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CONTENTS
PREFACE ............................................................................................................................... ............ XI
1. MATHEMATICAL PRELIMINARIES ...................................................................................................... 1
1 . 1 . Infinite Series.................................................................................................................. 1
1.2. Series of Functions ....................................................................................................... 21
1.3. Binomial Theorem ........................................................................................................ 33
1.4. Mathematical Induction............................................................................................... 40
1.5. Operations of Series Expansions of Functions .............................................................. 41
1.6. Some Important Series................................................................................................. 45
1.7. Vectors ......................................................................................................................... 46
1.8. Complex Numbers and Functions................................................................................. 53
1.9. Derivatives and Extrema .............................................................................................. 62
1.10. Evaluation of Integrals ................................................................................................. 65
1.11. Dirac Delta Functions ................................................................................................... 75
Additional Readings .................................................................................................... 82
2. DETERMINANTS AND MATRICES .................................................................................................... 83
2.1 Determinants ............................................................................................................... 83
2.2 Matrices ....................................................................................................................... 95
Additional Readings .................................................................................................. 121
3. VECTOR ANALYSIS .................................................................................................................... 123
3.1 Review of Basics Properties........................................................................................ 124
3.2 Vector in 3 ‐ D Spaces................................................................................................. 126
3.3 Coordinate Transformations ...................................................................................... 133
3 . 4 Rotations in 3 ........................................................................................................ 139
3.5 Differential Vector Operators..................................................................................... 143
3.6 Differential Vector Operators: Further Properties...................................................... 153
3.7 Vector Integrations .................................................................................................... 159
3.8 Integral Theorems...................................................................................................... 164
3.9 Potential Theory......................................................................................................... 170
3.10 Curvilinear Coordinates.............................................................................................. 182
Additional Readings .................................................................................................. 203
4. TENSOR AND DIFFERENTIAL FORMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
4.1 Tensor Analysis .......................................................................................................... 205
4.2 Pseudotensors, Dual Tensors ..................................................................................... 215
4.3 Tensor in General Coordinates ................................................................................... 218
4.4 Jacobians.................................................................................................................... 227
4.5 Differential Forms ...................................................................................................... 232
4.6 Differentiating Forms................................................................................................. 238
4.7 Integrating Forms ...................................................................................................... 243
Additional Readings .................................................................................................. 249
5. VECTOR SPACES ....................................................................................................................... 251
5.1 Vector in Function Spaces .......................................................................................... 251
5.2 Gram ‐ Schmidt Orthogonalization ............................................................................. 269
5.3 Operators ................................................................................................................... 275
5.4 Self‐Adjoint Operators ................................................................................................ 283
5.5 Unitary Operators ...................................................................................................... 287
5.6 Transformations of Operators.................................................................................... 292
5.7 Invariants ................................................................................................................... 294
5.8 Summary – Vector Space Notations........................................................................... 296
Additional Readings .................................................................................................. 297
6. EIGENVALUE PROBLEMS............................................................................................................. 299
6.1 Eigenvalue Equations ................................................................................................. 299
6.2 Matrix Eigenvalue Problems ...................................................................................... 301
6.3 Hermitian Eigenvalue Problems ................................................................................. 310
6.4 Hermitian Matrix Diagonalization ............................................................................. 311
6.5 Normal Matrices ........................................................................................................ 319
Additional Readings .................................................................................................. 328
7. ORDINARY DIFFERENTIAL EQUATIONS........................................................................................... 329
7.1 Introduction ............................................................................................................... 329
7.2 First ‐ Order Equations ............................................................................................... 331
7.3 ODEs with Constant Coefficients................................................................................ 342
7.4 Second‐Order Linear ODEs ......................................................................................... 343
7.5 Series Solutions‐ Frobenius‘ Method.......................................................................... 346
7.6 Other Solutions .......................................................................................................... 358
7 . 7 Inhomogeneous Linear ODEs ..................................................................................... 375
7.8 Nonlinear Differential Equations................................................................................ 377
Additional Readings .................................................................................................. 380
