TRB PG Assistant - Index of a point with respect to a closed curve, local properties of analytic functions, removable singularities, Taylor's theorem Online Test (MCQ Quiz)

1. 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.

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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)

8. 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

9. 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

10. 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

11. 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

12. 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

13. An analytic function is locally:

A) Constant B) Linear C) Representable by a convergent power series D) Differentiable only once

14. 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

15. 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

16. 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.

17. 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.

18. 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.

19. 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.

20. 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

21. 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

22. 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

23. 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

24. 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!

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 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.

27. 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

28. 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

29. 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.

30. 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

31. 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.

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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|

42. 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.

43. 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

44. 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

45. 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

46. 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

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:

A) The unbounded component B) The bounded component C) Any component D) The component containing infinity

48. 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

49. 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!

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

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