7.1 Conformal mapping

Conformal mapping transforms complex domains while preserving angles. It's a powerful tool in complex analysis, connecting geometric properties to analytic functions. Understanding conformal mapping helps solve problems in physics and engineering by simplifying complex geometries.

Key properties include angle preservation, one-to-one correspondence, and orientation preservation. These features allow us to study intricate domains by mapping them to simpler ones, maintaining essential geometric characteristics. Conformal mapping bridges theory and practical applications in various fields.

Definition of conformal mapping

• Conformal mapping is a transformation that preserves angles between curves in the complex plane
• It maps a domain in the complex plane to another domain while maintaining the local geometry
• Conformal maps are essential tools in complex analysis for studying the behavior of analytic functions and solving various problems in physics and engineering

Preservation of angles

• One of the key properties of conformal mapping is the preservation of angles between curves
• If two curves intersect at a certain angle in the original domain, their mapped counterparts will intersect at the same angle in the transformed domain
• This property allows for the study of complex geometries by mapping them to simpler domains while maintaining the essential geometric features

Local properties of conformal maps

Conformality at a point

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• A function is conformal at a point if it preserves angles between curves passing through that point
• The conformality at a point is determined by the behavior of the function in an infinitesimal neighborhood around the point
• The complex derivative of the function at the point must be non-zero for the mapping to be conformal

Conformality in a neighborhood

• A function is conformal in a neighborhood if it is conformal at every point within that neighborhood
• The conformality in a neighborhood ensures that the mapping preserves angles not only at a single point but also in the surrounding region
• This property allows for the study of local geometric properties of the mapped domain

Global properties of conformal maps

One-to-one correspondence

• A conformal mapping establishes a one-to-one correspondence between points in the original domain and the mapped domain
• Each point in the original domain is mapped to a unique point in the transformed domain, and vice versa
• This property ensures that the mapping is invertible, allowing for the study of the inverse mapping and its properties

Orientation preservation

• Conformal mappings preserve the orientation of curves in the complex plane
• If a curve is oriented counterclockwise in the original domain, its mapped counterpart will also be oriented counterclockwise in the transformed domain
• This property is crucial for studying the global geometry of the mapped domain and its topological properties

Analytic functions and conformal mapping

Cauchy-Riemann equations

• For a complex function $f(z) = u(x, y) + iv(x, y)$ to be conformal, it must satisfy the Cauchy-Riemann equations:
  • $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$
  • $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
• These equations ensure that the function is analytic (differentiable) in the complex plane
• Analytic functions have many useful properties, such as being infinitely differentiable and satisfying the maximum modulus principle

Harmonic functions

• The real and imaginary parts of an analytic function, $u(x, y)$ and $v(x, y)$, are harmonic functions
• Harmonic functions satisfy Laplace's equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ and $\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0$
• The properties of harmonic functions, such as the mean value property and the maximum principle, are useful in studying the behavior of conformal mappings

Examples of conformal mappings

Linear fractional transformations

• Linear fractional transformations, also known as Möbius transformations, are conformal mappings of the form $f(z) = \frac{az + b}{cz + d}$, where $ad - bc \neq 0$
• They map circles and lines to circles and lines, making them useful for studying the geometry of the complex plane
• Examples include the inversion map $f(z) = \frac{1}{z}$ and the rotation map $f(z) = e^{i\theta}z$

Exponential function

• The exponential function $f(z) = e^z$ is a conformal mapping from the complex plane to itself
• It maps horizontal lines to circles centered at the origin and vertical lines to rays emanating from the origin
• The periodicity of the exponential function, $e^{z + 2\pi i} = e^z$, leads to its use in studying periodic phenomena

Logarithmic function

• The logarithmic function $f(z) = \log z$ is the inverse of the exponential function and is also a conformal mapping
• It maps the right half-plane to a horizontal strip and the unit circle to a vertical line segment
• The logarithmic function is multi-valued, requiring a branch cut to define a single-valued branch

