Constructing a Conformal Equivalence

We aim to construct a conformal equivalence f between the “angle”
A = { z ∈ ℂ | z ≠ 0, 0 < arg(z) < π/3 }
and the unit disk
𝔻 = { w ∈ ℂ | |w| < 1 }.

1. Map the angle to a horizontal strip

Use the logarithm map:

z → w₁ = log(z) = ln|z| + i·arg(z),

where z ∈ A. Under this map:

• The boundary ray arg(z) = 0 maps to the real axis (Im(w₁) = 0).
• The boundary ray arg(z) = π/3 maps to the line Im(w₁) = π/3.
• The region A maps to the horizontal strip
  S = { w₁ ∈ ℂ | 0 < Im(w₁) < π/3 }.

2. Map the strip to the upper half-plane

The exponential stretching map:

w₁ → w₂ = exp(πi·w₁ / 3)

transforms the strip S to the upper half-plane
H = { w₂ ∈ ℂ | Im(w₂) > 0 }.

3. Map the upper half-plane to the unit disk

The Möbius transformation:

w₂ → w₃ = (w₂ – i) / (w₂ + i)

is a conformal equivalence between the upper half-plane H and the unit disk 𝔻.

4. Combine the maps

The full conformal map f: A → 𝔻 is the composition:

f(z) = [(exp(πi·log(z) / 3) – i) / (exp(πi·log(z) / 3) + i)].

5. Simplified expression

Expanding the steps, the final form of f(z) is:

f(z) = [(exp(πi(ln|z| + i·arg(z)) / 3) – i) / (exp(πi(ln|z| + i·arg(z)) / 3) + i)].

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