1 Introduction and Prerequisites
 1.1 Prerequisites
2 Linear vector spaces
 2.1 Definition of a linear vector space
  2.1.1 Problems
 2.2 Linear independence and basis vectors
  2.2.1 The scalar product
  2.2.2 Questions
 2.3 Function spaces
  2.3.1 Continuous basis functions: Fourier Transforms
  2.3.2 General orthogonality and completeness in function spaces
  2.3.3 Example from Quantum Mechanics
 2.4 Appendix
3 Operators, Eigenvectors and Eigenvalues
 3.1 Linear operators
  3.1.1 Domain, Codomain and Range
  3.1.2 Matrix representations of linear operators
  3.1.3 Adjoint operator and hermitian operators
 3.2 Eigenvalue equations
  3.2.1 Problems
 3.3 Sturm-Liouville equations
  3.3.1 How to bring an equation to Sturm-Liouville form
  3.3.2 A useful result
  3.3.3 Hermitian Sturm Liouville operators
  3.3.4 Second solutions, singularities
  3.3.5 Eigenvectors and eigenvalues
 3.4 Series solutions and orthogonal polynomials
  3.4.1 The quantum-mechanical oscillator and Hermite polynomials
  3.4.2 Legendre polynomials
  3.4.3 Bessel functions and the circular drum
4 Green functions
 4.1 General properties
  4.1.1 First example: Electrostatics
  4.1.2 The eigenstate method
  4.1.3 The continuity method
 4.2 Quantum mechanical scattering
 4.3 Time-dependent wave equation
  4.3.1 Solution for the Green function by Fourier transforms
  4.3.2 Wave equations in (2 + 1) dimensions
5 Variational calculus
 5.1 Functionals and stationary points
 5.2 Stationary points
 5.3 Special cases with examples: first integrals
  5.3.1 Functional of first derivative only
  5.3.2 No explicit dependence on x
 5.4 Generalisations
  5.4.1 Variable end points
  5.4.2 One endpoint free
  5.4.3 More than one function: Hamilton’s principle
  5.4.4 More dimensions: field equations
  5.4.5 Higher derivatives
 5.5 Constrained variational problems
  5.5.1 Lagrange’s undetermined multipliers
  5.5.2 Generalisation to functionals
  5.5.3 Eigenvalue problems
  5.5.4 The Rayleigh-Ritz method
A Contour Integration
 A.1 The Basics
 A.2 Contour Integration
 A.3 Residues
 A.4 Example 1: Simplest case
 A.5 Example 2: Complex exponentials
 A.6 Final case: poles on the real axis