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 ↓


Complex Numbers


Complex numbers are an extension of the idea of axis and dimension to the field of real numbers. In simple terms, they extend what can be expressed by numbers beyond the traditional real number line. Complex numbers are integral not only in mathematics but in a wide variety of fields including engineering, physics, and computer science.

Introduction to complex numbers

On the real number line, each number can be viewed as a point. Complex numbers introduce an additional dimension, known as the complex plane. Formally, a complex number is expressed as:

z = a + bi

Here, a and b are real numbers, and i is the imaginary unit, which satisfies the following equation:

i² = -1

Complex plane

The complex plane is a two-dimensional space where each point represents a complex number. The horizontal axis (real axis) corresponds to the real part of the complex number, Re(z) = a. The vertical axis (imaginary axis) corresponds to the imaginary part, Im(z) = b.

Im ↑ | b | ● (a, b) = a + bi | +----→ Re a

Types of forms: rectangular and polar

Rectangular form

The form z = a + bi is known as the rectangular form. It is straightforward and is often used in basic calculations.

Polar form

The polar form represents complex numbers using the magnitude r (also called the modulus) and the angle θ (the argument or phase). The relation is given as:

z = r(cosθ + i·sinθ)

The magnitude r is calculated as:

r = √(a² + b²)

The angle θ, which is the angle with the real axis, is calculated as:

θ = atan2(b, a)

Conversion between forms

To switch between rectangular and polar forms, use:

  • a = r cosθ
  • b = r sinθ

Visualization of operations on complex numbers

Add

Adding complex numbers involves adding their corresponding parts. If z₁ = a + bi and z₂ = c + di, then their sum is:

z₁ + z₂ = (a + c) + (b + d)i

Visually, this can be represented as a vector sum in the complex plane.

Im ↑ | b | ● (a, b) | | ↗ b+d + / | / | / d | / +---● (c, d) +---→ Re

Multiplication

When two complex numbers are multiplied, their moduli are combined and their angles are added:

z₁z₂ = r₁r₂ [cos(θ₁ + θ₂) + i·sin(θ₁ + θ₂)]

Again, if z₁ = a + bi and z₂ = c + di, then the product is:

z₁z₂ = (ac - bd) + (ad + bc)i

Complex conjugates

The complex conjugate of a complex number is a number whose real part is equal to a and the imaginary part has the opposite sign. The conjugate of z = a + bi is:

z* = a - bi

Division of complex numbers

To divide two complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator:

z₁ / z₂ = (z₁z₂*) / (z₂z₂*)

Euler's formula

The key to understanding complex numbers is Euler's formula:

e^(iθ) = cosθ + i·sinθ

Combining this with the polar form gives the exponential form of a complex number:

z = re^(iθ)

Applications of complex numbers

There are various important applications of complex numbers:

  • Engineering: Electrical circuits use complex impedances.
  • Physics: Quantum mechanics and relativity often involve complex numbers.
  • Signal processing: Fourier transforms use complex numbers for frequency analysis.

Conclusion

Complex numbers extend the potential of ordinary arithmetic, offering solutions to equations that have no real solutions. They represent a vital foundation for advanced areas of mathematics and science. Understanding the basics of complex numbers - how to manipulate them and how to look at them - is crucial for delving deeper into topics such as complex analysis, differential equations, and beyond.


Graduate → 7.1


U
username
0%
completed in Graduate

← Prev (7)
Complex Analysis
Next (7.1.1) →
Polar Form

Comments 



Complex Numbers

https://www.buddymath.com/g/7.1?bmal=en