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

1. Consider f(z) = e^z expanded around z₀ = 0. What is the Taylor series? — ∑ zⁿ/n!
2. What is the winding number of a simple closed curve with respect to a point inside the curve? — 1
3. What is the winding number of a simple closed curve with respect to a point outside the curve? — 0
4. Consider a curve γ that traverses the unit circle counterclockwise twice. What is the index of the origin (0,0) with respect to γ? — 2
5. 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? — 0
6. If f(z) = 1/(1-z) is expanded in a Taylor series around z₀ = 0, what is the radius of convergence? — 1
7. The index of a point z₀ with respect to a curve γ can be calculated using the integral: — 1/(2πi) ∫[γ] dz / (z - z₀)
8. The Laurent series expansion around an isolated singularity z₀ is unique for a given annulus: — R₁ < |z - z₀| < R₂
9. According to Taylor's Theorem, if f(z) is analytic in a disk |z - z₀| < R, then f(z) can be represented by: — A convergent power series ∑ aₙ(z - z₀)ⁿ
10. If f(z) has a pole of order m at z₀, then lim (z→z₀) (z - z₀)ᵐ f(z) is: — A finite non-zero complex number
11. If a closed curve γ does not pass through a point z₀, what is the index of z₀ with respect to γ? — Always 0
12. If f(z) is analytic and non-constant in a domain D, then f(D) is: — An open set
13. What are the coefficients aₙ in the Taylor series expansion of f(z) around z₀? — aₙ = f⁽ⁿ⁾(z₀) / n!
14. 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)? — aₙ = f⁽ⁿ⁾(z₀) / n!
15. 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: — c(z - z₀)ᵐ for some non-zero constant c
16. 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? — Cauchy's Integral Formula for derivatives
17. Which theorem relates the number of zeros and poles of an analytic function inside a closed curve to the integral of its derivative? — Argument Principle
18. An analytic function is locally: — Representable by a convergent power series
19. What is a key local property of analytic functions concerning their derivatives? — Derivatives are also analytic
20. What does the Cauchy's Integral Formula for derivatives state? — f⁽ⁿ⁾(z₀) = n! / (2πi) ∫[γ] f(z) / (z - z₀)ⁿ⁺¹ dz
21. The Taylor series expansion of f(z) around z₀ converges in the largest open disk centered at z₀ within which: — f(z) is analytic
22. Which condition guarantees that a singularity z₀ of f(z) is removable? — f(z) is bounded in a neighborhood of z₀ (excluding z₀).
23. What is the condition for a function f(z) to be analytic at a point z₀? — f(z) is differentiable at z₀ and in a neighborhood around z₀.
24. If f(z) is analytic at z₀, what is the value of f(z₀) in terms of its Taylor series expansion around z₀? — f(z₀) = a₀
25. What is the Identity Theorem for analytic functions? — If two analytic functions agree on a set with a limit point, they are identical.
26. 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? — Integral = 2πi * index(γ, z₀)
27. If f'(z₀) ≠ 0, what does this imply about the local mapping of an analytic function f(z) at z₀? — It is a rotation and scaling
28. If f(z) is analytic at z₀, what does this imply about the function in a neighborhood of z₀? — It can be represented by a power series
29. What is a consequence of analyticity regarding the mapping properties of a function? — It preserves angles
30. What is a removable singularity of a complex function f(z) at z₀? — lim (z→z₀) f(z) exists and is finite.
31. If f(z) has a removable singularity at z₀, its Laurent series expansion around z₀ will have: — No terms with negative powers of (z - z₀).
32. The index of a point z₀ with respect to a curve γ is zero if z₀ is: — Outside the region enclosed by γ
33. If f(z) = 1/z², what is the nature of the singularity at z=0? — Pole of order 2
34. If a function f(z) has a removable singularity at z₀, then f(z) can be made analytic at z₀ by: — Redefining f(z₀) to be lim (z→z₀) f(z)
35. Consider the function f(z) = sin(z)/z. What type of singularity does it have at z₀ = 0? — Removable singularity
36. If f(z) = 1/z, what type of singularity does it have at z₀ = 0? — Pole of order 1
37. If f(z) = e^(1/z), what type of singularity does it have at z₀ = 0? — Essential singularity
38. If f(z) = z² sin(1/z), what is the nature of the singularity at z=0? — Removable singularity
39. Consider the function f(z) = z⁻¹ sin(z). What is the nature of the singularity at z=0? — Removable singularity
40. The index of a point z₀ with respect to a closed curve γ is also known as the: — Winding number
41. If f(z) has a removable singularity at z₀, how can it be redefined to be analytic at z₀? — Set f(z₀) to be the limit of f(z) as z approaches z₀.
42. A point z₀ where a function f(z) fails to be analytic is called a(n): — Singularity
43. What is the relationship between the derivative of an analytic function and its Taylor series? — The derivative can be obtained by differentiating the Taylor series term by term.
44. What is the radius of convergence of the Taylor series expansion of f(z) around z₀? — The distance from z₀ to the nearest singularity of f(z).
45. 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? — The index remains constant
46. What is the definition of the index of a point z₀ with respect to a closed curve γ? — The number of times γ winds around z₀ in the counterclockwise direction.
47. 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: — Any component
48. Taylor's Theorem states that the power series representation of an analytic function is: — Unique
49. Taylor's Theorem is a generalization of the Maclaurin series when the expansion is centered at: — z₀ = 0
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