Mathematics for Natural Scientists II: Advanced Methods (Undergraduate Lecture Notes in Physics)

Preface to the Second Edition
Preface to the First Edition
	References
Famous Scientists Mentioned in the Book
Contents
1 Elements of Linear Algebra
	1.1 Vector Spaces
		1.1.1 Introduction to Multi-dimensional Complex Vector Spaces
		1.1.2 Analogy Between Functions and Vectors
		1.1.3 Orthogonalisation Procedure (Gram–Schmidt Method)
	1.2 Matrices: Definition and Properties
		1.2.1 Introduction of a Concept
		1.2.2 Operations with Matrices
		1.2.3 Inverse Matrix: An Introduction
		1.2.4 Linear Transformations and a Group
		1.2.5 Orthogonal and Unitary Transformations
	1.3 Determinant of a Square Matrix
		1.3.1 Formal Definition
		1.3.2 Properties of Determinants
		1.3.3 Relation to a Linear Antisymmetric Function
		1.3.4 Practical Method of Calculating Determinants. Minors
		1.3.5 Determinant of a Tridiagonal Matrix
	1.4 A Linear System of Algebraic Equations
		1.4.1 Cramer\'s Method
		1.4.2 Gaussian Elimination
	1.5 Rank of a Matrix
	1.6 Wronskian
		1.6.1 Linear Independence of Functions
		1.6.2 Linear ODE\'s: Abel\'s Formula
		1.6.3 Linear ODE Reverse Engineering
	1.7 Calculation of the Inverse Matrix
	1.8 Eigenvectors and Eigenvalues of a Square Matrix
		1.8.1 General Formulation
		1.8.2 Algebraic and Geometric Multiplicities
		1.8.3 Left Eigenproblem
		1.8.4 Hermitian (Symmetric) Matrices
		1.8.5 Normal Matrices
	1.9 Similarity Transformation
		1.9.1 Diagonalisation of Matrices
		1.9.2 General Case of Repeated Eigenvalues
		1.9.3 Simultaneous Diagonalisation of Two Matrices
	1.10 Spectral Theorem and Function of a Matrix
		1.10.1 Normal Matrices
		1.10.2 General Matrices
	1.11 Generalised Eigenproblem
	1.12 Famous Identities with the Matrix Exponential
	1.13 Quadratic Forms
	1.14 Extremum of a Function of nn Variables
	1.15 Trace of a Matrix
	1.16 Tridiagonalisation of a Matrix: The Lanczos Method
	1.17 Dividing Matrices Into Blocks
	1.18 Defective Matrices
		1.18.1 Jordan Normal Form
		1.18.2 Function of a General Matrix
	1.19 Systems of Linear Differential Equations
		1.19.1 General Consideration
		1.19.2 Homogeneous Systems with Constant Coefficients
		1.19.3 Non-homogeneous Systems
		1.19.4 The Case of the Matrix Having a Lower Rank
		1.19.5 Higher Order Linear Differential Equations
		1.19.6 Homogeneous Systems with Defective Matrices
	1.20 Pseudo-Inverse of a Matrix
	1.21 General 3D Rotation
	1.22 Examples in Physics
		1.22.1 Particle in a Magnetic Field
		1.22.2 Kinetics
		1.22.3 Vibrations in Molecules
		1.22.4 Least Square Method and MD Simulations
		1.22.5 Vibrations of Atoms in an Infinite Chain. A Point Defect
		1.22.6 States of an Electron in a Solid
		1.22.7 Time Propagation of a Wavefunction
		1.22.8 Slater Determinants and Fermionic Many-Body Wavefunctions
		1.22.9 Dirac Equation
		1.22.10 Classical 1D Ising Model
2 Complex Numbers and Functions
	2.1 Representation of Complex Numbers
	2.2 Functions on a Complex Plane
		2.2.1 Regions in the Complex Plane and Mapping
		2.2.2 Differentiation. Analytic Functions
	2.3 Main Elementary Functions
		2.3.1 Integer Power Function
		2.3.2 Integer Root Function
		2.3.3 Exponential and Hyperbolic Functions
		2.3.4 Logarithm
		2.3.5 Trigonometric Functions
		2.3.6 Inverse Trigonometric Functions
		2.3.7 General Power Function
