786 14.5 The First Order, Linear Differential Equation.. 15 First Order Linear Systems of Differential Equations 84615.1 Introduction.. 872 16 Theory of Linear Ordinary Differential Equa
Trang 20.1 Advice to Teachers xxv
0.2 Acknowledgments xxv
0.3 Warnings and Disclaimers xxvi
0.4 Suggested Use xxvii
0.5 About the Title xxvii
I Algebra 1 1 Sets and Functions 2 1.1 Sets 2
1.2 Single Valued Functions 4
1.3 Inverses and Multi-Valued Functions 6
1.4 Transforming Equations 9
1.5 Exercises 11
1.6 Hints 14
1.7 Solutions 16
Trang 32 Vectors 22
2.1 Vectors 22
2.1.1 Scalars and Vectors 22
2.1.2 The Kronecker Delta and Einstein Summation Convention 25
2.1.3 The Dot and Cross Product 26
2.2 Sets of Vectors in n Dimensions 33
2.3 Exercises 36
2.4 Hints 38
2.5 Solutions 40
II Calculus 47 3 Differential Calculus 48 3.1 Limits of Functions 48
3.2 Continuous Functions 53
3.3 The Derivative 56
3.4 Implicit Differentiation 61
3.5 Maxima and Minima 62
3.6 Mean Value Theorems 66
3.6.1 Application: Using Taylor’s Theorem to Approximate Functions 68
3.6.2 Application: Finite Difference Schemes 73
3.7 L’Hospital’s Rule 75
3.8 Exercises 81
3.8.1 Limits of Functions 81
3.8.2 Continuous Functions 81
3.8.3 The Derivative 82
3.8.4 Implicit Differentiation 84
3.8.5 Maxima and Minima 84
3.8.6 Mean Value Theorems 85
Trang 43.8.7 L’Hospital’s Rule 85
3.9 Hints 87
3.10 Solutions 93
3.11 Quiz 113
3.12 Quiz Solutions 114
4 Integral Calculus 116 4.1 The Indefinite Integral 116
4.2 The Definite Integral 122
4.2.1 Definition 122
4.2.2 Properties 123
4.3 The Fundamental Theorem of Integral Calculus 125
4.4 Techniques of Integration 127
4.4.1 Partial Fractions 127
4.5 Improper Integrals 130
4.6 Exercises 134
4.6.1 The Indefinite Integral 134
4.6.2 The Definite Integral 134
4.6.3 The Fundamental Theorem of Integration 136
4.6.4 Techniques of Integration 136
4.6.5 Improper Integrals 137
4.7 Hints 138
4.8 Solutions 141
4.9 Quiz 150
4.10 Quiz Solutions 151
5 Vector Calculus 154 5.1 Vector Functions 154
5.2 Gradient, Divergence and Curl 155
5.3 Exercises 163
Trang 55.4 Hints 166
5.5 Solutions 168
5.6 Quiz 177
5.7 Quiz Solutions 178
III Functions of a Complex Variable 179 6 Complex Numbers 180 6.1 Complex Numbers 180
6.2 The Complex Plane 184
6.3 Polar Form 188
6.4 Arithmetic and Vectors 193
6.5 Integer Exponents 195
6.6 Rational Exponents 197
6.7 Exercises 201
6.8 Hints 208
6.9 Solutions 211
7 Functions of a Complex Variable 239 7.1 Curves and Regions 239
7.2 The Point at Infinity and the Stereographic Projection 242
7.3 A Gentle Introduction to Branch Points 246
7.4 Cartesian and Modulus-Argument Form 246
7.5 Graphing Functions of a Complex Variable 249
7.6 Trigonometric Functions 252
7.7 Inverse Trigonometric Functions 259
7.8 Riemann Surfaces 268
7.9 Branch Points 270
7.10 Exercises 286
Trang 67.11 Hints 297
7.12 Solutions 302
8 Analytic Functions 360 8.1 Complex Derivatives 360
8.2 Cauchy-Riemann Equations 367
8.3 Harmonic Functions 372
8.4 Singularities 377
8.4.1 Categorization of Singularities 377
8.4.2 Isolated and Non-Isolated Singularities 381
8.5 Application: Potential Flow 383
8.6 Exercises 388
8.7 Hints 396
8.8 Solutions 399
9 Analytic Continuation 437 9.1 Analytic Continuation 437
9.2 Analytic Continuation of Sums 440
9.3 Analytic Functions Defined in Terms of Real Variables 442
9.3.1 Polar Coordinates 446
9.3.2 Analytic Functions Defined in Terms of Their Real or Imaginary Parts 450
9.4 Exercises 454
9.5 Hints 456
9.6 Solutions 457
10 Contour Integration and the Cauchy-Goursat Theorem 462 10.1 Line Integrals 462
