Preface 

xi  


1  (102) 

Definitions, Elementary Approach 


1  (6) 

Rotation of the Coordinate Axes 


7  (5) 


12  (6) 


18  (7) 

Triple Scalar Product, Triple Vector Product 


25  (7) 


32  (6) 


38  (5) 


43  (6) 


49  (5) 


54  (6) 


60  (4) 


64  (4) 


68  (11) 

Gauss' Law, Poisson's Equation 


79  (4) 


83  (12) 


95  (8) 


101  (2) 

Vector Analysis in Curved Coordinates and Tensors 


103  (62) 

Orthogonal Coordinates in R3 


103  (7) 

Differential Vector Operators 


110  (4) 

Special Coordinate Systems: Introduction 


114  (1) 

Circular Cylinder Coordinates 


115  (8) 

Spherical Polar Coordinates 


123  (10) 


133  (6) 

Contraction, Direct Product 


139  (2) 


141  (1) 

Pseudotensors, Dual Tensors 


142  (9) 


151  (9) 

Tensor Derivative Operators 


160  (5) 


163  (2) 

Determinants and Matrices 


165  (76) 


165  (11) 


176  (19) 


195  (13) 

Hermitian Matrices, Unitary Matrices 


208  (7) 

Diagonalization of Matrices 


215  (16) 


231  (10) 


239  (2) 


241  (80) 

Introduction to Group Theory 


241  (5) 

Generators of Continuous Groups 


246  (15) 


261  (5) 

Angular Momentum Coupling 


266  (12) 

Homogeneous Lorentz Group 


278  (5) 

Lorentz Covariance of Maxwell's Equations 


283  (8) 


291  (13) 


304  (17) 


319  (2) 


321  (82) 


321  (4) 


325  (14) 


339  (3) 


342  (6) 


348  (4) 


352  (11) 


363  (7) 


370  (6) 

Bernoulli Numbers, EulerMaclaurin Formula 


376  (13) 


389  (7) 


396  (7) 


401  (2) 

Functions of a Complex Variable I Analytic Properties, Mapping 


403  (52) 


404  (9) 

CauchyRiemann Conditions 


413  (5) 

Cauchy's Integral Theorem 


418  (7) 

Cauchy's Integral Formula 


425  (5) 


430  (8) 


438  (5) 


443  (8) 


451  (4) 


453  (2) 

Functions of a Complex Variable II 


455  (44) 


455  (27) 


482  (7) 

Method of Steepest Descents 


489  (10) 


497  (2) 

The Gamma Function (Factorial Function) 


499  (36) 

Definitions, Simple Properties 


499  (11) 

Digamma and Polygamma Functions 


510  (6) 


516  (4) 


520  (7) 

Incomplete Gamma Function 


527  (8) 


533  (2) 


535  (86) 

Partial Differential Equations 


535  (8) 

FirstOrder Differential Equations 


543  (11) 


554  (8) 


562  (3) 

Series SolutionsFrobenius' Method 


565  (13) 


578  (14) 

Nonhomogeneous EquationGreen's Function 


592  (19) 

Heat Flow, or Diffusion, PDE 


611  (10) 


618  (3) 

SturmLiouville TheoryOrthogonal Functions 


621  (54) 


622  (12) 


634  (8) 

GramSchmidt Orthogonalization 


642  (7) 

Completeness of Eigenfunctions 


649  (13) 

Green's FunctionEigenfunction Expansion 


662  (13) 


674  (1) 


675  (66) 

Bessel Functions of the First Kind, Jv(x) 


675  (19) 


694  (5) 


699  (8) 


707  (6) 

Modified Bessel Functions, Iv(x) and Kv(x) 


713  (6) 


719  (6) 

Spherical Bessel Functions 


725  (16) 


739  (2) 


741  (76) 


741  (8) 


749  (7) 


756  (11) 


767  (4) 

Associated Legendre Functions 


771  (15) 


786  (7) 

Orbital Angular Momentum Operators 


793  (4) 

Addition Theorem for Spherical Harmonics 


797  (6) 


803  (3) 

Legendre Functions of the Second Kind 


806  (7) 

Vector Spherical Harmonics 


813  (4) 


816  (1) 


817  (64) 


817  (20) 


837  (11) 


848  (11) 


859  (4) 

Confluent Hypergeometric Functions 


863  (6) 


869  (12) 


879  (2) 


881  (50) 


881  (7) 

Advantages, Uses of Fourier Series 


888  (4) 

Applications of Fourier Series 


892  (11) 

Properties of Fourier Series 


903  (7) 


910  (4) 

Discrete Fourier Transform 


914  (5) 

Fourier Expansions of Mathieu Functions 


919  (12) 


929  (2) 


931  (74) 


931  (5) 

Development of the Fourier Integral 


936  (2) 

Fourier TransformsInversion Theorem 


938  (8) 

Fourier Transform of Derivatives 


946  (5) 


951  (4) 


955  (6) 


961  (4) 


965  (6) 

Laplace Transform of Derivatives 


971  (8) 


979  (11) 

Convolution (Faltungs) Theorem 


990  (4) 

Inverse Laplace Transform 


994  (11) 


1003  (2) 


1005  (32) 


1005  (7) 

Integral Transforms, Generating Functions 


1012  (6) 

Neumann Series, Separable (Degenerate) Kernels 


1018  (11) 


1029  (8) 


1036  (1) 


1037  (42) 

A Dependent and an Independent Variable 


1038  (6) 

Applications of the Euler Equation 


1044  (8) 

Several Dependent Variables 


1052  (4) 

Several Independent Variables 


1056  (2) 

Several Dependent and Independent Variables 


1058  (2) 


1060  (5) 

Variation with Constraints 


1065  (7) 

RayleighRitz Variational Technique 


1072  (7) 


1076  (3) 

Nonlinear Methods and Chaos 


1079  (30) 


1079  (1) 


1080  (5) 

Sensitivity to Initial Conditions and Parameters 


1085  (3) 

Nonlinear Differential Equations 


1088  (21) 


1107  (2) 


1109  (44) 

Definitions, Simple Properties 


1109  (7) 


1116  (12) 


1128  (2) 


1130  (4) 

Gauss' Normal Distribution 


1134  (4) 


1138  (15) 


1150  (1) 


1150  (3) 
Index 

1153  