8. STURM – LIOUVILLE THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
8.1 Introduction ............................................................................................................... 381
8.2 Hermitian Operators .................................................................................................. 384
8.3 ODE Eigenvalue Problems .......................................................................................... 389
8.4 Variation Methods ..................................................................................................... 395
8.5 Summary, Eigenvalue Problems................................................................................. 398
Additional Readings .................................................................................................. 399
9. PARTIAL DIFFERENTIAL EQUATIONS .............................................................................................. 401
9.1 Introduction ............................................................................................................... 401
9.2 First ‐ Order Equations ............................................................................................... 403
9.3 Second – Order Equations .......................................................................................... 409
9.4 Separation of Variables............................................................................................. 414
9.5 Laplace and Poisson Equations .................................................................................. 433
9.6 Wave Equations ......................................................................................................... 435
9.7 Heat – Flow, or Diffution PDE..................................................................................... 437
9.8 Summary.................................................................................................................... 444
Additional Readings .................................................................................................. 445
10. GREEN’ FUNCTIONS .................................................................................................................. 447
10.1 One – Dimensional Problems .................................................................................... 448
10.2 Problems in Two and Three Dimensions .................................................................... 459
Additional Readings .................................................................................................. 467
11. COMPLEX VARIABLE THEORY ...................................................................................................... 469
11.1 Complex Variables and Functions .............................................................................. 470
11.2 Cauchy – Riemann Conditions.................................................................................... 471
11.3 Cauchy’s Integral Theorem ........................................................................................ 477
11.4 Cauchy’s Integral Formula ......................................................................................... 486
11.5 Laurent Expansion...................................................................................................... 492
11.6 Singularities ............................................................................................................... 497
11.7 Calculus of Residues ................................................................................................... 509
11.8 Evaluation of Definite Integrals.................................................................................. 522
11.9 Evaluation of Sums..................................................................................................... 544
11.10 Miscellaneous Topics .................................................................................................. 547
Additional Readings .................................................................................................. 550
12. FURTHER TOPICS IN ANALYSIS ..................................................................................................... 551
12.1 Orthogonal Polynomials............................................................................................. 551
12.2 Bernoulli Numbers ..................................................................................................... 560
12.3 Euler – Maclaurin Integration Formula ...................................................................... 567
12.4 Dirichlet Series ........................................................................................................... 571
1 2 . 5 Infinite Products ......................................................................................................... 574
12.6 Asymptotic Series....................................................................................................... 577
12.7 Method of Steepest Descents..................................................................................... 585
12.8 Dispertion Relations ................................................................................................... 591
Additional Readings .................................................................................................. 598
13. GAMMA FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
13.1 Definitions, Properties................................................................................................ 599
13.2 Digamma and Polygamma Functions ........................................................................ 610
13.3 The Beta Function ...................................................................................................... 617
13.4 Stirling’s Series ........................................................................................................... 622
13.5 Riemann Zeta Function .............................................................................................. 626
13.6 Other Ralated Function .............................................................................................. 633
Additional Readings .................................................................................................. 641
14. BESSEL FUNCTIONS ................................................................................................................... 643
14.1 Bessel Functions of the First kind, Jν(x)....................................................................... 643
14.2 Orthogonality............................................................................................................. 661
14.3 Neumann Functions, Bessel Functions of the Second kind ........................................ 667
14.4 Hankel Functions........................................................................................................ 674
14.5 Modified Bessel Functions, Iν(x) and Kν(x)................................................................ 680
14.6 Asymptotic Expansions .............................................................................................. 688
14.7 Spherical Bessel Functions ......................................................................................... 698
Additional Readings .................................................................................................. 713
15. LEGENDRE FUNCTIONS............................................................................................................... 715
15.1 Legendre Polynomials ................................................................................................ 716
15.2 Orthogonality............................................................................................................. 724
15.3 Physical Interpretation of Generating Function ......................................................... 736
15.4 Associated Legendre Equation ................................................................................... 741
15.5 Spherical Harmonics................................................................................................... 756
15.6 Legendre Functions of the Second Kind...................................................................... 766