10.2 Contour Integrals 464
10.2.1 Maximum Modulus Integral Bound 466
10.3 The Cauchy-Goursat Theorem 467
Trang 710.4 Contour Deformation 469
10.5 Morera’s Theorem 471
10.6 Indefinite Integrals 473
10.7 Fundamental Theorem of Calculus via Primitives 474
10.7.1 Line Integrals and Primitives 474
10.7.2 Contour Integrals 474
10.8 Fundamental Theorem of Calculus via Complex Calculus 475
10.9 Exercises 478
10.10Hints 482
10.11Solutions 483
11 Cauchy’s Integral Formula 493 11.1 Cauchy’s Integral Formula 494
11.2 The Argument Theorem 501
11.3 Rouche’s Theorem 502
11.4 Exercises 505
11.5 Hints 509
11.6 Solutions 511
12 Series and Convergence 525 12.1 Series of Constants 525
12.1.1 Definitions 525
12.1.2 Special Series 527
12.1.3 Convergence Tests 529
12.2 Uniform Convergence 536
12.2.1 Tests for Uniform Convergence 537
12.2.2 Uniform Convergence and Continuous Functions 539
12.3 Uniformly Convergent Power Series 539
12.4 Integration and Differentiation of Power Series 547
12.5 Taylor Series 550
Trang 812.5.1 Newton’s Binomial Formula 553
12.6 Laurent Series 555
12.7 Exercises 560
12.7.1 Series of Constants 560
12.7.2 Uniform Convergence 566
12.7.3 Uniformly Convergent Power Series 566
12.7.4 Integration and Differentiation of Power Series 568
12.7.5 Taylor Series 569
12.7.6 Laurent Series 571
12.8 Hints 574
12.9 Solutions 582
13 The Residue Theorem 626 13.1 The Residue Theorem 626
13.2 Cauchy Principal Value for Real Integrals 634
13.2.1 The Cauchy Principal Value 634
13.3 Cauchy Principal Value for Contour Integrals 639
13.4 Integrals on the Real Axis 643
13.5 Fourier Integrals 647
13.6 Fourier Cosine and Sine Integrals 649
13.7 Contour Integration and Branch Cuts 652
13.8 Exploiting Symmetry 655
13.8.1 Wedge Contours 655
13.8.2 Box Contours 658
13.9 Definite Integrals Involving Sine and Cosine 659
13.10Infinite Sums 662
13.11Exercises 666
13.12Hints 680
13.13Solutions 686
Trang 9IV Ordinary Differential Equations 772
14.1 Notation 773
14.2 Example Problems 775
14.2.1 Growth and Decay 775
14.3 One Parameter Families of Functions 777
14.4 Integrable Forms 779
14.4.1 Separable Equations 780
14.4.2 Exact Equations 782
14.4.3 Homogeneous Coefficient Equations 786
14.5 The First Order, Linear Differential Equation 791
14.5.1 Homogeneous Equations 791
14.5.2 Inhomogeneous Equations 792
14.5.3 Variation of Parameters 795
14.6 Initial Conditions 796
14.6.1 Piecewise Continuous Coefficients and Inhomogeneities 797
14.7 Well-Posed Problems 801
14.8 Equations in the Complex Plane 803
14.8.1 Ordinary Points 803
14.8.2 Regular Singular Points 806
14.8.3 Irregular Singular Points 812
14.8.4 The Point at Infinity 814
14.9 Additional Exercises 816
14.10Hints 819
14.11Solutions 822
14.12Quiz 843
14.13Quiz Solutions 844
Trang 1015 First Order Linear Systems of Differential Equations 846
15.1 Introduction 846
15.2 Using Eigenvalues and Eigenvectors to find Homogeneous Solutions 847
15.3 Matrices and Jordan Canonical Form 852
15.4 Using the Matrix Exponential 860
15.5 Exercises 865
15.6 Hints 870
15.7 Solutions 872
16 Theory of Linear Ordinary Differential Equations 900 16.1 Exact Equations 900
16.2 Nature of Solutions 901
16.3 Transformation to a First Order System 905
16.4 The Wronskian 905
16.4.1 Derivative of a Determinant 905
16.4.2 The Wronskian of a Set of Functions 906
16.4.3 The Wronskian of the Solutions to a Differential Equation 908
16.5 Well-Posed Problems 911
16.6 The Fundamental Set of Solutions 913
16.7 Adjoint Equations 915
16.8 Additional Exercises 919
16.9 Hints 920
16.10Solutions 922
16.11Quiz 928
16.12Quiz Solutions 929
