Analytic mappings - conformality, arcs and closed curves, conformal mapping, linear fractional transformations, cross ratio and symmetry - Question Bank
1. Symmetry with respect to a circle or a line in the complex plane is preserved under:
2. If f(z) is conformal at z₀, then the angle between any two smooth curves passing through z₀ is preserved in magnitude and:
3. The cross-ratio of four points is invariant under the group of:
4. What is the image of the unit circle |z|=1 under the mapping f(z) = z + 1?
5. Which theorem states that if f(z) is analytic and non-constant in a region R, then f maps open sets to open sets?
6. The inverse of a linear fractional transformation f(z) = (az+b)/(cz+d) is given by:
7. What is the condition for a linear fractional transformation f(z) = (az+b)/(cz+d) to map the real axis to itself?
8. A mapping f(z) = u + iv is conformal at z₀ if and only if u and v satisfy the:
9. What is the argument of the angle of magnification at a point z₀ where f(z) is conformal?
10. What is the angle of magnification at a point z₀ where f(z) is conformal?
11. At z₀ = 0, the mapping f(z) = z² is:
12. The transformation f(z) = z² is conformal at z₀ if:
13. What is the image of the unit circle |z|=1 under the mapping f(z) = z + 1/z?
14. The condition for four points z₁, z₂, z₃, z₄ to be concyclic or collinear is that their cross-ratio is:
15. Which of the following is NOT a property of the cross-ratio (z₁, z₂, z₃, z₄)?
16. Consider the cross-ratio (z₁, z₂, z₃, z₄). If we permute z₃ and z₄, how does the cross-ratio change?
17. What are the fixed points of the transformation f(z) = 1/z?
18. A linear fractional transformation is conformal everywhere except possibly at:
19. A mapping is called 'sense-preserving' if it preserves:
20. What is the Jacobian of the transformation w = f(z) = u(x,y) + iv(x,y) at a point z₀ where f'(z₀) ≠ 0?
21. If f(z) is analytic and non-constant in a region R, then the image of any open set in R under f is:
22. What is the image of the upper half-plane Im(z) > 0 under the mapping f(z) = iz?
23. The set of all linear fractional transformations forms a group under:
24. The property that a linear fractional transformation preserves the cross-ratio is fundamental to:
25. If the cross-ratio (z₁, z₂, z₃, z₄) = 1, what can be said about the points z₁, z₂, z₃, z₄?
26. What is the cross-ratio of the points 0, 1, i, ∞?
27. If f(z) is a linear fractional transformation, then f(∞) is defined as:
28. How many points uniquely determine a linear fractional transformation?
29. A linear fractional transformation maps the extended complex plane (C ∪ {∞}) to:
30. What is the image of the real axis under the mapping f(z) = 1/z?
31. What is the image of the unit circle |z| = 1 under the mapping f(z) = 1/z?
32. The transformation f(z) = 1/z maps circles and lines to:
33. Which property is NOT preserved by a conformal mapping?
34. A mapping f(z) is conformal at z₀ if it is analytic at z₀ and:
35. Consider a conformal mapping f(z). If two curves C₁ and C₂ intersect at a point z₀ at an angle θ, what is the angle between their images f(C₁) and f(C₂) under the mapping?
36. What is a 'closed curve' in complex analysis?
37. What is meant by an 'arc' in the context of complex analysis?
38. The cross-ratio of four points on a circle or a line is always:
39. What is the value of the cross-ratio (z₁, z₂, z₃, z₄) if z₁, z₂, z₃ are collinear?
40. A key property of the cross-ratio is that it is invariant under:
41. What is the formula for the cross-ratio (z₁, z₂, z₃, z₄)?
42. The cross-ratio of four distinct complex numbers z₁, z₂, z₃, z₄ is denoted by:
43. What is the condition ad - bc ≠ 0 crucial for in a linear fractional transformation?
44. What is the general form of a linear fractional transformation?
45. A linear fractional transformation is also known as a:
46. Which of the following transformations is NOT a linear fractional transformation?
47. What does it mean for a mapping to be conformal at a point?
48. If a function f(z) is analytic in a region R, and f'(z) ≠ 0 for all z in R, then f(z) is conformal in R. What property does conformality preserve?
49. What is the primary condition for a function f(z) to be conformal at a point z₀?