Mathematical analysis

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Mathematical analysis includes the theories of

• Numerical analysis
  • Numerical approximations

• Real analysis
  • Calculus
    • Univariable calculus (univariate calculus)
    • Multivariable calculus (multivariate calculus)
  • Real numbers (real line, continuum)
  • Functions of real variables
    • Functions of one real variable
    • Functions of several real variables
  • Limits
    • Convergence and divergence
  • Infinite sequences
  • Infinite series
  • Infinite products
  • Continued fractions
  • Differentiation and real differentiability
    • Partial differentiation
  • Antidifferentiation (indefinite integration)
  • Measure theory
  • Definite integration
  • Differential equations
    • Partial differential equations
  • Integral equations

• Vector calculus
  • Vectors
  • Vector functions

• Tensor calculus
  • Tensors
  • Tensor functions

• Complex analysis
  • Complex numbers (complex plane)
  • Functions of complex variables
  • Complex differentiation and complex differentiability (analaticity, holomorphicity)
  • Complex integration (path integration)
  • Analytic functions (holomorphic functions)

• p-adic analysis
  • p-adic numbers

• Non-standard analysis
  • Hyperreal numbers
    • Infinitesimals (infinitely small numbers)
    • Infinitely large numbers
  • Hyperreal functions

• Functional analysis
  • Calculus of variations
  • Spaces of functions
  • Banach spaces
  • Hilbert spaces
  • Functionals

• Harmonic analysis
  • Spaces of periodic functions
  • Fourier series

• Topological analysis
  • Topological spaces (spaces of mathematical objects with a definition of nearness)
    • Metric spaces (spaces of mathematical objects with a definition of distance)

Applied analytical techniques

• Analytic number theory
  • Transcendental number theory
• Analytic combinatorics
  • Generating functions (formal power series)
  • Asymptotic analysis
• Differential geometry
• Differential topology