17 Techniques for Linear Differential Equations 930 17.1 Constant Coefficient Equations 930
17.1.1 Second Order Equations 931
17.1.2 Real-Valued Solutions 935
Trang 1117.1.3 Higher Order Equations 937
17.2 Euler Equations 940
17.2.1 Real-Valued Solutions 942
17.3 Exact Equations 945
17.4 Equations Without Explicit Dependence on y 946
17.5 Reduction of Order 947
17.6 *Reduction of Order and the Adjoint Equation 948
17.7 Additional Exercises 951
17.8 Hints 957
17.9 Solutions 960
18 Techniques for Nonlinear Differential Equations 984 18.1 Bernoulli Equations 984
18.2 Riccati Equations 986
18.3 Exchanging the Dependent and Independent Variables 990
18.4 Autonomous Equations 992
18.5 *Equidimensional-in-x Equations 995
18.6 *Equidimensional-in-y Equations 997
18.7 *Scale-Invariant Equations 1000
18.8 Exercises 1001
18.9 Hints 1004
18.10Solutions 1006
19 Transformations and Canonical Forms 1018 19.1 The Constant Coefficient Equation 1018
19.2 Normal Form 1021
19.2.1 Second Order Equations 1021
19.2.2 Higher Order Differential Equations 1022
19.3 Transformations of the Independent Variable 1024
19.3.1 Transformation to the form u” + a(x) u = 0 1024
Trang 1219.3.2 Transformation to a Constant Coefficient Equation 1025
19.4 Integral Equations 1027
19.4.1 Initial Value Problems 1027
19.4.2 Boundary Value Problems 1029
19.5 Exercises 1032
19.6 Hints 1034
19.7 Solutions 1035
20 The Dirac Delta Function 1041 20.1 Derivative of the Heaviside Function 1041
20.2 The Delta Function as a Limit 1043
20.3 Higher Dimensions 1045
20.4 Non-Rectangular Coordinate Systems 1046
20.5 Exercises 1048
20.6 Hints 1050
20.7 Solutions 1052
21 Inhomogeneous Differential Equations 1059 21.1 Particular Solutions 1059
21.2 Method of Undetermined Coefficients 1061
21.3 Variation of Parameters 1065
21.3.1 Second Order Differential Equations 1065
21.3.2 Higher Order Differential Equations 1068
21.4 Piecewise Continuous Coefficients and Inhomogeneities 1071
21.5 Inhomogeneous Boundary Conditions 1074
21.5.1 Eliminating Inhomogeneous Boundary Conditions 1074
21.5.2 Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions 1076
21.5.3 Existence of Solutions of Problems with Inhomogeneous Boundary Conditions 1077
21.6 Green Functions for First Order Equations 1079
21.7 Green Functions for Second Order Equations 1082
Trang 1321.7.1 Green Functions for Sturm-Liouville Problems 1092
21.7.2 Initial Value Problems 1095
21.7.3 Problems with Unmixed Boundary Conditions 1098
21.7.4 Problems with Mixed Boundary Conditions 1100
21.8 Green Functions for Higher Order Problems 1104
21.9 Fredholm Alternative Theorem 1109
21.10Exercises 1117
21.11Hints 1123
21.12Solutions 1126
21.13Quiz 1164
21.14Quiz Solutions 1165
22 Difference Equations 1166 22.1 Introduction 1166
22.2 Exact Equations 1168
22.3 Homogeneous First Order 1169
22.4 Inhomogeneous First Order 1171
22.5 Homogeneous Constant Coefficient Equations 1174
22.6 Reduction of Order 1177
22.7 Exercises 1179
22.8 Hints 1180
22.9 Solutions 1181
23 Series Solutions of Differential Equations 1184 23.1 Ordinary Points 1184
23.1.1 Taylor Series Expansion for a Second Order Differential Equation 1188
23.2 Regular Singular Points of Second Order Equations 1198
23.2.1 Indicial Equation 1201
23.2.2 The Case: Double Root 1203
23.2.3 The Case: Roots Differ by an Integer 1206
Trang 1423.3 Irregular Singular Points 1216
23.4 The Point at Infinity 1216
23.5 Exercises 1219
23.6 Hints 1224
23.7 Solutions 1225
23.8 Quiz 1248
23.9 Quiz Solutions 1249
24 Asymptotic Expansions 1251 24.1 Asymptotic Relations 1251
