PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Analysis → Complex Analysis ↓


Conformal Mappings


Conformal mappings are a very interesting topic in complex analysis, especially for their applications in various fields of science and engineering. This article will cover the essential concepts, properties, applications, and examples of conformal mapping, enriched with numerous illustrative examples and mathematical expressions to provide an in-depth understanding of the topic.

Introduction to conformal mapping

In the simplest terms, a conformal mapping (or conformal transformation) is a function that preserves angles. Specifically, it preserves the angles between curves at any given point, although the curves themselves may be stretched, shrunk, or deformed. These mappings are defined on domains within the complex plane and appear frequently in complex analysis.

Let us consider a function f defined on a domain D in the complex plane. If this function is holomorphic (complex differentiable at every point) and has a non-zero derivative everywhere in D, then it can be considered conformal. Mathematically:

f : D → ℂ

f is conformal if:

1. f is holomorphic on D 2. f' ≠ 0 for all points in D

Mathematical formulation

Consider two curves given by parametric equations in the complex plane that intersect at the point z_0. The function f maps these two curves to new curves, preserving the angle between them. If the original curves are described by z(t) and w(t) near z_0, then their image curves under f are f(z(t)) and f(w(t)).

If the angle between the original curves is θ, then the angle between the image curves will also be θ, which confirms that f is conformal at z_0.

Properties of conformal mapping

Conformal mapping has several important properties:

Angle protection

By definition, conformal mapping preserves the angle between curves. For two intersecting curves, the angle measured in their tangent directions remains unchanged under the mapping. This property is essential in applications that require the maintenance of geometric shapes and configurations.

Local similarity

Conformal mappings not only preserve angles, but also act locally like similarity mappings, which means that locally, they look almost like a combination of translation, rotation, and scaling.

Preservation of orientation

Conformal maps preserve orientation if the mapping process maintains the clockwise or counterclockwise order of the points.

Riemann mapping theorem

One of the deepest results related to conformal mapping is the Riemann mapping theorem. It states that if you have a non-empty open simplicial-adjoint proper subset of the complex plane, then there exists a binary conformal map between this subset and the open unit disk in the complex plane.

Applications of conformal mapping

Conformal mapping is used in a variety of scientific fields, including:

Fluid dynamics

In fluid dynamics, conformal mapping is leveraged to simplify problems involving potential flow around objects by transforming boundaries into more tractable shapes.

Engineering and CAD

In engineering, particularly in computer-aided design (CAD), conformal mappings are used to transform geometry while preserving angles, making them valuable in areas requiring shape optimization.

Electrostatics and magnetostatics

In electrostatics and magnetostatics, potential fields can often be analyzed using conformal mapping, in which complex domains are transformed into simpler forms while maintaining the properties of the field.

Examples of conformal mapping

To illustrate the concept of conformal mapping, let us look at some classic examples in the complex plane.

Example 1: Linear mapping

Consider the function f(z) = az + b, where a and b are complex constants and a ≠ 0 This simple transformation is a conformal mapping because it represents a combination of scaling (by |a|), rotation (by arg(a)), and translation (by b).

Change details:

Scaling: The magnitude of a object scales distances.
Rotation: The argument to a rotates the points.
Translation: b moves all points by a constant vector.

Example 2: Multiplicative inverse

The mapping given by f(z) = 1/z is conformal on the complex plane except at the origin. This function exhibits interesting properties; it inverts points radially and reflects them across the real axis.

Geometric effects:

Points closer to the origin are moved away, maintaining their angle of incidence with lines passing through the origin. A circle centered at the origin is transformed into another circle (which does not pass through the origin).

Example 3: Möbius transformation

The Möbius transformation takes the form:

f(z) = (az + b) / (cz + d)

where ad - bc ≠ 0 These are more general than linear transformations and exhibit interesting properties such as preserving angles and mapping circles to circles or lines.

Example calculation:

Assume a = 1, b = 0, c = 1, and d = 1. The function becomes:

f(z) = (z) / (z + 1)

It maps the complex plane minus -1 onto the unit circle, leaving the point 1.

Conclusion

Conformal mapping is a fascinating and useful concept in understanding and working with complex variables. Despite the distortions they introduce, these functions crucially preserve the angles between curves, providing a means of transforming complex domains into more analytically manageable forms. From fluid mechanics to electromagnetic field theory, the impact of conformal mapping is widely felt, echoing the rich interconnection of mathematics with physics.


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Conformal Mappings

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