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

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Complex Analysis


Complex analysis is a fascinating and essential field of mathematics, primarily concerned with complex numbers and functions of complex variables. Although it sounds complicated, it is an extension of real analysis with some important and interesting differences.

Complex numbers

Complex numbers form the basis of complex analysis. A complex number is of the form a + bi, where a and b are real numbers, and i is an imaginary unit having the property i2 = -1.

For example, the complex number 3 + 4i has real part 3 and imaginary part 4.

Visualization of complex numbers

3 + 4i

The red dot represents the complex number 3 + 4i on the complex plane. The horizontal line is the real axis, and the vertical line is the imaginary axis.

Complex tasks

Complex functions are functions that take complex numbers as input and return complex numbers. A simple example of this is f(z) = z2 + 1.

Domain and range

Just like real functions, complex functions have a domain and a range. The domain is the set of inputs where the function is defined. The range is the set of possible outputs.

f(z) = z^2 + 1

For the above function, if the domain is all complex numbers, then the range will also include all complex numbers.

Analytical functions

An important type of complex functions are analytic functions, also known as holomorphic functions. These functions are differentiable in their domain, just as real functions are differentiable.

The concept of differentiability in complex analysis is stronger than in real analysis. A function that is complexly differentiable at a point is automatically differentiable in some neighborhood of that point.

Cauchy–Riemann equations

For a complex function f(z) = u(x, y) + iv(x, y) to be differentiable, the functions u and v must satisfy the Cauchy–Riemann equations:

ux = vy uy = -vx

These equations provide a set of necessary conditions for a function to be analytic.

Example

Consider f(z) = z2, which can be expressed as f(x + iy) = (x + iy)2 = x2 - y2 + 2xyi. Then u(x, y) = x2 - y2 and v(x, y) = 2xy.

Check out the Cauchy-Riemann equation:

ux = 2x, vy = 2x uy = -2y, vx = 2y

These satisfy the equations, so f(z) = z2 is analytic.

Specification

Singularities are points where a function is not analytic. There are several classifications of singularities:

  • Removable singularities: points where a function can be defined or redefined so that it becomes analytic.
  • Pole: The point where a function goes to infinity.
  • Essential singularities: points where the function behaves irregularly.

Example of a pole

The function f(z) = 1/z has a pole at z = 0 because it becomes undefined (tends to infinity) as z approaches 0.

Contour integration

Contour integration is a key technique used to evaluate complex integrals. The integral of a complex function is calculated along a path or contour in the complex plane.

The line integral of a complex function over a contour C is given by:

∫C f(z) dz

Cauchy's integration theorem

A fundamental theorem in contour integration is Cauchy's integral theorem. It states that if a function f(z) is analytic everywhere inside and on a closed contour C, then:

∫C f(z) dz = 0

Example of contour integration

C

Consider the integral of f(z) = z over the contour line C, a circle of radius 1 centered at the origin.

∫C z dz = 0

Since f(z) = z is analytic everywhere, Cauchy's integration theorem tells us that this integral is zero.

Residue theorem

The residue theorem is a powerful tool in complex analysis. It is used to evaluate complex integrals using residues, which are special values associated with singularities of the function.

If f(z) has isolated singularities inside the contour line C, then:

∫C f(z) dz = 2πi ∑ Res(f, ak)

where Res(f, ak) is the residue of f at the singularity ak.

Applications of complex analysis

Complex analysis has wide applications in both pure and applied mathematics, including:

  • Physics: in quantum mechanics, electromagnetism, and fluid dynamics.
  • Engineering: especially in control theory and signal processing.
  • Number theory: Complex analysis is important for understanding the distribution of prime numbers via the Riemann zeta function.
  • Other areas: Used in differential equations, dynamical systems, and various areas of computational mathematics.

Conclusion

Complex analysis not only builds on the fundamental concepts learned in calculus and real analysis, forming a gateway to more advanced areas of mathematics, but it also has numerous applications in scientific and engineering disciplines. Its combination of geometric insights with algebraic techniques makes it a uniquely powerful tool for solving many kinds of problems.


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