9.2 Statement of the Riemann mapping theorem

Conformal mappings are transformations that preserve angles between curves while mapping complex domains. They're crucial in complex analysis, establishing one-to-one correspondences between points in original and mapped domains. These mappings are biholomorphic functions, both analytic and with non-zero derivatives.

The Riemann mapping theorem states conditions for conformal mappings between complex domains. It requires domains to be simply connected and not the entire complex plane. This theorem guarantees the existence of conformal mappings to canonical domains like the unit disk or upper half-plane.

Definition of conformal mapping

• Conformal mapping is a transformation that preserves angles between curves at their intersection points while mapping one complex domain to another
• Establishes a one-to-one correspondence between points in the original domain and the mapped domain
• Conformal mappings are biholomorphic functions, meaning they are complex-valued functions that are analytic (differentiable) and have a non-zero derivative at every point in their domain

Angle preservation

Top images from around the web for Angle preservation

• 

  complex analysis - Conformal Maps onto the Unit Disc in $\mathbb{C}$ - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

• 

  Understanding the slit plane and the complex $\sqrt{z}$ - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  complex analysis - Conformal Maps onto the Unit Disc in $\mathbb{C}$ - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

1 of 3

Top images from around the web for Angle preservation

• 

  complex analysis - Conformal Maps onto the Unit Disc in $\mathbb{C}$ - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

• 

  Understanding the slit plane and the complex $\sqrt{z}$ - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  complex analysis - Conformal Maps onto the Unit Disc in $\mathbb{C}$ - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

1 of 3

• The primary characteristic of conformal mappings is the preservation of angles between curves at their intersection points
• 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
• Angle preservation is a local property, meaning it holds true for infinitesimal regions around each point

One-to-one correspondence

• Conformal mappings establish 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
• One-to-one correspondence ensures that the mapping is invertible, meaning there exists an inverse mapping that can transform the mapped domain back to the original domain

Biholomorphic functions

• Conformal mappings are biholomorphic functions, which are complex-valued functions that satisfy two conditions:
  1. They are analytic (differentiable) at every point in their domain
  2. They have a non-zero derivative at every point in their domain
• Being analytic ensures that the function is smooth and well-behaved, while having a non-zero derivative guarantees that the mapping is locally invertible

Conditions for Riemann mapping theorem

• The Riemann mapping theorem states the conditions under which a conformal mapping exists between two complex domains
• Two key conditions must be satisfied for the theorem to apply:
  1. The domain must be simply connected
  2. The domain must not be the entire complex plane

Simply connected domains

• A domain is simply connected if any closed curve within the domain can be continuously deformed to a point without leaving the domain
• Intuitively, a simply connected domain has no holes or obstacles that would prevent a closed curve from being shrunk to a point
• Examples of simply connected domains include the unit disk, the upper half-plane, and any domain that can be obtained by removing a finite number of points from the complex plane

Domain not entire complex plane

• The Riemann mapping theorem does not apply when the domain is the entire complex plane
• This is because the complex plane is not simply connected, as any closed curve encircling the point at infinity cannot be continuously deformed to a point
• Conformal mappings between the entire complex plane and other domains are not guaranteed to exist

Existence of conformal mapping

• The Riemann mapping theorem guarantees the existence of a conformal mapping between any simply connected domain (other than the entire complex plane) and certain canonical domains
• Two commonly used canonical domains are the unit disk and the upper half-plane

Mapping to unit disk

• The Riemann mapping theorem ensures that for any simply connected domain $D$ (other than the entire complex plane), there exists a conformal mapping $f$ from $D$ to the unit disk $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$
• The mapping $f$ maps the domain $D$ onto the unit disk, preserving angles and establishing a one-to-one correspondence between points in $D$ and points in $\mathbb{D}$

Mapping to upper half-plane

• Similarly, the Riemann mapping theorem guarantees the existence of a conformal mapping $g$ from any simply connected domain $D$ (other than the entire complex plane) to the upper half-plane $\mathbb{H} = \{z \in \mathbb{C} : \text{Im}(z) > 0\}$
• The mapping $g$ maps the domain $D$ onto the upper half-plane, preserving angles and establishing a one-to-one correspondence between points in $D$ and points in $\mathbb{H}$

Uniqueness up to automorphism

• The conformal mapping guaranteed by the Riemann mapping theorem is unique up to composition with an automorphism of the canonical domain
• For the unit disk, the automorphisms are the Möbius transformations of the form $\phi(z) = e^{i\theta} \frac{z - a}{1 - \overline{a}z}$, where $|a| < 1$ and $\theta \in \mathbb{R}$
• For the upper half-plane, the automorphisms are the Möbius transformations of the form $\psi(z) = \frac{az + b}{cz + d}$, where $a, b, c, d \in \mathbb{R}$ and $ad - bc > 0$

Construction of conformal mapping

• While the Riemann mapping theorem guarantees the existence of conformal mappings, it does not provide an explicit formula for constructing them
• Several methods have been developed to construct conformal mappings for specific domains, including the Riemann mapping function, Schwarz-Christoffel mapping, and numerical methods

