Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Complex Analysis → Analytic Functions ↓


Conformal Mappings


Introduction

Conformal mappings are a fundamental concept in complex analysis, a branch of mathematics that studies functions of complex numbers. These mappings have the magical ability to preserve angles and the shape of infinitesimal figures, making them incredibly useful in various fields such as engineering, physics, and computer science. In this article, we will explore conformal mappings, look at their properties, examples, and applications. We will discuss this topic in depth with simple language and illustrative examples to make it accessible to undergraduate-level learners.

Understanding complex numbers

Before diving into conformal mapping, it is essential to have a basic understanding of complex numbers. A complex number has a real part and an imaginary part, and is usually represented as z = x + yi, where x and y are real numbers, and i is the square root of -1.

z = x + yi

Complex numbers can be visualized in the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

What is conformal mapping?

A conformal mapping is a function that locally preserves angles. This means that if you have two intersecting curves in the complex plane, a conformal mapping will transform these curves into new curves that still intersect at the same angle. However, it is important to note that while angles are preserved, conformal mappings can distort the shape.

Conformal mappings are typically analytic functions, which are complex functions that are differentiable at every point in their domain. The requirement for differentiability is important because it ensures that the function behaves well enough to preserve angles.

Properties of conformal mapping

Several key properties define conformal mapping:

  • Angle preservation: As mentioned, conformal mapping preserves the angles between intersecting curves in the domain.
  • Analyticity: Conformal mappings are usually analytic functions.
  • Local scale factor: Angles are preserved, but shapes may change in size. At any point, there is a local scale factor that determines how lengths are increased or decreased.
  • Open mapping: If a function is conformal and non-constant, then it maps open sets to open sets.

Visualization of conformal mapping

Let us consider a simple example. The function f(z) = z^2 is a common example to illustrate conformal mapping.

f(z) = z^2

Example: Consider two intersecting lines in the complex plane. Applying the mapping f(z) = z^2, you will get the parabolas in the image. Despite the transformation, the angles at the intersection are preserved.

Some important conformal maps

Identity map

The simplest of all mappings is the identity map: f(z) = z. This map leaves the complex plane unchanged, and, thus, trivially preserves angles.

f(z) = z

Exponential map

The exponential function, f(z) = e^z, is conformal everywhere in the complex plane. This function transforms lines and some curves in the complex plane into other curves, such as spirals.

f(z) = e^z

Logarithmic map

The logarithmic function, f(z) = log(z), is another example, although it is not defined on the whole complex plane because of branch cuts. It is mainly used for mapping annular fields.

f(z) = log(z)

Applications of conformal mapping

Conformal mappings are applied in a variety of fields. In fluid dynamics, they help simplify complex flow problems. In cartography, they are used to preserve angles in a map projection, although the size and scale may vary. Another application is in electrical engineering, particularly to design circuits that require specific path layouts.

Example: Fluid dynamics

In fluid dynamics, conformal mapping helps visualize and calculate potential flows around objects. By mapping a complex flow pattern into a simpler one while preserving important features such as flow angle, engineers can analyze and design efficient systems.

Example: Electrical engineering

In electrical engineering, conformal mapping can optimize the paths of power lines and reduce losses in transmission. By carefully mapping the physical layout of a circuit, engineers can ensure that current flows in a way that minimizes resistance and heat generation.

Conclusion

Conformal mappings are a fascinating and powerful concept in complex analysis, with many practical applications. By preserving angles while allowing transformations of scale and shape, they provide invaluable tools for mathematics, engineering, physics, and beyond. Understanding conformal mappings can open the door to solving complex problems more efficiently and elegantly.


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Conformal Mappings

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