Power functions

• Power functions of the form $f(z) = z^n$, where $n$ is a non-zero complex number, are conformal mappings
• They map circles and lines to algebraic curves, depending on the value of $n$
• Examples include the square root function $f(z) = \sqrt{z}$ and the cube root function $f(z) = \sqrt[3]{z}$

Conformal mapping of basic regions

Upper half-plane

• The upper half-plane, defined by $\{z \in \mathbb{C} : \text{Im}(z) > 0\}$, is a fundamental domain in complex analysis
• Conformal mappings can be used to map the upper half-plane to other domains, such as the unit disk or the infinite strip
• The Cayley transform $f(z) = \frac{z - i}{z + i}$ maps the upper half-plane to the unit disk

Unit disk

• The unit disk, defined by $\{z \in \mathbb{C} : |z| < 1\}$, is another important domain in complex analysis
• Conformal mappings can be used to map the unit disk to other domains, such as the upper half-plane or the infinite strip
• The Möbius transformation $f(z) = \frac{z - a}{1 - \bar{a}z}$, where $|a| < 1$, maps the unit disk to itself

Infinite strip

• The infinite strip, defined by $\{z \in \mathbb{C} : 0 < \text{Im}(z) < \pi\}$, is a domain often encountered in applications
• Conformal mappings can be used to map the infinite strip to other domains, such as the upper half-plane or the unit disk
• The exponential function $f(z) = e^z$ maps the infinite strip to the right half-plane

Applications of conformal mapping

Fluid dynamics

• Conformal mapping is used in fluid dynamics to study the flow of incompressible fluids around obstacles
• By mapping the flow domain to a simpler geometry, such as the upper half-plane, the velocity potential and stream function can be determined more easily
• The Joukowsky transform $f(z) = z + \frac{1}{z}$ is used to map the flow around a circular cylinder to the flow around an airfoil

Electrostatics

• Conformal mapping is applied in electrostatics to solve problems involving the distribution of electric potential and field in complex geometries
• By mapping the problem domain to a simpler geometry, such as the upper half-plane, the Laplace equation for the electric potential can be solved more easily
• The Schwarz-Christoffel transformation is used to map polygonal domains to the upper half-plane

Heat conduction

• Conformal mapping is used in heat conduction to study the temperature distribution in bodies with complex shapes
• By mapping the heat conduction domain to a simpler geometry, such as the unit disk, the heat equation can be solved more easily
• The Möbius transformation $f(z) = \frac{z - a}{1 - \bar{a}z}$ is used to map the unit disk to itself, which can be useful in studying radially symmetric heat conduction problems

Conformal mapping vs other transformations

Isometric vs conformal maps

• Isometric maps, also known as distance-preserving maps, preserve the distances between points in the domain
• Conformal maps, on the other hand, preserve angles between curves but may not preserve distances
• While isometric maps are more restrictive, conformal maps offer more flexibility in studying complex geometries

Quasi-conformal maps

• Quasi-conformal maps are a generalization of conformal maps that allow for bounded distortion of angles
• They are useful in situations where strict conformality is not required or cannot be achieved
• Quasi-conformal maps have applications in various fields, such as geometry, dynamics, and topology

Numerical methods for conformal mapping

Schwarz-Christoffel transformation

• The Schwarz-Christoffel transformation is a conformal mapping that maps the upper half-plane to the interior of a polygon
• It is given by the formula $f(z) = A \int_{z_0}^z \prod_{k=1}^n (w - z_k)^{-\alpha_k/\pi} dw + B$, where $z_k$ are the prevertices and $\alpha_k$ are the interior angles of the polygon
• Numerical methods, such as the Schwarz-Christoffel toolbox in MATLAB, are used to compute the parameters of the transformation and evaluate the mapping

Boundary element method

• The boundary element method is a numerical technique for solving boundary value problems in complex analysis
• It involves discretizing the boundary of the domain into elements and solving a system of integral equations to determine the boundary values of the analytic function
• The boundary element method can be used to compute conformal mappings by solving the appropriate boundary value problem, such as the Dirichlet problem or the Neumann problem