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 Numbers


Complex numbers are fascinating mathematical objects that extend the familiar concept of one-dimensional numbers to the two-dimensional complex plane. They are used in many fields of mathematics, physics, engineering, and beyond because they encompass a rich structure that allows us to solve equations that have no solutions in the realm of real numbers.

What are complex numbers?

At the root of the complex numbers is the imaginary unit, denoted by i, which is defined by the following property:

i² = -1

A complex number is expressed as:

z = a + bi

where a and b are real numbers. The real part of z is a, and the imaginary part is b.

Visual representation of complex numbers

Since complex numbers have two components, they can be viewed as points or vectors in a plane, called the complex plane. The horizontal axis represents the real part, and the vertical axis represents the imaginary part.

z = a + by Real axis Imaginary axis

In the above example, the complex number z = a + bi is represented by the red dot in the complex plane, where its position is determined by the coordinates (a, b).

Basic operations with complex numbers

Complex numbers can be added, subtracted, multiplied and divided in the same way as real numbers, with the difference being that all operations take into account the imaginary unit i and its property i² = -1.

Addition and subtraction

To add or subtract complex numbers, add the real parts and the imaginary parts separately:

(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) - (c + di) = (a - c) + (b - d)i

Example: Let z1 = 3 + 4i and z2 = 1 + 2i.

  • Sum: z1 + z2 = (3 + 1) + (4 + 2)i = 4 + 6i
  • Subtraction: z1 - z2 = (3 - 1) + (4 - 2)i = 2 + 2i

Multiplication

Multiplication of complex numbers is performed using the distributive law:

(a + bi) * (c + di) = ac + adi + bci + bdi²
(a + bi) * (c + di) = (ac - bd) + (ad + bc)i

Notice how bdi² becomes bd(-1), which is -bd.
Example: Multiply z1 = 2 + 3i and z2 = 1 + 4i.

  • (2 + 3i) * (1 + 4i) = 2*1 + 2*4i + 3i*1 + 3*4i² = 2 + 8i + 3i - 12
  • = (2 - 12) + (8 + 3)i = -10 + 11i

Division

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of the complex number z = a + bi is overline{z} = a - bi.

frac{a + bi}{c + di} * frac{c - di}{c - di} = frac{(a + bi)(c - di)}{c² + d²}

Apply the product:

frac{(ac + bd) + (bc - ad)i}{c² + d²}

Example: z1 = 4 + 2i Divide by z2 = 1 - i.

  • The conjugate of z2 is 1 + i.
  • frac{4 + 2i}{1 - i} * frac{1 + i}{1 + i} = frac{(4 + 2i)(1 + i)}{1² + 1²} = frac{4 + 4i + 2i + 2i²}{2}
  • Since i² = -1, therefore 2i² = -2. Thus, frac{4 + 4i + 2i - 2}{2} = frac{2 + 6i}{2} = 1 + 3i.

Modulus and argument of complex numbers

The modulus (or absolute value) of a complex number z = a + bi is its distance from the origin in the complex plane, calculated as:

|z| = sqrt{a² + b²}

The argument of z, denoted by arg(z), is the angle that the line (denoting z) makes with the positive real axis, usually measured in radians.

Example: Find the modulus and argument of z = 3 + 4i.

  • Modulus: |z| = sqrt{3² + 4²} = sqrt{9 + 16} = 5
  • Reason: If tan θ = frac{4}{3}, then θ ≈ 0.93 (in radians).

Polar and exponential forms

Complex numbers can also be represented in polar form, which highlights their geometric nature:

z = r(cos θ + i sin θ)

where r is the modulus of z, and θ is the argument.

Using Euler's formula e^{iθ} = cos θ + i sin θ, the polar form of a complex number can be expressed in its exponential form:

z = re^{iθ}

This form is especially useful for multiplying and dividing complex numbers, as well as raising them to powers.

Complex conjugates and division

The complex conjugate of a number z = a + bi is overline{z} = a - bi. Conjugation represents z on the real axis in the complex plane.

The product of a complex number and its conjugate is always a non-negative real number:

z * overline{z} = (a + bi)(a - bi) = a² + b²

Example: Find the conjugate and check the property for z = 5 + 12i.

  • Conjugate: overline{z} = 5 - 12i
  • Product: z * overline{z} = (5 + 12i)(5 - 12i) = 5² + 12² = 25 + 144 = 169

Applications of complex numbers

Complex numbers are widely used in the fields of engineering, physics, and applied mathematics. They are important in solving differential equations, analyzing electrical circuits, and modeling waves and oscillations. A particularly interesting application is in quantum mechanics, where complex numbers are used to describe the state of a quantum system.

For example, in electrical engineering, alternating currents and voltages are often represented as complex numbers. This allows engineers to analyze AC circuits in a simpler way using Ohm’s law and other principles. The imaginary unit is used to take into account the phase difference between alternating quantities.

Conclusion

Complex numbers are an integral part of modern mathematics and engineering. They extend the concept of numbers by providing a way to perform operations in two dimensions with both real and imaginary parts, and provide tools for solving complex problems in a variety of fields. Understanding the basic operations with complex numbers and visualizing them in the complex plane can greatly benefit learners in mastering this concept.


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Complex Numbers

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