Mathematical Tools for Physics

Contents

Introduction................. iv

Bibliography................ vi

1 Basic Stuff................. 1
Trigonometry
Parametric Differentiation
Gaussian Integrals
erf and Gamma
Differentiating
Integrals
Polar Coordinates
Sketching Graphs

2 Infinite Series................ 30
The Basics
Deriving Taylor Series
Convergence
Series of Series
Power series, two variables
Stirling’s Approximation
Useful Tricks
Diffraction
Checking Results

3 Complex Algebra.............. 65
Complex Numbers
Some Functions
Applications of Euler’s Formula
Logarithms
Mapping

4 Differential Equations............ 83

Linear Constant-Coefficient

Forced Oscillations

Series Solutions

Trigonometry via ODE’s

Green’s Functions

Separation of Variables

Simultaneous Equations

Simultaneous ODE’s

Legendre’s Equation

5 Fourier Series............... 118

Examples

Computing Fourier Series

Choice of Basis

Periodically Forced ODE’s

Return to Parseval

Gibbs Phenomenon

6 Vector Spaces............... 142

The Underlying Idea

Axioms

Examples of Vector Spaces

Linear Independence

Norms

Scalar Product

Bases and Scalar Products

Gram-Schmidt Orthogonalization

Cauchy-Schwartz inequality

Infinite Dimensions

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