Applied Mathematics

by Mahan, Gerald D.

• ISBN13: 9780306466830
• ISBN10: 030646683X
• Format: Hardcover
• Copyright: 2001-12-01
• Publisher: Kluwer Academic Pub

Summary

This is a textbook for a two-semester graduate course in Mathematical Methods in Physics. Most universities give this course, which is often taught jointly with the Engineering or Mathematics Departments. General topics include: group theory, linear equations, matrices, series, functions of complex variables, conformal mapping, special functions, and partial differential equations. Each chapter has numerous homework problems. The section on transforms includes those on Fourier and Laplace, as well as the modern topic of wavelets. The chapters on partial differential equations include: Laplace's, Poisson's, Helmholtz, diffusion, and wave equations. Problems are treated in one, two, and three dimensions.

Table of Contents

Determinants	1	(12)
Cramer's Rule	3	(1)
Gaussian Elimination	4	(3)
Special Determinants	7	(6)
Matrices	13	(34)
Several Theorems	14	(1)
Linear Equations	15	(2)
Inverse of a Matrix	17	(1)
Eigenvalues and Eigenvectors	18	(5)
Unitary Transformations	23	(2)
Hon-Hermitian Matrices	25	(7)
A Special Matrix	32	(2)
Gram--Schmidt	34	(4)
Chains	38	(9)
Group Theory	47	(26)
Basic Properties of Groups	47	(4)
Group Representations	51	(3)
Characters	54	(4)
Direct Product Groups	58	(2)
Representations	58	(2)
Basis Functions	60	(1)
Angular Momentum	61	(3)
Products of Representations	64	(1)
Quantum Mechanics	64	(4)
Eigenvalues	64	(1)
Representations	65	(2)
Matrix Elements	67	(1)
Double Groups	68	(5)
Complex Variables	73	(46)
Introduction	73	(2)
Analytic Functions	75	(5)
Multivalued Functions	80	(4)
Contour Integrals	84	(15)
Meromorphic Functions	99	(2)
Higher Poles	101	(1)
Integrals Involving Branch Cuts	102	(7)
Approximate Evaluation of Integrals	109	(10)
Steepest Descent	109	(1)
Saddle Point Integrals	110	(9)
Series	119	(22)
Taylor Series	119	(2)
Convergence	121	(5)
Laurent Series	126	(2)
Meromorphic Functions	128	(1)
Asymptotic Series	129	(3)
Summing Series	132	(3)
Pade Approximants	135	(6)
Conformal Mapping	141	(36)
Laplace's Equation	141	(3)
Mapping	144	(5)
Examples	149	(8)
Schwartz--Christoffel Transformations	157	(10)
van der Pauw	167	(10)
Currents	168	(2)
Resistance	170	(2)
Mapping	172	(5)
Markov Averaging	177	(18)
Random Walk	177	(6)
Speckle	183	(5)
Inhomogeneous Broadening	188	(7)
Fourier Transforms	195	(22)
Fourier Transforms	195	(7)
Unbounded Space	195	(2)
Half-Space	197	(2)
Finite Systems in 1D	199	(3)
Laplace Transforms	202	(3)
Wavelets	205	(12)
Continous Wavelet Transform	207	(3)
Discrete Transforms	210	(7)
Equations of Physics	217	(20)
Boundary and Initial Conditions	218	(1)
Boltzmann Equation	218	(7)
Moment Equations	219	(4)
Diffusion Equations	223	(1)
Fluid Equations	223	(2)
Solving Differential Equations	225	(10)
Homogeneous Linear Equations	225	(2)
Inhomogeneous Linear Equations	227	(2)
Nonlinear Equations	229	(6)
Elliptic Integrals	235	(2)
One Dimension	237	(28)
Introduction	237	(3)
Diffusion Equation	240	(12)
Wave Equation	252	(13)
Two Dimensions	265	(32)
Rectangular Coordinates	265	(8)
Laplace's Equation	265	(3)
Diffusion Equation	268	(4)
Wave Equation	272	(1)
Polar Coordinates	273	(24)
Laplace's Equation	274	(5)
Helmholtz Equation	279	(7)
Hankel Transforms	286	(11)
Three Dimensions	297	(36)
Cartesian Coordinates	297	(2)
Cylindrical Coordinates	299	(6)
Spherical Coordinates	305	(10)
Laplace's Equation	306	(3)
Diffusion and Wave Equations	309	(6)
Problems Inside a Sphere	315	(4)
Vector Wave Equation	319	(14)
Bulk Waves	319	(1)
Boundary Conditions	320	(13)
Odds and Ends	333	(32)
Hypergeometric Functions	333	(10)
Continued Fractions	337	(2)
Solving Equations With Series	339	(4)
Orthogonal Polynomials	343	(4)
Parabolic Cylinder Functions	344	(1)
Hermite Polynomials	344	(2)
Laguerre Polynomials	346	(1)
Sturm--Liouville	347	(5)
Green's Functions	352	(5)
Singular Integral Equations	357	(8)
Index	365