Analytic mappings - conformality, arcs and closed curves, conformal mapping, linear fractional transformations, cross ratio and symmetry - One Line Questions

1. What is the formula for the cross-ratio (z₁, z₂, z₃, z₄)? — ((z₁-z₃)(z₂-z₄)) / ((z₁-z₄)(z₂-z₃))
2. The cross-ratio of four distinct complex numbers z₁, z₂, z₃, z₄ is denoted by: — [z₁, z₂, z₃, z₄]
3. What is the Jacobian of the transformation w = f(z) = u(x,y) + iv(x,y) at a point z₀ where f'(z₀) ≠ 0? — |f'(z₀)|²
4. What is the value of the cross-ratio (z₁, z₂, z₃, z₄) if z₁, z₂, z₃ are collinear? — ∞
5. If f(z) is a linear fractional transformation, then f(∞) is defined as: — a/c (if c ≠ 0)
6. What are the fixed points of the transformation f(z) = 1/z? — 1 and -1
7. What is the cross-ratio of the points 0, 1, i, ∞? — i
8. What is meant by an 'arc' in the context of complex analysis? — A continuous path that is not necessarily closed.
9. If f(z) is analytic and non-constant in a region R, then the image of any open set in R under f is: — An open set
10. What is a 'closed curve' in complex analysis? — A curve where the starting point and ending point are the same.
11. The cross-ratio of four points on a circle or a line is always: — A real number
12. What is the condition for a linear fractional transformation f(z) = (az+b)/(cz+d) to map the real axis to itself? — a, b, c, d are real
13. The set of all linear fractional transformations forms a group under: — Composition
14. Symmetry with respect to a circle or a line in the complex plane is preserved under: — Linear fractional transformations
15. A linear fractional transformation is also known as a: — Möbius transformation
16. Which theorem states that if f(z) is analytic and non-constant in a region R, then f maps open sets to open sets? — Open Mapping Theorem
17. The transformation f(z) = 1/z maps circles and lines to: — Circles or lines
18. At z₀ = 0, the mapping f(z) = z² is: — Not conformal
19. A mapping is called 'sense-preserving' if it preserves: — The orientation of angles
20. 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? — Angles between intersecting curves
21. What is the primary condition for a function f(z) to be conformal at a point z₀? — f'(z₀) ≠ 0
22. A mapping f(z) is conformal at z₀ if it is analytic at z₀ and: — f'(z₀) ≠ 0
23. Which of the following transformations is NOT a linear fractional transformation? — f(z) = z²
24. What is the general form of a linear fractional transformation? — f(z) = (az + b) / (cz + d), where ad - bc ≠ 0
25. The inverse of a linear fractional transformation f(z) = (az+b)/(cz+d) is given by: — f⁻¹(w) = (-dw+b)/(cw-a)
26. The cross-ratio of four points is invariant under the group of: — Linear fractional transformations
27. What is the condition ad - bc ≠ 0 crucial for in a linear fractional transformation? — It ensures the transformation is invertible.
28. Which of the following is NOT a property of the cross-ratio (z₁, z₂, z₃, z₄)? — It is zero if any two of the points coincide.
29. What does it mean for a mapping to be conformal at a point? — It preserves angles but not necessarily distances.
30. Consider the cross-ratio (z₁, z₂, z₃, z₄). If we permute z₃ and z₄, how does the cross-ratio change? — It is inverted (1/value).
31. A mapping f(z) = u + iv is conformal at z₀ if and only if u and v satisfy the: — Cauchy-Riemann equations
32. If f(z) is conformal at z₀, then the angle between any two smooth curves passing through z₀ is preserved in magnitude and: — Sense
33. The property that a linear fractional transformation preserves the cross-ratio is fundamental to: — Mapping circles and lines to circles and lines
34. The condition for four points z₁, z₂, z₃, z₄ to be concyclic or collinear is that their cross-ratio is: — Real
35. What is the angle of magnification at a point z₀ where f(z) is conformal? — |f'(z₀)|
36. What is the argument of the angle of magnification at a point z₀ where f(z) is conformal? — arg(f'(z₀))
37. Which property is NOT preserved by a conformal mapping? — The area of a region.
38. A linear fractional transformation maps the extended complex plane (C ∪ {∞}) to: — The extended complex plane (C ∪ {∞})
39. What is the image of the real axis under the mapping f(z) = 1/z? — The real axis
40. What is the image of the unit circle |z|=1 under the mapping f(z) = z + 1/z? — The real axis
41. What is the image of the unit circle |z| = 1 under the mapping f(z) = 1/z? — The unit circle |w| = 1
42. What is the image of the unit circle |z|=1 under the mapping f(z) = z + 1? — The circle |w-1|=1
43. What is the image of the upper half-plane Im(z) > 0 under the mapping f(z) = iz? — The lower half-plane Im(w) < 0
44. If the cross-ratio (z₁, z₂, z₃, z₄) = 1, what can be said about the points z₁, z₂, z₃, z₄? — They are not necessarily related in any specific geometric way.
45. A key property of the cross-ratio is that it is invariant under: — Linear Fractional Transformations
46. How many points uniquely determine a linear fractional transformation? — Three
47. A linear fractional transformation is conformal everywhere except possibly at: — The poles of the transformation
48. The transformation f(z) = z² is conformal at z₀ if: — z₀ ≠ 0
49. 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? — θ
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