3. Mathematics

• 3.1. Integration
  • 3.1.1. General Case
  • 3.1.2. Special Cases
  • 3.1.3. Motivation
  • 3.1.4. Example
• 3.2. Complex Numbers
  • 3.2.1. Real and Imaginary Part
  • 3.2.2. Argument Function
  • 3.2.3. Exponential
  • 3.2.4. Logarithm
  • 3.2.5. Power
  • 3.2.6. Examples
  • 3.2.7. Complex Conjugate
  • 3.2.8. Complex Derivatives
  • 3.2.9. Testing Identities Using Computer Code
• 3.3. Residue Theorem
  • 3.3.1. Computation of Residues
  • 3.3.2. Useful Formulas
  • 3.3.3. Complex Substitution
• 3.4. Fourier Transform
  • 3.4.1. Shift Theorem
  • 3.4.2. Scaling
  • 3.4.3. Derivative
  • 3.4.4. Convolution
  • 3.4.5. Radial Fourier Transform
  • 3.4.6. Examples
• 3.5. Fourier Transform of a Periodic Function (e.g. in a Crystal)
  • 3.5.1. One Dimension (Fourier Series)
• 3.6. Discrete Fourier Transform
• 3.7. Fast Fourier Transform (FFT)
  • 3.7.1. Decimation In Frequency (DIF)
• 3.8. Laplace Transform
• 3.9. Hilbert Transform
• 3.10. Periodic Functions
• 3.11. Polar Coordinates
  • 3.11.1. Example
• 3.12. Spherical Coordinates
  • 3.12.1. Example I
  • 3.12.2. Example II
• 3.13. Argument function, atan2
• 3.14. Multiple Argument Formulas
  • 3.14.1. sin(a x)
  • 3.14.2. cos(a x)
• 3.15. Delta Function
• 3.16. Distributions
• 3.17. Variations and Functional Derivatives
  • 3.17.1. Functions of One Variable
  • 3.17.2. Functions of several variables
  • 3.17.3. Functionals
  • 3.17.4. Examples
• 3.18. Dirac Notation
• 3.19. Homogeneous Functions (Euler’s Theorem)
  • 3.19.1. Example 1
  • 3.19.2. Example 2
• 3.20. Green Functions
  • 3.20.1. Introduction
  • 3.20.2. Boundary Conditions
  • 3.20.3. Symmetry
  • 3.20.4. Examples
• 3.21. Binomial Coefficients
• 3.22. Double Sums
• 3.23. Triangle Inequality
• 3.24. Gamma Function
• 3.25. Incomplete Gamma Function
  • 3.25.1. Example
• 3.26. Factorial
• 3.27. Double Factorial
  • 3.27.1. Example
• 3.28. Fermi-Dirac Integral
• 3.29. Legendre Polynomials
  • 3.29.1. Example I
  • 3.29.2. Example II
  • 3.29.3. Example III
  • 3.29.4. Example IV
• 3.30. Spherical Harmonics
  • 3.30.1. Examples
• 3.31. Gaunt Coefficients
  • 3.31.1. Example I
  • 3.31.2. Example II
  • 3.31.3. Example III
  • 3.31.4. Example IV
  • 3.31.5. Example V
  • 3.31.6. Example VI
• 3.32. Wigner 3j Symbols
  • 3.32.1. Examples
• 3.33. Multipole Expansion
• 3.34. Hypergeometric Functions
  • 3.34.1. Convergence Conditions
  • 3.34.2. Elementary and Special Functions
  • 3.34.3. Example I
  • 3.34.4. Example II
  • 3.34.5. Example III
• 3.35. Feynman Parameters
  • 3.35.1. Example 1
  • 3.35.2. Example 2
• 3.36. Groups
  • 3.36.1. Theory
  • 3.36.2. Crystallographic Point Groups
  • 3.36.3. Applications of finite groups
  • 3.36.4. Continuous Groups
  • 3.36.5. Literature
• 3.37. Wigner D Function
  • 3.37.1. Derivation
• 3.38. Ordinary Differential Equations
  • 3.38.1. Finite Difference Formulas
  • 3.38.2. Integrating ODE
  • 3.38.3. Radial Poisson Equation
• 3.39. Linear Algebra
  • 3.39.1. Scalar Product
  • 3.39.2. Projections
• 3.40. Differential Geometry
  • 3.40.1. Manifolds
  • 3.40.2. Examples
• 3.41. Operators
  • 3.41.1. Introduction
  • 3.41.2. Spectrum
  • 3.41.3. Derivative Operator
  • 3.41.4. Sturm–Liouville Operator
  • 3.41.5. Angular Momentum Operator
• 3.42. Variational Formulation of PDEs
  • 3.42.1. Poisson Equation
  • 3.42.2. Radial Schrödinger Equation