Key Conformal Mapping Techniques

Why This Matters

Conformal mappings are the power tools of complex analysis—they let you transform complicated domains into simpler ones while preserving the local geometry that makes analytic functions so well-behaved. You're being tested on your ability to recognize which transformation fits which problem, understand why angle preservation matters, and apply these techniques to boundary value problems, fluid flow, and geometric function theory. The Riemann mapping theorem tells us that conformal equivalence is surprisingly common, but the real skill is knowing how to construct the actual map.

Don't just memorize formulas—know what geometric action each transformation performs and when to deploy it. Can you explain why Möbius transformations are the "rigid motions" of the extended complex plane? Do you understand why Schwarz-Christoffel mappings involve those specific integral forms? These conceptual connections are what separate strong exam performance from mere formula recall.

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Foundational Transformations: The Building Blocks

These elementary mappings form the basis for constructing more complex conformal maps. Master these first—they appear in nearly every application.

Power Functions

• Maps sectors to half-planes—the function $f(z) = z^n$ multiplies angles by $n$, making it ideal for "opening up" wedge-shaped domains
• Branch cuts required when $n$ is non-integer; the function becomes multi-valued, requiring careful choice of a principal branch
• Local behavior near zeros of analytic functions resembles $z^n$, making power functions essential for understanding singularity structure

Exponential and Logarithmic Mappings

• Exponential $f(z) = e^z$ maps horizontal strips to annuli—a strip of height $2\pi$ becomes the punctured plane, demonstrating the periodic nature of $e^z$
• Logarithm inverts this action, mapping annular regions back to strips; requires branch cut selection since $\log(z)$ is inherently multi-valued
• Periodicity and covering maps are central applications—these functions reveal how the complex plane "wraps around" itself

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Möbius Transformations: The Automorphisms of the Sphere

Möbius transformations are the conformal self-maps of the extended complex plane $\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$. They form a group under composition, and every conformal automorphism of the Riemann sphere has this form.

Möbius Transformations

• General form $f(z) = \frac{az + b}{cz + d}$ with $ad - bc \neq 0$; the nonzero determinant condition ensures the map is invertible
• Circle-preserving property—maps circles and lines (viewed as circles through $\infty$) to circles and lines, making them ideal for problems involving circular boundaries
• Uniquely determined by three points—if you know where three points map, the entire transformation is fixed; this is the three-point theorem

Bilinear Transformations

• Same as Möbius transformations—the terms are synonymous; "bilinear" emphasizes the linear structure in both numerator and denominator
• Disk automorphisms have the special form $f(z) = e^{i\theta}\frac{z - a}{1 - \bar{a}z}$ for $|a| < 1$, mapping the unit disk to itself
• Cross-ratio preservation is the key invariant—$\frac{(z_1 - z_3)(z_2 - z_4)}{(z_1 - z_4)(z_2 - z_3)}$ remains unchanged under any Möbius transformation

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Polygon Mappings: Schwarz-Christoffel and Applications

When your target domain has straight edges, Schwarz-Christoffel is your go-to technique. The formula encodes the polygon's geometry directly into the mapping's derivative.

Schwarz-Christoffel Transformations

• Maps upper half-plane $\mathbb{H}$ to polygonal regions via $f(z) = A\int^z \prod_{k=1}^{n}(\zeta - x_k)^{\alpha_k - 1}d\zeta + B$, where $x_k$ are prevertices and $\alpha_k \pi$ are interior angles
• Derivative encodes turning angles—the exponents $\alpha_k - 1$ determine how much the boundary "turns" at each vertex; this is why the formula works
• Applications in potential theory include electrostatics in polygonal conductors and groundwater flow in angular aquifers

Joukowski Transformation

• Formula $f(z) = \frac{1}{2}\left(z + \frac{1}{z}\right)$ maps circles passing through $z = \pm 1$ to airfoil shapes; circles centered at origin become ellipses
• Critical points at $z = \pm 1$ where conformality fails—these map to the sharp trailing edge of the airfoil, which is physically significant
• Aerodynamic applications made this transformation famous; potential flow around a cylinder transforms to flow around a wing

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Theoretical Foundations: Existence and Structure

These results tell us when conformal maps exist and what properties they must have—essential for understanding the big picture.

Riemann Mapping Theorem

• Any simply connected proper subset of $\mathbb{C}$ is conformally equivalent to the unit disk—this is arguably the most important theorem in geometric function theory
• Non-constructive existence proof—the theorem guarantees a map exists but doesn't tell you how to find it; that's where explicit techniques become essential
• Uniqueness up to disk automorphisms—once you fix a point and a direction, the map is unique; three real parameters determine it completely

Preservation of Angles Under Conformal Mappings

• Angle preservation is equivalent to $f'(z) \neq 0$—at points where the derivative vanishes, angles get multiplied by the order of the zero
• Local shape preservation means infinitesimal circles map to infinitesimal circles; this is why conformal maps are useful for local geometric analysis
• Physical significance in fluid dynamics (streamlines meet boundaries at correct angles) and electrostatics (field lines remain orthogonal to equipotentials)

Inverse Transformations

• Inverse of a conformal map is conformal—if $f$ is analytic with $f'(z) \neq 0$, then $f^{-1}$ exists locally and is also analytic
• Inverse function theorem guarantees this; the derivative of the inverse is $\frac{1}{f'(f^{-1}(w))}$
• Practical importance for boundary value problems—solve in the simple domain, then map the solution back to the original region

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Quick Reference Table

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Self-Check Questions

1. Both power functions and exponential mappings involve multi-valuedness. What is the key geometric difference in how they transform domains?

2. You need to map the upper half-plane to a rectangle. Which technique would you use, and what information about the rectangle determines the mapping?

3. Compare the Joukowski transformation and a general Schwarz-Christoffel mapping: what geometric feature do both produce, and why is Joukowski preferred for airfoils?

4. The Riemann mapping theorem guarantees a conformal map from any simply connected domain to the unit disk. Why doesn't this make explicit mapping techniques unnecessary? (Think about what the theorem does and doesn't provide.)

5. If $f(z)$ is conformal and $f'(z_0) = 0$, what happens to angles at $z_0$? Give an example of a mapping where this occurs and explain the geometric consequence.