Analytical Methods in Physics

Preface
Contents
List of Figures
Complex Analysis
	1 Lecture 1: Complex Numbers
		1.1 Representations of Complex Numbers
		1.2 Application to Classical Mechanics
	2 Lecture 2: Complex Series and Functions
		2.1 Complex Series
		2.2 Elementary Functions of One Complex Variable
	3 Lecture 3: Analytic Functions
		3.1 Complex Differentiation
		3.2 Cauchy-Riemann Conditions
	4 Lecture 4: Complex Integration
		4.1 Line Integrals
		4.2 Cauchy's Theorem
		4.3 Cauchy's Integral Formula
	5 Lecture 5: Generalized Cauchy's Integral Formula
		5.1 Generalized Cauchy's Integral Theorem
		5.2 Cauchy's Inequality
	6 Lecture 6: Taylor Expansion
		6.1 Taylor Expansion
	7 Lecture 7: Laurent Expansion
		7.1 Laurent Expansion
		7.2 Definitions: Poles and Residue
		7.3 Examples
	8 Lecture 8: Methods of Finding the Residue
		8.1 Finding the Residue
		8.2 Residue Theorem
	9 Lecture 9: Definite Integrals
		9.1 Mapping to Unit Circle
		9.2 Extend to Complex Plane
		9.3 Jordan's Lemma
		9.4 Improper Integrals
Variational Calculus
	1 Lecture 10: Calculus of Variations: Euler Stationarity Condition
		1.1 Euler Condition
	2 Lecture 11: Cycloids and Geodesics
		2.1 Curve Representation x= x(y)
		2.2 Polar Representation θ= θ(r), r=r(θ)
		2.3 Geodesics on Quadratic Surfaces
	3 Lecture 12: Fermat's Principle and Hamilton's Principle
		3.1 Fermat's Principle
		3.2 Hamilton's Principle
Ordinary Differential Equations
	1 Lecture 13: First Order ODEs
		1.1 First Order ODE's
		1.2 Linear Second Order Ode's
	2 Lecture 14: Linear Second Order ODEs
		2.1 Second Order Ode's with Constant Coefficients
	3 Lecture 15: Green Function Method
		3.1 Dirac Delta Function
		3.2 Linear Second Order Ode's: Green Function Method
	4 Lecture 16: Green Function Method Examples
		4.1 Green Function Method
		4.2 Initial Value Problem
		4.3 Boundary Value Problem
	5 Lecture 17: Power Series Method
		5.1 Power Series Method
	6 Lecture 18: Frobenius Method
Fourier Series
	1 Lecture 19: Fourier Series for 2π-Periodicity
		1.1 Functions with 2π Periodicity
	2 Lecture 20: Fourier Series for 2L-Periodicity
		2.1 Even and Odd Extensions
		2.2 Completeness Relation
Integral Transforms
	1 Lecture 21: Fourier Transform
		1.1 Fourier Transform
	2 Lecture 22: Laplace Transform
		2.1 Laplace Transform
		2.2 Inverse Laplace Transform Integral (Supplementary)
	3 Lecture 23: Convolution of Integral Transforms
		3.1 Integral Transform of a Convolution
		3.2 Boundary Value Problem
		3.3 Initial Value Problem
Partial Differential Equations
	1 Lecture 24: Separation of Variable Method
		1.1 Definitions
		1.2 Separation of Variable Method
		1.3 Diffusion Equation in 2D: Cartesian Coordinates
	2 Lecture 25: Separation of Variable Method Non-Cartesian Coordinates
		2.1 Laplace Equation: Spherical Coordinates
		2.2 Diffusion Equation in Polar Coordinates
	3 Lecture 26: Integral Transform Method
		3.1 Laplace Transform Method
		3.2 Fourier Transform Method
		3.3 Green's Function Method
Appendix  References