Questions: Conformal Mappings

Question 1

The function f(z) = z³ has a critical point at z = 0, where f'(0) = 0. What happens to angles at this point under the map?

• A. Angles are preserved, since z³ is holomorphic everywhere
• B. Angles are doubled, since z³ is a degree-3 map
• C. Angles are tripled — multiplied by 3 — because the zero of f' has order 2 (k=2)
• D. The map is not defined at z = 0 and angles are undefined there

Answer: (C)

At a zero of f' of order k, the map locally behaves like z^{k+1} and multiplies all angles by k+1. For f(z) = z³, we have f'(z) = 3z², which has a zero of order k=2 at z=0. So angles at z=0 are multiplied by k+1 = 3. Being holomorphic everywhere is necessary but not sufficient for conformality — you also need f'(z) ≠ 0. Option A conflates holomorphicity with conformality, a common mistake; the key additional requirement is the nonvanishing derivative.

Question 2

Why is Riemann's mapping theorem useful for solving boundary value problems on oddly shaped regions?

• A. It guarantees that any region can be mapped to another region with the same area, making computations equivalent
• B. It provides an explicit formula for the conformal map, which can be used to compute solutions directly
• C. It allows you to transform a problem on a complicated region to the unit disk, where solutions are known, and then pull back through the map — because Laplace's equation is preserved under conformal coordinates
• D. It shows that any harmonic function on a complicated region can be extended to the complex plane

Answer: (C)

The key facts working together are: (1) any simply connected region can be conformally mapped to the unit disk (Riemann's theorem), and (2) Laplace's equation ∇²u = 0 is preserved under conformal coordinate changes. This means a harmonic function on the disk pulls back to a harmonic function on the original region. Since Poisson's formula gives explicit solutions on the disk, you can solve the problem there and transfer the solution back. Riemann's theorem does not provide the explicit map — finding it requires additional work — but its existence guarantees the strategy is always available.

Question 3

Laplace's equation is preserved under conformal changes of coordinates, which is what makes conformal maps useful for solving potential problems.

• T. True
• F. False

Answer: True

This is the critical analytical fact underlying all applications of conformal maps to physics and engineering. If u is harmonic (∇²u = 0) in the w-plane, and w = f(z) is conformal, then u ∘ f is harmonic in the z-plane. This means you can transform a boundary value problem for the Laplacian from a complicated domain to a simple one (like the unit disk), solve it there, and pull the solution back. Without this preservation property, conformal maps would be geometrically interesting but analytically useless for differential equations.

Question 4

Orientation-reversing maps, like complex conjugation f(z) = z̄, are not angle-preserving and therefore cannot be conformal.

• T. True
• F. False

Answer: False

Orientation-reversing maps do preserve angles — but they reverse orientation. Complex conjugation maps the angle θ to −θ, so the absolute angle between two curves is preserved even though their signed orientation flips. A conformal map in the strict sense is angle-preserving and orientation-preserving, which is why we require holomorphicity rather than just angle-preservation. But the statement as written — that orientation-reversing maps cannot preserve angles — is false. This is noted in the Common Misconceptions: 'orientation-reversing maps (like conjugation) also preserve angles.'

Question 5

Why is the condition f'(z₀) ≠ 0 essential for conformality, and what goes wrong geometrically when f'(z₀) = 0?

Model answer: A holomorphic function acts near z₀ by multiplying all displacement vectors by f'(z₀). When f'(z₀) ≠ 0, this means every direction is rotated by the same angle arg(f'(z₀)), so the angle between any two tangent vectors is unchanged. When f'(z₀) = 0, the first-order behavior vanishes and the map locally looks like z^{k+1} near the critical point — which multiplies angles by k+1 rather than preserving them. Geometrically, the map 'folds' the plane at that point: distinct directions get mapped to the same direction, angles between curves are distorted.

Intuitively, conformality requires that the linearization of f at z₀ (given by the derivative) be a nonzero complex number — a scaled rotation. Zero derivative means the linearization is the zero map, which collapses all directions. The map then requires looking at higher-order terms, which are no longer pure rotations but power maps that multiply angles. Critical points are geometrically the places where the conformal structure breaks down, and recognizing them is essential when using conformal maps in practice.