Complex Analysis

  1. Special Functions
    1. Gamma Function
      1. Definition and Properties
        1. Generalization of Factorial Function
          1. Analytic Continuation and Reflection Formula
            1. Relation to Pi Function
            2. Euler's Integral Representation
              1. Functional Equations
                1. Applications
                  1. Probability and Statistics (Gamma Distribution)
                    1. Complex Integration
                      1. Analytic Continuation of Factorials
                    2. Zeta Function
                      1. Riemann Zeta Function
                        1. Definition and Properties
                          1. Analytic Continuation and Zeros
                            1. Functional Equation
                              1. Riemann Hypothesis
                              2. Applications in Number Theory
                                1. Prime Number Theorem
                                  1. Euler Product Formula
                                  2. Dirichlet L-functions
                                  3. Elliptic Functions
                                    1. Definitions and Properties
                                      1. Doubly Periodic Functions
                                      2. Jacobi Elliptic Functions
                                        1. Definitions and Relations
                                          1. Applications in Mechanics
                                          2. Weierstrass Elliptic Functions
                                            1. Applications in Algebraic Geometry
                                              1. Relation to Elliptic Curves
                                                1. Complex Tori
                                              2. Hypergeometric Functions
                                                1. Definitions and Series Representation
                                                  1. Gauss's Hypergeometric Function
                                                    1. Differential Equation and Solutions
                                                      1. Special Values and Transformations
                                                      2. Generalized Hypergeometric Functions
                                                        1. pFq Functions
                                                        2. Applications
                                                          1. Solutions to Differential Equations
                                                            1. Quantum Mechanics and Wave Functions
                                                          2. Other Special Functions
                                                            1. Bessel Functions
                                                              1. Definitions and Ordinary Differential Equation
                                                                1. Relations and Recurrence
                                                                  1. Applications in Wave Propagation and Vibrations
                                                                  2. Legendre Functions
                                                                    1. Definitions and Orthogonal Polynomials
                                                                      1. Applications in Gravitational and Electrostatic Potentials
                                                                      2. Hermite Functions
                                                                        1. Definitions and Differential Equation
                                                                          1. Applications in Quantum Mechanics (Harmonic Oscillator)
                                                                          2. Laguerre Functions
                                                                            1. Definitions and Orthogonality
                                                                              1. Applications in Quantum Mechanics (Hydrogen Atom)
                                                                              2. Chebyshev Functions
                                                                                1. Definitions and Approximations
                                                                                  1. Applications in Numerical Analysis and Approximation Theory
                                                                              5. Conformal Mappings
                                                                              First Page
                                                                              7. Riemann Surfaces