Riemann mapping function

• The Riemann mapping function is a constructive proof of the Riemann mapping theorem that provides an explicit formula for the conformal mapping from a simply connected domain to the unit disk
• The construction involves solving the Dirichlet problem for the Laplace equation with boundary conditions determined by the domain
• The resulting harmonic function is then used to define the conformal mapping

Schwarz-Christoffel mapping

• The Schwarz-Christoffel mapping is a conformal mapping that maps the upper half-plane onto a polygonal domain
• The mapping is given by an integral formula that depends on the vertices and interior angles of the polygon
• Schwarz-Christoffel mappings can be used to construct conformal mappings between polygonal domains and the upper half-plane or the unit disk

Numerical methods

• In many cases, explicit formulas for conformal mappings are not available, and numerical methods must be employed to approximate the mapping
• Numerical methods for constructing conformal mappings include:
  • Finite element methods, which discretize the domain and solve the Laplace equation numerically
  • Boundary integral methods, which reformulate the problem as an integral equation on the boundary of the domain
  • Iterative methods, such as the Schwarz-Christoffel toolbox, which use a combination of analytical and numerical techniques to approximate the mapping

Applications of Riemann mapping theorem

• The Riemann mapping theorem has numerous applications in various fields, including complex analysis, fluid dynamics, and electrostatics

Complex analysis

• In complex analysis, the Riemann mapping theorem is a fundamental result that underlies many other important theorems and techniques
• Conformal mappings are used to simplify complex analysis problems by transforming domains into more tractable forms (unit disk or upper half-plane)
• The theorem is also used in the study of univalent functions, which are analytic functions that are one-to-one in their domain

Fluid dynamics

• Conformal mappings are used in fluid dynamics to study the flow of ideal fluids (inviscid and irrotational) around obstacles
• By conformally mapping the domain exterior to the obstacle onto a simpler domain (half-plane or disk), the flow problem can be solved more easily
• The Joukowsky transform, a specific conformal mapping, is used to design airfoil shapes for aerodynamic applications

Electrostatics

• In electrostatics, conformal mappings are used to solve problems involving electric fields and potentials in two-dimensional domains
• The Laplace equation, which governs the electric potential in the absence of charges, is invariant under conformal mappings
• By conformally mapping the domain of interest onto a simpler domain, the electric field and potential can be determined more easily

Generalizations and extensions

• The Riemann mapping theorem has been generalized and extended in various ways to accommodate more complex situations and domains

Multiply connected domains

• The Riemann mapping theorem does not apply directly to multiply connected domains, which are domains with holes or obstacles
• However, the theorem can be generalized to certain classes of multiply connected domains, such as domains with a finite number of holes
• The generalization involves the use of the Koebe uniformization theorem, which states that any finitely connected domain can be conformally mapped onto a canonical multiply connected domain

Riemann surfaces

• Riemann surfaces are generalized complex domains that allow for multi-valued functions and branch points
• The uniformization theorem, a generalization of the Riemann mapping theorem, states that any simply connected Riemann surface is conformally equivalent to one of three canonical domains: the complex plane, the unit disk, or the Riemann sphere
• The uniformization theorem is a powerful tool in the study of algebraic curves and their associated Riemann surfaces

Uniformization theorem

• The uniformization theorem is a far-reaching generalization of the Riemann mapping theorem that applies to arbitrary Riemann surfaces
• It states that any simply connected Riemann surface is conformally equivalent to one of three canonical domains: the complex plane, the unit disk, or the Riemann sphere
• The uniformization theorem has deep connections to various areas of mathematics, including complex analysis, algebraic geometry, and group theory

Historical context

• The Riemann mapping theorem is named after Bernhard Riemann, a German mathematician who made significant contributions to complex analysis and differential geometry

Bernhard Riemann

• Bernhard Riemann (1826-1866) was a German mathematician who introduced many fundamental concepts in complex analysis and differential geometry
• Riemann's work on complex analysis, particularly his theory of Riemann surfaces and the Riemann mapping theorem, laid the foundation for the modern understanding of complex analysis
• Riemann's contributions to mathematics also include the Riemann zeta function, Riemannian geometry, and the Riemann hypothesis, one of the most famous unsolved problems in mathematics

Development of complex analysis

• The Riemann mapping theorem played a crucial role in the development of complex analysis as a distinct branch of mathematics
• Prior to Riemann, complex analysis was primarily focused on the study of complex-valued functions and their properties
• Riemann's work on the mapping theorem and Riemann surfaces shifted the focus to the geometric aspects of complex analysis and the study of conformal mappings
• The theorem opened up new avenues for research and led to the development of many important concepts and techniques in complex analysis

Impact on mathematics

• The Riemann mapping theorem has had a profound impact on various branches of mathematics, including complex analysis, differential geometry, and algebraic geometry
• The theorem and its generalizations have been used to solve a wide range of problems in these fields, from the classification of Riemann surfaces to the study of minimal surfaces
• The ideas and techniques introduced by Riemann and the mapping theorem continue to inspire new research and discoveries in mathematics to this day