁽ⁿ⁾(z₀) = 1 / (2πi) ∫[γ] f(z) / (z - z₀)ⁿ dz
17. If f(z) = 1/(1-z) is expanded in a Taylor series around z₀ = 0, what is the radius of convergence?
A) 1
B) ∞
C) 0
D) 2
18. The Laurent series expansion around an isolated singularity z₀ is unique for a given annulus:
A) 1/R₂ < |z - z₀| < 1/R₁
B) |z - z₀| < R
C) |z - z₀| > R
D) R₁ < |z - z₀| < R₂
19. If f(z) has a removable singularity at z₀, its Laurent series expansion around z₀ will have:
A) No terms with negative powers of (z - z₀).
B) Only terms with negative powers of (z - z₀).
C) A finite number of terms with negative powers.
D) An infinite number of terms with negative powers.
20. Taylor's Theorem is a generalization of the Maclaurin series when the expansion is centered at:
A) z₀ = 0
B) z₀ = 1
C) z₀ = ∞
D) z₀ = -1
21. What is the relationship between the derivative of an analytic function and its Taylor series?
A) The derivative can be obtained by differentiating the Taylor series term by term.
B) The derivative is the constant term of the series.
C) The derivative is the sum of the series.
D) The derivative is unrelated to the Taylor series.
22. Consider f(z) = e^z expanded around z₀ = 0. What is the Taylor series?
A) ∑ zⁿ/n!
B) ∑ zⁿ
C) ∑ (z-1)ⁿ/n!
D) ∑ zⁿ/n
23. If f(z) is analytic at z₀, what is the value of f(z₀) in terms of its Taylor series expansion around z₀?
A) f(z₀) = a₀
B) f(z₀) = a₁
C) f(z₀) = 0
D) f(z₀) = 1
24. What is the radius of convergence of the Taylor series expansion of f(z) around z₀?
A) The distance from z₀ to the nearest singularity of f(z).
B) The distance from z₀ to the origin.
C) Infinity.
D) Zero.
25. Taylor's Theorem states that the power series representation of an analytic function is:
A) Unique
B) Not necessarily unique
C) Dependent on the radius of convergence
D) Dependent on the chosen point z₀
26. What are the coefficients aₙ in the Taylor series expansion of f(z) around z₀?
A) aₙ = f⁽ⁿ⁾(z₀) / n!
B) aₙ = f⁽ⁿ⁾(z₀)
C) aₙ = f(z₀) / n!
D) aₙ = f⁽ⁿ⁾(0) / n!
27. According to Taylor's Theorem, if f(z) is analytic in a disk |z - z₀| < R, then f(z) can be represented by:
A) A convergent power series ∑ aₙ(z - z₀)ⁿ
B) A convergent Laurent series
C) A finite sum of terms
D) A Fourier series
28. If f(z) = e^(1/z), what type of singularity does it have at z₀ = 0?
A) Removable singularity
B) Pole
C) Essential singularity
D) Branch point
29. If f(z) = 1/z, what type of singularity does it have at z₀ = 0?
A) Removable singularity
B) Pole of order 1
C) Essential singularity
D) Isolated singularity
30. Consider the function f(z) = sin(z)/z. What type of singularity does it have at z₀ = 0?
A) Removable singularity
B) Pole
C) Essential singularity
D) Branch point
31. Which condition guarantees that a singularity z₀ of f(z) is removable?
A) f(z) is bounded in a neighborhood of z₀ (excluding z₀).
B) f(z) has a pole at z₀.
C) f(z) is undefined at z₀.
D) lim (z→z₀) f(z) = 0.
32. If f(z) has a removable singularity at z₀, how can it be redefined to be analytic at z₀?
A) Set f(z₀) to be the limit of f(z) as z approaches z₀.
B) Set f(z₀) to infinity.
C) Remove the point z₀ from the domain.
D) Define f(z₀) as the derivative at z₀.
33. What is a removable singularity of a complex function f(z) at z₀?
A) lim (z→z₀) f(z) exists and is finite.
B) lim (z→z₀) |f(z)| = ∞.
C) f(z) has a simple pole at z₀.
D) f(z) is not defined at z₀.
34. What is the Identity Theorem for analytic functions?
A) If two analytic functions agree on a set with a limit point, they are identical.
B) If an analytic function is zero everywhere, it is the zero function.
C) An analytic function is identical to its derivative.
D) If an analytic function is constant, its derivative is zero.
35. If f'(z₀) ≠ 0, what does this imply about the local mapping of an analytic function f(z) at z₀?
A) It is a rotation and scaling
B) It is a translation only
C) It is a reflection
D) It collapses the neighborhood to a point
36. What is a consequence of analyticity regarding the mapping properties of a function?
A) It maps lines to lines
B) It preserves angles
C) It maps circles to ellipses
D) It always maps open sets to closed sets
37. An analytic function is locally:
A) Constant
B) Linear
C) Representable by a convergent power series
D) Differentiable only once
38. If f(z) is analytic at z₀, what does this imply about the function in a neighborhood of z₀?
A) It is constant
B) It has a removable singularity
C) It can be represented by a power series
D) It has a pole
39. What is a key local property of analytic functions concerning their derivatives?
A) Derivatives can be discontinuous
B) Derivatives are always zero
C) Derivatives are also analytic
D) Derivatives are constant
40. Which theorem relates the number of zeros and poles of an analytic function inside a closed curve to the integral of its derivative?
A) Cauchy's Integral Theorem
B) Liouville's Theorem
C) Argument Principle
D) Taylor's Theorem
41. If f(z) is analytic and non-zero in a simply connected domain D, and γ is a closed curve in D, what is the index of any point z₀ inside γ with respect to γ?
A) 0
B) 1
C) Depends on f(z)
D) Undefined
42. The index of a point z₀ with respect to a closed curve γ is also known as the:
A) Residue
B) Winding number
C) Argument principle
D) Cauchy principal value
43. Let f(z) be analytic in a domain D. If γ is a closed curve in D and z₀ is a point not on γ, what is the relationship between the index of z₀ and the integral of f'(z)/f(z) along γ?
A) Integral = 2πi * index(γ, z₀)
B) Integral = index(γ, z₀) / (2πi)
C) Integral = 0
D) Integral = πi * index(γ, z₀)
44. If γ is a closed curve and z₀ is a point not on γ, what can be said about the index of z₀ with respect to γ if γ is continuously deformed without passing through z₀?
A) The index changes
B) The index remains constant
C) The index becomes 0
D) The index becomes 1
45. Consider a curve γ that traverses the unit circle counterclockwise twice. What is the index of the origin (0,0) with respect to γ?
A) 0
B) 1
C) 2
D) 4
46. What is the winding number of a simple closed curve with respect to a point outside the curve?
A) 0
B) 1
C) -1
D) Undefined
47. What is the winding number of a simple closed curve with respect to a point inside the curve?
A) 0
B) 1
C) -1
D) Depends on the orientation of the curve
48. If a closed curve γ does not pass through a point z₀, what is the index of z₀ with respect to γ?
A) Always 0
B) Always 1
C) Depends on the function being analytic
D) Undefined
49. What is the definition of the index of a point z₀ with respect to a closed curve γ?
A) The number of times γ winds around z₀ in the counterclockwise direction.
B) The total length of the curve γ divided by the distance from z₀ to the curve.
C) The average value of the derivative of the function along γ.
D) The number of zeros of the function inside the curve γ.
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