	2.4 Integration in the Complex Plane
		2.4.1 Definition
		2.4.2 Integration of Analytic Functions
		2.4.3 Fundamental Theorem of Algebra
		2.4.4 Fresnel Integral
	2.5 Complex Functional Series
		2.5.1 Numerical Series
		2.5.2 General Functional Series
		2.5.3 Power Series
		2.5.4 The Laurent Series
		2.5.5 Zeros and Singularities of Functions
	2.6 Analytic Continuation
	2.7 Residues
		2.7.1 Definition
		2.7.2 Functional Series: An Example
		2.7.3 Applications of Residues in Calculating Real Axis Integrals
	2.8 Linear Differential Equations: Series Solutions
	2.9 Selected Applications in Physics
		2.9.1 Dispersion Relations
		2.9.2 Propagation of Electro-Magnetic Waves in a Material
		2.9.3 Electron Tunnelling in Quantum Mechanics
		2.9.4 Propagation of a Quantum State
		2.9.5 Matsubara Sums
		2.9.6 A Beautiful Counting Problem
		2.9.7 Landau-Zener Problem
3 Fourier Series
	3.1 Trigonometric Series: An Intuitive Approach
	3.2 Dirichlet Conditions
	3.3 Gibbs–Wilbraham Phenomenon
	3.4 Integration and Differentiation of the Fourier Series
	3.5 Parseval\'s Theorem
	3.6 Summing Inverse Powers of Integers
	3.7 Complex (Exponential) form of the Fourier Series
	3.8 Application to Differential Equations
	3.9 A More Rigorous Approach to the Fourier Series
		3.9.1 Convergence `on Average\'
		3.9.2 A More Rigorous Approach to the Fourier Series: Dirichlet Theorem
		3.9.3 Expansion of a Function via Orthogonal Functions
	3.10 Investigation of the Convergence of the FS
		3.10.1 The Second Mean-Value Integral Theorem
		3.10.2 First Estimates of the Fourier Coefficients
		3.10.3 More Detailed Investigation of FS Convergence
		3.10.4 Improvements of Convergence of the FS
	3.11 Applications of Fourier Series in Physics
		3.11.1 Expansions of Functions Describing Crystal Properties
		3.11.2 Ewald\'s Formula
		3.11.3 Born and von Karman Boundary Conditions
		3.11.4 Atomic Force Microscopy
		3.11.5 Applications in Quantum Mechanics
		3.11.6 Free Electron Gas
4 Special Functions
	4.1 Dirac Delta Function
	4.2 The Gamma Function
		4.2.1 Definition and Main Properties
		4.2.2 Beta Function
	4.3 Orthogonal Polynomials
		4.3.1 Legendre Polynomials
		4.3.2 General Theory of Orthogonal Polynomials
	4.4 Differential Equation of Generalised Hypergeometric Type
		4.4.1 Transformation to a Standard Form
		4.4.2 Solutions of the Standard Equation
		4.4.3 Classical Orthogonal Polynomials
	4.5 Associated Legendre Function
		4.5.1 Bound Solutions of the Associated Legendre Equation
		4.5.2 Orthonormality of Associated Legendre Functions
		4.5.3 Laplace Equation in Spherical Coordinates
	4.6 Bessel Equation
		4.6.1 Bessel Differential Equation and its Solutions
		4.6.2 Half-Integer Bessel Functions
		4.6.3 Recurrence Relations for Bessel Functions
		4.6.4 Generating Function and Integral Representation for Bessel Functions
		4.6.5 Orthogonality and Functional Series Expansion
	4.7 Selected Applications in Physics
		4.7.1 Schrödinger Equation for a Harmonic Oscillator
		4.7.2 Schrödinger Equation for the Hydrogen Atom
		4.7.3 A Free Electron in a Cylindrical Ptential Well
		4.7.4 Stirling\'s Formula and Phase Transitions
		4.7.5 Band Structure of a Solid
		4.7.6 Oscillations of a Circular Membrane
		4.7.7 Multipole Expansion of the Electrostatic Potential
	4.8 Van Hove Singularities
5 Fourier Transform
	5.1 The Fourier Integral
