Section
Complex Analysis
Holomorphic functions, contour integration, singularities, conformal maps, and the projective geometry of the Riemann sphere.
Core idea
Complex analysis studies functions governed by complex differentiability. Its one-variable theory combines local power-series rigidity, global contour integration, conformal geometry, and the projective symmetry of the Riemann sphere.
Click any term to expand its definition inline.
---
Complex differentiability
Basic objects and criteria
- Complex numbers
- Complex domain
- Complex derivative
- Cauchy–Riemann equations
- Cauchy–Riemann criterion
- Holomorphic map and holomorphic function
- Entire function
- Real and complex power series
- Holomorphic functions are analytic
Rigidity theorems
- Identity theorem
- Maximum modulus principle
- Open mapping theorem
- Liouville's theorem
- Fundamental theorem of algebra
- Normal family
- Montel theorem
- Holomorphic germ
- Analytic continuation
- Monodromy theorem
---
Contours and integral formulas
---
Laurent theory and meromorphic functions
- Laurent series
- Classification of isolated singularities
- Casorati–Weierstrass theorem
- Great Picard theorem
- Little Picard theorem
- Order of a zero or pole
- Logarithmic derivative
- Residue
- Meromorphic function
- Residue theorem
- Argument principle
- Rouché's theorem
---
Conformal mapping
- Conformal map
- Riemann mapping theorem
- Riemann surface
- Complex manifold
- Category of complex manifolds
---
Riemann sphere and Möbius geometry
- Riemann sphere
- Rational function
- Möbius transformation
- Anti-Möbius transformation
- Möbius transformation group
- Automorphisms of the Riemann sphere
- Generalized circle
- Cross-ratio
- Cross-ratio invariance under Möbius transformations
- Sharp three-transitivity of the Möbius group
- Cross-ratio-preserving bijections are Möbius
- Schwarzian derivative
- Schwarzian chain rule
- Möbius characterization by the Schwarzian
- Equal Schwarzians differ by Möbius postcomposition
- Schwarzian derivative and second-order linear ODEs
- Complex projective structure
- Holomorphic projective connection
- Projective connections form an affine space
---
Adjacent geometric structures
- Projective line
- Complex projective space
- Complex coordinate chart
- Biholomorphism
- Sheaf of holomorphic functions
Potential theory and several complex variables
- Harmonic function
- Maximum principle for harmonic functions
- Upper-semicontinuous function
- Subharmonic function
- Pluriharmonic function
- Relations among H, SH, PSH, and PH
- Subharmonicity of the logarithmic modulus
- Entire function of several complex variables
- Levi form of a function
- Plurisubharmonic function
- Strictly plurisubharmonic function
- Complex Monge–Ampère operator
- Hörmander L2 theorem for the d-bar equation
- Holomorphic L2 mean-value estimate
Quaternionic pluripotential theory
- Cauchy–Fueter operators
- Quaternionic Hessian
- Quaternionic plurisubharmonic function
- Strictly quaternionic plurisubharmonic function
- Quaternionic Monge–Ampère measure
- Mixed quaternionic Monge–Ampère measure
- Continuity of quaternionic Monge–Ampère measures
- Quaternionic Błocki formula
- Quaternionic Monge–Ampère equation
- Strictly quaternionically pseudoconvex domain
- Dirichlet theorem for the quaternionic Monge–Ampère equation
Octonionic pluripotential theory on the plane
- Affine octonionic line
- Octonionic Radon transform
- Octonionic Hessian
- Octonionic plurisubharmonic function
- Octonionic Monge–Ampère measure
Paper-guided expansion
References
- Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw–Hill, 1979.
- John B. Conway, Functions of One Complex Variable I, 2nd ed., Springer, 1978. Publisher record.
- Otto Forster, Lectures on Riemann Surfaces, Springer, 1981. Publisher record.