Applied Mathematics Book

1	Determinants	1
1.1	Cramer's Rule	3
1.2	Gaussian Elimination	4
1.3	Special Determinants	7
2	Matrices	13
2.1	Several Theorems	14
2.2	Linear Equations	15
2.3	Inverse of a Matrix	17
2.4	Eigenvalues and Eigenvectors	18
2.5	Unitary Transformations	23
2.6	Hon-Hermitian Matrices	25
2.7	A Special Matrix	32
2.8	Gram-Schmidt	34
2.9	Chains	38
3	Group Theory	47
3.1	Basic Properties of Groups	47
3.2	Group Representations	51
3.3	Characters	54
3.4	Direct Product Groups	58
3.5	Basis Functions	60
3.6	Angular Momentum	61
3.7	Products of Representations	64
3.8	Quantum Mechanics	64
3.9	Double Groups	68
4	Complex Variables	73
4.1	Introduction	73
4.2	Analytic Functions	75
4.3	Multivalued Functions	80
4.4	Contour Integrals	84
4.5	Meromorphic Functions	99
4.6	Higher Poles	101
4.7	Integrals Involving Branch Cuts	102
4.8	Approximate Evaluation of Integrals	109
5	Series	119
5.1	Taylor Series	119
5.2	Convergence	121
5.3	Laurent Series	126
5.4	Meromorphic Functions	128
5.5	Asymptotic Series	129
5.6	Summing Series	132
5.7	Pade Approximants	135
6	Conformal Mapping	141
6.1	Laplace's Equation	141
6.2	Mapping	144
6.3	Examples	149
6.4	Schwartz-Christoffel Transformations	157
6.5	van der Pauw	167
7	Markov Averaging	177
7.1	Random Walk	177
7.2	Speckle	183
7.3	Inhomogeneous Broadening	188
8	Fourier Transforms	195
8.1	Fourier Transforms	195
8.2	Laplace Transforms	202
8.3	Wavelets	205
9	Equations of Physics	217
9.1	Boundary and Initial Conditions	218
9.2	Boltzmann Equation	218
9.3	Solving Differential Equations	225
9.4	Elliptic Integrals	235
10	One Dimension	237
10.1	Introduction	237
10.2	Diffusion Equation	240
10.3	Wave Equation	252
11	Two Dimensions	265
11.1	Rectangular Coordinates	265
11.2	Polar Coordinates	273
12	Three Dimensions	297
12.1	Cartesian Coordinates	297
12.2	Cylindrical Coordinates	299
12.3	Spherical Coordinates	305
12.4	Problems Inside a Sphere	315
12.5	Vector Wave Equation	319
13	Odds and Ends	333
13.1	Hypergeometric Functions	333
13.2	Orthogonal Polynomials	343
13.3	Sturm-Liouville	347
13.4	Green's Functions	352
13.5	Singular Integral Equations	357
	Index	365