Taylor Series, Laurent Series and Residues - Question Bank
1. The Laurent series expansion of a function f(z) about an isolated singularity z₀ is unique in any annulus 0 < |z - z₀| < R where f(z) is analytic. This uniqueness is guaranteed by:
2. If a function f(z) is analytic in a simply connected domain D, and C is any simple closed contour in D, then by Cauchy's Integral Theorem:
3. What is the residue of f(z) = e^(iz) / (z² + a²) at z = ia, where a > 0?
4. Consider the function f(z) = 1 / (z² + 4). What are the poles of this function?
5. If the integral ∫_C f(z) dz = 0 for every simple closed contour C in a domain D, then f(z) is:
6. What is the residue of f(z) = z² / (z-1)³ at z = 1?
7. The Laurent series expansion of f(z) = 1/(z-1)(z-2) in the annulus 1 < |z| < 2 contains terms of the form:
8. What is the residue of f(z) = cos(z) / z³ at z = 0?
9. If f(z) has a pole of order 1 at z₀, the residue is given by lim_(z→z₀) (z - z₀) f(z). This is also known as the residue at a:
10. What is the principal part of the Laurent series of f(z) = 1/sin(z) around z=0?
11. The Taylor series expansion of a function f(z) about z₀ is unique if the function is analytic in a neighborhood of z₀. This uniqueness is a consequence of:
12. Which of the following functions is analytic everywhere in the complex plane?
13. What is the residue of f(z) = z e^(1/z) at z = 0?
14. If f(z) has a pole of order m at z₀, then (z - z₀)ᵐ f(z) has a removable singularity at z₀. What is the limit of (z - z₀)ᵐ f(z) as z approaches z₀?
15. The integral of a function around a closed contour is related to the residues inside the contour. This is the statement of which theorem?
16. What is the residue of f(z) = 1 / (z(z-2)²) at z = 2?
17. If f(z) has an essential singularity at z₀, the Laurent series expansion about z₀ has:
18. The Taylor series of f(z) = 1/(1-z) about z=0 is Σ zⁿ. What is the radius of convergence?
19. What is the condition for a function f(z) to be analytic in a domain D?
20. If a function f(z) has a pole of order m at z₀, its Laurent series expansion in the punctured neighborhood 0 < |z - z₀| < R will have:
21. What is the residue of f(z) = e^z / z at z = 0?
22. Consider the integral ∫_(-∞ to ∞) dx / (x² + 1). Which contour is typically used to evaluate this integral using residues?
23. The Cauchy principal value of an integral is often used when dealing with improper integrals. How does it relate to residues?
24. What is the residue of f(z) = z / (z - 1)(z + 2) at z = 1?
25. If lim_(z→z₀) (z - z₀)ᵐ f(z) = L, where L is a finite non-zero complex number, and lim_(z→z₀) (z - z₀)ᵐ⁺¹ f(z) = 0, then z₀ is a pole of order:
26. What is the order of the pole at z=0 for the function f(z) = 1/z²?
27. Which of the following is NOT a type of isolated singularity for a complex function?
28. The Taylor series expansion for e^z about z=0 is Σ[n=0 to ∞] zⁿ/n!. What is the radius of convergence for this series?
29. In the context of Taylor series, what does the term 'analytic' imply about a function?
30. What is the residue of f(z) = sin(z) / z³ at z = 0?
31. If f(z) has a removable singularity at z₀, what is its residue at z₀?
32. What is the residue of f(z) = 1 / (z² + 1) at z = i?
33. Consider the function f(z) = 1 / (z - a). What is the residue of f(z) at z = a?
34. If a function f(z) is analytic inside and on a simple closed contour C, except for a finite number of isolated singularities inside C, the Residue Theorem is applicable. What is the value of the integral ∫_C f(z) dz in this case?
35. For a pole z₀ of order m > 1, the residue can be calculated using the formula: Res(f, z₀) = (1 / (m-1)!) * lim_(z→z₀) d^(m-1)/dz^(m-1) [ (z - z₀)ᵐ f(z) ]. What is the value of m for a simple pole?
36. How can the residue at a simple pole z₀ of a function f(z) = P(z)/Q(z) be calculated if P(z₀) ≠ 0, Q(z₀) = 0, and Q'(z₀) ≠ 0?
37. The Residue Theorem states that ∫_C f(z) dz = 2πi * (Sum of residues of f(z) at poles inside C). What condition must the poles satisfy?
38. What is the residue of a function f(z) at an isolated singularity z₀?
39. A singularity z₀ is called a removable singularity if the principal part of the Laurent series expansion of f(z) about z₀ is:
40. If the principal part of the Laurent series of f(z) about z₀ contains infinitely many terms, what type of singularity is z₀?
41. If the principal part of the Laurent series of f(z) about z₀ contains only a finite number of terms, what type of singularity is z₀?
42. What is the significance of the principal part of a Laurent series in classifying singularities?
43. The coefficients c_n in the Laurent series expansion of f(z) about z₀ in an annulus 0 < |z - z₀| < R are given by the integral formula: c_n = (1/2πi) ∫_C f(ζ) / (ζ - z₀)ⁿ⁺¹ dζ. What is C in this formula?
44. A Laurent series expansion of a function f(z) about a point z₀ is given by Σ[n=-∞ to ∞] c_n (z - z₀)ⁿ. The part of the series with negative powers of (z - z₀) is called the:
45. Which type of series expansion is used to represent a function in an annulus (a region between two concentric circles)?
46. If a function f(z) has a removable singularity at z₀, how does its Taylor series behave around z₀?
47. For a function f(z) to be analytic at a point z₀ and in a disk |z - z₀| < R, its Taylor series expansion about z₀ converges to f(z) within that disk. What is the minimum value of R called?
48. The Taylor series expansion of a function f(z) about a point z₀ is given by Σ[n=0 to ∞] a_n (z - z₀)ⁿ. What is the formula for the coefficient a_n?
49. What is the fundamental theorem that allows a function to be represented by a power series in a neighborhood of a point?