All Modules
University Math Lab: Systems & Structures FREE
Loading simulation...
Conformal Mappings
Learning Objectives
- Understand angle-preserving maps
- Apply Möbius transformations
- Map between domains
Key Concepts
Conformal Map
Analytic function with non-zero derivative — preserves angles
Möbius Transformation
f(z) = (az + b)/(cz + d) where ad - bc ≠ 0
Theory
**Conformal map:** A function f(z) is conformal at z₀ if f'(z₀) ≠ 0.
- Preserves angles between curves
- Preserves orientation
**Common conformal maps:**
- f(z) = z² (doubles angles at origin)
- f(z) = eᶻ (maps strips to sectors)
- f(z) = (z-1)/(z+1) (Möbius transformation)
**Möbius Transformation:** f(z) = (az+b)/(cz+d), maps circles/lines to circles/lines.
Examples
f(z) = 1/z. Image of unit circle?
Solution: Unit circle maps to itself (|1/z| = 1 when |z| = 1)