Additional Readings .................................................................................................. 771
16. ANGULAR MOMENTUM ............................................................................................................. 773
16.1 Angular Momentum Operators.................................................................................. 774
16.2 Angular Momentum Coupling.................................................................................... 784
16.3 Spherical Tensors ....................................................................................................... 796
16.4 Vector Spherical Harmonics ....................................................................................... 809
Additional Readings .................................................................................................. 814
17. GROUP THEORY ....................................................................................................................... 815
17.1 Introduction to Group Theory .................................................................................... 815
17.2 Representation of Groups .......................................................................................... 821
17.3 Symmetry and Physics................................................................................................ 826
17.4 Discrete Groups.......................................................................................................... 830
1 7 . 5 Direct Products........................................................................................................... 837
17.6 Simmetric Group ........................................................................................................ 840
17.7 Continous Groups....................................................................................................... 845
17.8 Lorentz Group ............................................................................................................ 862
17.9 Lorentz Covariance of Maxwell’s Equantions............................................................. 866
17.10 Space Groups ............................................................................................................. 869
Additional Readings .................................................................................................. 870
18. MORE SPECIAL FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871
18.1 Hermite Functions...................................................................................................... 871
18.2 Applications of Hermite Functions ............................................................................. 878
18.3 Laguerre Functions..................................................................................................... 889
18.4 Chebyshev Polynomials .............................................................................................. 899
18.5 Hypergeometric Functions ......................................................................................... 911
18.6 Confluent Hypergeometric Functions......................................................................... 917
18.7 Dilogarithm ................................................................................................................ 923
18.8 Elliptic Integrals.......................................................................................................... 927
Additional Readings .................................................................................................. 932
19. FOURIER SERIES........................................................................................................................ 935
19.1 General Properties ..................................................................................................... 935
19.2 Application of Fourier Series ...................................................................................... 949
19.3 Gibbs Phenomenon .................................................................................................... 957
Additional Readings .................................................................................................. 962
20. INTEGRAL TRANSFORMS............................................................................................................. 963
20.1 Introduction ............................................................................................................... 963
20.2 Fourier Transforms..................................................................................................... 966
20.3 Properties of Fourier Transforms ............................................................................... 980
20.4 Fourier Convolution Theorem..................................................................................... 985
20.5 Signal – Proccesing Applications ................................................................................ 997
20.6 Discrete Fourier Transforms..................................................................................... 1002
20.7 Laplace Transforms.................................................................................................. 1008
20.8 Properties of Laplace Transforms............................................................................. 1016
20.9 Laplace Convolution Transforms.............................................................................. 1034
20.10 Inverse Laplace Transforms...................................................................................... 1038
Additional Readings ................................................................................................ 1045
21. INTEGRAL EQUATIONS ............................................................................................................. 1047
21.1 Introduction ............................................................................................................. 1047
21.2 Some Special Methods ............................................................................................. 1053
21.3 Neumann Series ....................................................................................................... 1064
21.4 Hilbert – Schmidt Theory.......................................................................................... 1069
Additional Readings ................................................................................................ 1079
22. CALCULUS OF VARIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
2 2 . 1 Euler Equation.......................................................................................................... 1081
22.2 More General Variations.......................................................................................... 1096
22.3 Constrained Minima/Maxima .................................................................................. 1107
22.4 Variation with Constraints ....................................................................................... 1111
Additional Readings ................................................................................................ 1124
23. PROBABILITY AND STATISTICS.................................................................................................... 1125
23.1 Probability: Definitions, Simple Properties............................................................... 1126
23.2 Random Variables.................................................................................................... 1134
23.3 Binomial Distribution ............................................................................................... 1148
23.4 Poisson Distribution ................................................................................................. 1151
23.5 Gauss’ Nomal Distribution ....................................................................................... 1155
23.6 Transformation of Random Variables ...................................................................... 1159
23.7 Statistics................................................................................................................... 1165
Additional Readings ................................................................................................ 1179
