Ebook: Advanced Analytic Methods in Applied Mathematics, Science, and Engineering

LuBan Press, 2007. — 54 p.The Algebra of Operators
Calculus
Ordinary Differential Equations
Complex Analysis
Complex Numbers and Complex Variables
Analytic Functions
The Cauchy Integral Theorem
Evaluation of Real Integrals
Branch Points and Branch Cuts
Fourier Integrals and Fourier Series
The Laplace Transform
First-Order Partial Differential Equations
Trivial Example
Linear Homogeneous PDEs
Quasi-Linear PDEs
General Case
Second-Order Partial Differential Equations
The Laplace Equation
The Wave Equation
The Heat Equation
Separation of Variables
The Laplace Equation
The Wave Equation with Two Spatial Variables
The Schrödinger Equation
Singular Points of Ordinary Differential Equations
Taylor Series Solutions
Frobenius method
Solutions Near an Irregular Singular Point
Appendix: The Gamma Function
The WKB Approximation
WKB in the Zeroth and the First Order
Solutions Near an Irregular Singular Point
Higher-Order WKB Approximation
Turning Points
Asymptotic Expansions of Integrals
Integral Representation
The Laplace Method
Method of Stationary Phase
The Saddle Point Method
Appendix A: Gaussian Integrals
Appendix B: Infinite Contours
Boundary Layers and Singular Perturbation
Regular Perturbation
Boundary Layer Theory
Turning Points
Turning Point at an Endpoint
Interior Turning Points
Other Problems
Small Nonlinear Oscillations
Summing Leading Terms
Renormalized Perturbation—The Improved Poincare Method
The Two-Scale Method
The Renormalized Two-Scale Method
The Renormalization Group
Appendix of Useful Formulae
Bibliography
Index