		5.1.1 Intuitive Approach
		5.1.2 Alternative Forms of the Fourier Integral
		5.1.3 A More Rigorous Derivation of Fourier Integral
	5.2 Fourier Transform
		5.2.1 General Idea
		5.2.2 Fourier Transform of Derivatives
		5.2.3 Fourier Transform of an Integral
		5.2.4 Convolution Theorem
		5.2.5 Parseval\'s Theorem
		5.2.6 Poisson Summation Formula
	5.3 Applications of the Fourier Transform in Physics
		5.3.1 Various Notations and Multiple Fourier Transform
		5.3.2 Retarded Potentials
		5.3.3 Green\'s Function of a Differential Equation
		5.3.4 Time Correlation Functions
		5.3.5 Fraunhofer Diffraction
6 Laplace Transform
	6.1 Definition
	6.2 Method of Partial Fractions
	6.3 Detailed Consideration of the LT
		6.3.1 Analyticity of the LT
		6.3.2 Relation to the Fourier Transform
		6.3.3 Inverse Laplace Transform
	6.4 Properties of the Laplace Transform
		6.4.1 Derivatives of Originals and Images
		6.4.2 Shift in Images and Originals
		6.4.3 Integration of Images and Originals
		6.4.4 Convolution Theorem
	6.5 Solution of Ordinary Differential Equations (ODEs)
	6.6 Applications in Physics
		6.6.1 Application of the LT Method in Electronics
		6.6.2 Harmonic Particle with Memory
		6.6.3 Probabilities of Hops
		6.6.4 Inverse NC-AFM Problem
7 Curvilinear Coordinates
	7.1 Definition of Curvilinear Coordinates
	7.2 Unit Base Vectors
	7.3 Line Elements and Line Integral
		7.3.1 Line Element
		7.3.2 Line Integrals
	7.4 Surface Normal and Surface Integrals
	7.5 Volume Element and Jacobian in 3D
	7.6 Change of Variables in Multiple Integrals
	7.7 Multi-variable Gaussian
	7.8 upper NN-Dimensional Sphere
	7.9 Gradient of a Scalar Field
	7.10 Divergence of a Vector Field
	7.11 Laplacian
	7.12 Curl of a Vector Field
	7.13 Some Applications in Physics
		7.13.1 Partial Differential Equations of Mathematical Physics
		7.13.2 Classical Mechanics of a Particle
		7.13.3 Distribution Function of a Set of Particles
8 Partial Differential Equations of Mathematical Physics
	8.1 General Consideration
		8.1.1 Characterisation of Second-Order PDEs
		8.1.2 Initial and Boundary Conditions
	8.2 Wave Equation
		8.2.1 One-Dimensional String
		8.2.2 Propagation of Sound
		8.2.3 General Solution of PDE
		8.2.4 Uniqueness of Solution
		8.2.5 Fourier Method
		8.2.6 Forced Oscillations of the String
		8.2.7 General Boundary Problem
		8.2.8 Oscillations of a Rectangular Membrane
		8.2.9 General Remarks on the Applicability of the Fourier Method
	8.3 Heat-Conduction Equation
		8.3.1 Uniqueness of the Solution
		8.3.2 Fourier Method
		8.3.3 Stationary Boundary Conditions
		8.3.4 Heat Transport with Internal Sources
		8.3.5 Solution of the General Boundary Heat-Conduction Problem
	8.4 Problems Without Boundary Conditions
	8.5 Application of Fourier Method to Laplace Equation
	8.6 Method of Integral Transforms
		8.6.1 Fourier Transform
		8.6.2 Laplace Transform
9 Calculus of Variations
	9.1 Functions of a Single Variable
		9.1.1 Functionals Involving a Single Function
		9.1.2 Functionals Involving More Than One Function
		9.1.3 Functionals Containing Higher Derivatives
		9.1.4 Variation with Constraints Given by Zero Functions
		9.1.5 Variation with Constraints Given by Integrals
	9.2 Functions of Many Variables
	9.3 Applications in Physics
		9.3.1 Mechanics
		9.3.2 Functional Derivatives
		9.3.3 Many-Electron Theory
Index