INDEX ............................................................................................................................... ............ 1181
1. MATHEMATICAL PRELIMINARIES ...................................................................................................... 1
1 . 1 . Infinite Series.................................................................................................................. 1
1.2. Series of Functions ....................................................................................................... 21
1.3. Binomial Theorem ........................................................................................................ 33
1.4. Mathematical Induction............................................................................................... 40
1.5. Operations of Series Expansions of Functions .............................................................. 41
1.6. Some Important Series................................................................................................. 45
1.7. Vectors ......................................................................................................................... 46
1.8. Complex Numbers and Functions................................................................................. 53
1.9. Derivatives and Extrema .............................................................................................. 62
1.10. Evaluation of Integrals ................................................................................................. 65
1.11. Dirac Delta Functions ................................................................................................... 75
Additional Readings .................................................................................................... 82
2. DETERMINANTS AND MATRICES .................................................................................................... 83
2.1 Determinants ............................................................................................................... 83
2.2 Matrices ....................................................................................................................... 95
Additional Readings .................................................................................................. 121
3. VECTOR ANALYSIS .................................................................................................................... 123
3.1 Review of Basics Properties........................................................................................ 124
3.2 Vector in 3 ‐ D Spaces................................................................................................. 126
3.3 Coordinate Transformations ...................................................................................... 133
3 . 4 Rotations in 3 ........................................................................................................ 139
3.5 Differential Vector Operators..................................................................................... 143
3.6 Differential Vector Operators: Further Properties...................................................... 153
3.7 Vector Integrations .................................................................................................... 159
3.8 Integral Theorems...................................................................................................... 164
3.9 Potential Theory......................................................................................................... 170
3.10 Curvilinear Coordinates.............................................................................................. 182
Additional Readings .................................................................................................. 203
4. TENSOR AND DIFFERENTIAL FORMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
4.1 Tensor Analysis .......................................................................................................... 205
4.2 Pseudotensors, Dual Tensors ..................................................................................... 215
4.3 Tensor in General Coordinates ................................................................................... 218
4.4 Jacobians.................................................................................................................... 227
4.5 Differential Forms ...................................................................................................... 232
4.6 Differentiating Forms................................................................................................. 238
4.7 Integrating Forms ...................................................................................................... 243
Additional Readings .................................................................................................. 249
5. VECTOR SPACES ....................................................................................................................... 251
5.1 Vector in Function Spaces .......................................................................................... 251
5.2 Gram ‐ Schmidt Orthogonalization ............................................................................. 269
5.3 Operators ................................................................................................................... 275
5.4 Self‐Adjoint Operators ................................................................................................ 283
5.5 Unitary Operators ...................................................................................................... 287
5.6 Transformations of Operators.................................................................................... 292
5.7 Invariants ................................................................................................................... 294
5.8 Summary – Vector Space Notations........................................................................... 296
Additional Readings .................................................................................................. 297
6. EIGENVALUE PROBLEMS............................................................................................................. 299
6.1 Eigenvalue Equations ................................................................................................. 299
6.2 Matrix Eigenvalue Problems ...................................................................................... 301
6.3 Hermitian Eigenvalue Problems ................................................................................. 310
6.4 Hermitian Matrix Diagonalization ............................................................................. 311
6.5 Normal Matrices ........................................................................................................ 319
Additional Readings .................................................................................................. 328
7. ORDINARY DIFFERENTIAL EQUATIONS........................................................................................... 329
7.1 Introduction ............................................................................................................... 329
7.2 First ‐ Order Equations ............................................................................................... 331
7.3 ODEs with Constant Coefficients................................................................................ 342
7.4 Second‐Order Linear ODEs ......................................................................................... 343
7.5 Series Solutions‐ Frobenius‘ Method.......................................................................... 346
7.6 Other Solutions .......................................................................................................... 358
7 . 7 Inhomogeneous Linear ODEs ..................................................................................... 375
7.8 Nonlinear Differential Equations................................................................................ 377
Additional Readings .................................................................................................. 380
8. STURM – LIOUVILLE THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
