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


Complex analysis is a branch of mathematics that explores functions of complex numbers. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the x-axis for real numbers and the y-axis for imaginary numbers. A complex number is usually expressed as z = x + yi, where x and y are real numbers, and i is an imaginary unit with the property i² = -1.

Complex plane

The complex plane is the area on which complex numbers are plotted. The horizontal axis represents the real part x, and the vertical axis represents the imaginary part yi. Representing complex numbers as points or vectors in the complex plane helps in visualizing operations such as addition, subtraction, and multiplication.

z = x + y Imaginary axis Real axis

In the above example, the point z = x + yi (marked in red) in the complex plane represents a complex number.

Addition and subtraction of complex numbers

Adding and subtracting complex numbers is simple. You add or subtract the corresponding real and imaginary parts separately. For example, if z₁ = x₁ + y₁i and z₂ = x₂ + y₂i, then:

z₁ + z₂ = (x₁ + x₂) + (y₁ + y₂)i
z₁ – z₂ = (x₁ – x₂) + (y₁ – y₂)i

Example

Suppose we have two complex numbers, z₁ = 3 + 4i and z₂ = 1 + 2i. Then their sum and difference are calculated as follows:

z₁ + z₂ = (3 + 1) + (4 + 2)i = 4 + 6i
z₁ – z₂ = (3 – 1) + (4 – 2)i = 2 + 2i

Multiplication of complex numbers

Multiplication involves distributing the terms and applying the property i² = -1. For example:

z₁ * z₂ = (x₁ + y₁i)(x₂ + y₂i)
        = x₁x₂ + x₁y₂i + y₁x₂i + y₁y₂i²
        = (x₁x₂ – y₁y₂) + (x₁y₂ + y₁x₂)i

Here, the expansion of multiplication is similar to that of distributive algebra, with an additional rule simplifying i² to -1.

Example

Let's take two complex numbers z₁ = 2 + 3i and z₂ = 4 + i, then:

z₁ * z₂ = (2 + 3i)(4 + i)
        = 2*4 + 2*i + 3i*4 + 3i²
        = 8 + 2i + 12i – 3
        = 5 + 14i

Complex conjugates

For a complex number z = x + yi, its complex conjugate, denoted as z̅, is the number x - yi. Conjugates help to rationalize denominators and find modulus.

Example

For the complex number z = 5 + 3i, its conjugate is z̅ = 5 - 3i.

Division of complex numbers

To divide by a complex number, multiply the numerator and denominator by the conjugate of the denominator. For example, dividing by z₁ = x₁ + y₁i z₂ = x₂ + y₂i:

z₁ / z₂ = (x₁ + y₁i) / (x₂ + y₂i)
        = (x₁ + y₁i) * (x₂ - y₂i) / ((x₂ + y₂i) * (x₂ - y₂i))
        = [(x₁x₂ + y₁y₂) + (y₁x₂ – x₁y₂)i] / (x₂² + y₂²)

Example

Let's divide z₁ = 7 + i by z₂ = 2 - 3i:

z₁ / z₂ = (7 + i) * (2 + 3i) / ((2 - 3i) * (2 + 3i))
        = (14 + 21i + 2i - 3) / (4 + 9)
        = (11 + 23i) / 13
        = 11/13 + (23/13)i

Polar form

Every complex number can also be expressed in polar coordinates. The polar form reveals the magnitude and angle of the vector representing the complex number. If z = x + yi then its polar form is:

z = r(cosθ + isinθ)

where r = √(x² + y²) is the modulus, and θ = atan2(y, x) is the argument (or angle).

Example

For the complex number z = 3 + 4i, its modulus is r = √(3² + 4²) = 5 The angle θ is atan2(4, 3). Thus, the polar form is:

z = 5(cosθ + isinθ)

Euler's formula

Euler's formula is an important link between complex analysis and trigonometry, which is expressed as:

e^(iθ) = cosθ + isinθ

Using Euler's formula, the polar form of a complex number simplifies to:

z = re^(iθ)

Applications of complex analysis

Complex analysis has many applications in fields such as engineering, physics and number theory. It is used in signal processing, fluid dynamics and electromagnetism. The calculus of complex functions provides powerful tools for solving physical problems in thermodynamics and quantum mechanics.

Conclusion

Complex analysis is not only a beautiful and interesting field of mathematics, but it also brings forth real-world applications that demonstrate its impact beyond pure mathematics. By studying complex functions, integrals, and series expansions, one can uncover deep mathematical properties and powerful problem-solving techniques.


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Complex Analysis

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