Conformal Mapping
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📖Definition
A conformal mapping preserves angles. Analytic functions are conformal where f'(z) ≠ 0. It transforms complex regions to simpler ones.
📐Formulas
w = f(z), f'(z₀) ≠ 0
Condition for conformality
w = (az + b)/(cz + d), ad - bc ≠ 0
Möbius transformation (LFT)
w = e^z
Strip to wedge
w = (z - i)/(z + i)
Cayley transform (half-plane → unit disk)
✏️Examples
예제 1
Describe how w = z² maps first quadrant.
예제 2
Explain why Möbius maps circles to circles.
📜History
Discovered by: Bernhard Riemann (1851)
Riemann Mapping Theorem: Simply connected regions are conformally equivalent to the unit disk.
⚡Applications
Fluid Mechanics
Airfoil flow (Joukowski)
Electromagnetism
Electrostatic problems
Cartography
Map projections
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