8.1 Introduction ............................................................................................................... 381
8.2 Hermitian Operators .................................................................................................. 384
8.3 ODE Eigenvalue Problems .......................................................................................... 389
8.4 Variation Methods ..................................................................................................... 395
8.5 Summary, Eigenvalue Problems................................................................................. 398
Additional Readings .................................................................................................. 399
9. PARTIAL DIFFERENTIAL EQUATIONS .............................................................................................. 401
9.1 Introduction ............................................................................................................... 401
9.2 First ‐ Order Equations ............................................................................................... 403
9.3 Second – Order Equations .......................................................................................... 409
9.4 Separation of Variables............................................................................................. 414
9.5 Laplace and Poisson Equations .................................................................................. 433
9.6 Wave Equations ......................................................................................................... 435
9.7 Heat – Flow, or Diffution PDE..................................................................................... 437
9.8 Summary.................................................................................................................... 444
Additional Readings .................................................................................................. 445
10. GREEN’ FUNCTIONS .................................................................................................................. 447
10.1 One – Dimensional Problems .................................................................................... 448
10.2 Problems in Two and Three Dimensions .................................................................... 459
Additional Readings .................................................................................................. 467
11. COMPLEX VARIABLE THEORY ...................................................................................................... 469
11.1 Complex Variables and Functions .............................................................................. 470
11.2 Cauchy – Riemann Conditions.................................................................................... 471
11.3 Cauchy’s Integral Theorem ........................................................................................ 477
11.4 Cauchy’s Integral Formula ......................................................................................... 486
11.5 Laurent Expansion...................................................................................................... 492
11.6 Singularities ............................................................................................................... 497
11.7 Calculus of Residues ................................................................................................... 509
11.8 Evaluation of Definite Integrals.................................................................................. 522
11.9 Evaluation of Sums..................................................................................................... 544
11.10 Miscellaneous Topics .................................................................................................. 547
Additional Readings .................................................................................................. 550
12. FURTHER TOPICS IN ANALYSIS ..................................................................................................... 551
12.1 Orthogonal Polynomials............................................................................................. 551
12.2 Bernoulli Numbers ..................................................................................................... 560
12.3 Euler – Maclaurin Integration Formula ...................................................................... 567
12.4 Dirichlet Series ........................................................................................................... 571
1 2 . 5 Infinite Products ......................................................................................................... 574
12.6 Asymptotic Series....................................................................................................... 577
12.7 Method of Steepest Descents..................................................................................... 585
12.8 Dispertion Relations ................................................................................................... 591
Additional Readings .................................................................................................. 598
13. GAMMA FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
13.1 Definitions, Properties................................................................................................ 599
13.2 Digamma and Polygamma Functions ........................................................................ 610
13.3 The Beta Function ...................................................................................................... 617
13.4 Stirling’s Series ........................................................................................................... 622
13.5 Riemann Zeta Function .............................................................................................. 626
13.6 Other Ralated Function .............................................................................................. 633
Additional Readings .................................................................................................. 641
14. BESSEL FUNCTIONS ................................................................................................................... 643
14.1 Bessel Functions of the First kind, Jν(x)....................................................................... 643
14.2 Orthogonality............................................................................................................. 661
14.3 Neumann Functions, Bessel Functions of the Second kind ........................................ 667
14.4 Hankel Functions........................................................................................................ 674
14.5 Modified Bessel Functions, Iν(x) and Kν(x)................................................................ 680
14.6 Asymptotic Expansions .............................................................................................. 688
14.7 Spherical Bessel Functions ......................................................................................... 698
Additional Readings .................................................................................................. 713
15. LEGENDRE FUNCTIONS............................................................................................................... 715
15.1 Legendre Polynomials ................................................................................................ 716
15.2 Orthogonality............................................................................................................. 724
15.3 Physical Interpretation of Generating Function ......................................................... 736
15.4 Associated Legendre Equation ................................................................................... 741
15.5 Spherical Harmonics................................................................................................... 756
15.6 Legendre Functions of the Second Kind...................................................................... 766
Additional Readings .................................................................................................. 771
16. ANGULAR MOMENTUM ............................................................................................................. 773
16.1 Angular Momentum Operators.................................................................................. 774
16.2 Angular Momentum Coupling.................................................................................... 784
16.3 Spherical Tensors ....................................................................................................... 796
16.4 Vector Spherical Harmonics ....................................................................................... 809
Additional Readings .................................................................................................. 814
17. GROUP THEORY ....................................................................................................................... 815
17.1 Introduction to Group Theory .................................................................................... 815
17.2 Representation of Groups .......................................................................................... 821
17.3 Symmetry and Physics................................................................................................ 826
17.4 Discrete Groups.......................................................................................................... 830
1 7 . 5 Direct Products........................................................................................................... 837
17.6 Simmetric Group ........................................................................................................ 840
17.7 Continous Groups....................................................................................................... 845
17.8 Lorentz Group ............................................................................................................ 862
17.9 Lorentz Covariance of Maxwell’s Equantions............................................................. 866
17.10 Space Groups ............................................................................................................. 869
Additional Readings .................................................................................................. 870
18. MORE SPECIAL FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871
18.1 Hermite Functions...................................................................................................... 871
18.2 Applications of Hermite Functions ............................................................................. 878
18.3 Laguerre Functions..................................................................................................... 889
18.4 Chebyshev Polynomials .............................................................................................. 899
18.5 Hypergeometric Functions ......................................................................................... 911
18.6 Confluent Hypergeometric Functions......................................................................... 917
18.7 Dilogarithm ................................................................................................................ 923
18.8 Elliptic Integrals.......................................................................................................... 927
Additional Readings .................................................................................................. 932
19. FOURIER SERIES........................................................................................................................ 935
19.1 General Properties ..................................................................................................... 935
19.2 Application of Fourier Series ...................................................................................... 949
19.3 Gibbs Phenomenon .................................................................................................... 957
Additional Readings .................................................................................................. 962
20. INTEGRAL TRANSFORMS............................................................................................................. 963
20.1 Introduction ............................................................................................................... 963
20.2 Fourier Transforms..................................................................................................... 966
20.3 Properties of Fourier Transforms ............................................................................... 980
20.4 Fourier Convolution Theorem..................................................................................... 985
20.5 Signal – Proccesing Applications ................................................................................ 997
20.6 Discrete Fourier Transforms..................................................................................... 1002
20.7 Laplace Transforms.................................................................................................. 1008
20.8 Properties of Laplace Transforms............................................................................. 1016
20.9 Laplace Convolution Transforms.............................................................................. 1034
20.10 Inverse Laplace Transforms...................................................................................... 1038
Additional Readings ................................................................................................ 1045
21. INTEGRAL EQUATIONS ............................................................................................................. 1047
21.1 Introduction ............................................................................................................. 1047
21.2 Some Special Methods ............................................................................................. 1053
21.3 Neumann Series ....................................................................................................... 1064
21.4 Hilbert – Schmidt Theory.......................................................................................... 1069
Additional Readings ................................................................................................ 1079
22. CALCULUS OF VARIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
2 2 . 1 Euler Equation.......................................................................................................... 1081
22.2 More General Variations.......................................................................................... 1096
22.3 Constrained Minima/Maxima .................................................................................. 1107
22.4 Variation with Constraints ....................................................................................... 1111
Additional Readings ................................................................................................ 1124
23. PROBABILITY AND STATISTICS.................................................................................................... 1125
23.1 Probability: Definitions, Simple Properties............................................................... 1126
23.2 Random Variables.................................................................................................... 1134
23.3 Binomial Distribution ............................................................................................... 1148
23.4 Poisson Distribution ................................................................................................. 1151
23.5 Gauss’ Nomal Distribution ....................................................................................... 1155
23.6 Transformation of Random Variables ...................................................................... 1159
23.7 Statistics................................................................................................................... 1165
Additional Readings ................................................................................................ 1179
INDEX ............................................................................................................................... ............ 1181
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