24.2 Leading Order Behavior of Differential Equations 1255
24.3 Integration by Parts 1263
24.4 Asymptotic Series 1270
24.5 Asymptotic Expansions of Differential Equations 1272
24.5.1 The Parabolic Cylinder Equation 1272
25 Hilbert Spaces 1278 25.1 Linear Spaces 1278
25.2 Inner Products 1280
25.3 Norms 1281
25.4 Linear Independence 1283
25.5 Orthogonality 1283
25.6 Gramm-Schmidt Orthogonalization 1284
25.7 Orthonormal Function Expansion 1287
25.8 Sets Of Functions 1288
25.9 Least Squares Fit to a Function and Completeness 1294
25.10Closure Relation 1297
25.11Linear Operators 1302
25.12Exercises 1303
25.13Hints 1304
Trang 1525.14Solutions 1305
26 Self Adjoint Linear Operators 1307 26.1 Adjoint Operators 1307
26.2 Self-Adjoint Operators 1308
26.3 Exercises 1311
26.4 Hints 1312
26.5 Solutions 1313
27 Self-Adjoint Boundary Value Problems 1314 27.1 Summary of Adjoint Operators 1314
27.2 Formally Self-Adjoint Operators 1315
27.3 Self-Adjoint Problems 1318
27.4 Self-Adjoint Eigenvalue Problems 1318
27.5 Inhomogeneous Equations 1323
27.6 Exercises 1326
27.7 Hints 1327
27.8 Solutions 1328
28 Fourier Series 1330 28.1 An Eigenvalue Problem 1330
28.2 Fourier Series 1333
28.3 Least Squares Fit 1337
28.4 Fourier Series for Functions Defined on Arbitrary Ranges 1341
28.5 Fourier Cosine Series 1344
28.6 Fourier Sine Series 1345
28.7 Complex Fourier Series and Parseval’s Theorem 1346
28.8 Behavior of Fourier Coefficients 1349
28.9 Gibb’s Phenomenon 1358
28.10Integrating and Differentiating Fourier Series 1358
Trang 1628.11Exercises 1363
28.12Hints 1371
28.13Solutions 1373
29 Regular Sturm-Liouville Problems 1420 29.1 Derivation of the Sturm-Liouville Form 1420
29.2 Properties of Regular Sturm-Liouville Problems 1422
29.3 Solving Differential Equations With Eigenfunction Expansions 1433
29.4 Exercises 1439
29.5 Hints 1443
29.6 Solutions 1445
30 Integrals and Convergence 1470 30.1 Uniform Convergence of Integrals 1470
30.2 The Riemann-Lebesgue Lemma 1471
30.3 Cauchy Principal Value 1472
30.3.1 Integrals on an Infinite Domain 1472
30.3.2 Singular Functions 1473
31 The Laplace Transform 1475 31.1 The Laplace Transform 1475
31.2 The Inverse Laplace Transform 1477
31.2.1 ˆf (s) with Poles 1480
31.2.2 ˆf (s) with Branch Points 1484
31.2.3 Asymptotic Behavior of ˆf (s) 1488
31.3 Properties of the Laplace Transform 1489
31.4 Constant Coefficient Differential Equations 1492
31.5 Systems of Constant Coefficient Differential Equations 1495
31.6 Exercises 1497
31.7 Hints 1504
Trang 1731.8 Solutions 1507
32 The Fourier Transform 1539 32.1 Derivation from a Fourier Series 1539
32.2 The Fourier Transform 1541
32.2.1 A Word of Caution 1544
32.3 Evaluating Fourier Integrals 1545
32.3.1 Integrals that Converge 1545
32.3.2 Cauchy Principal Value and Integrals that are Not Absolutely Convergent 1548
32.3.3 Analytic Continuation 1550
32.4 Properties of the Fourier Transform 1552
32.4.1 Closure Relation 1552
32.4.2 Fourier Transform of a Derivative 1553
32.4.3 Fourier Convolution Theorem 1554
32.4.4 Parseval’s Theorem 1557
32.4.5 Shift Property 1559
32.4.6 Fourier Transform of x f(x) 1559
32.5 Solving Differential Equations with the Fourier Transform 1560
32.6 The Fourier Cosine and Sine Transform 1562
32.6.1 The Fourier Cosine Transform 1562
32.6.2 The Fourier Sine Transform 1563
32.7 Properties of the Fourier Cosine and Sine Transform 1564
32.7.1 Transforms of Derivatives 1564
32.7.2 Convolution Theorems 1566
32.7.3 Cosine and Sine Transform in Terms of the Fourier Transform 1568
32.8 Solving Differential Equations with the Fourier Cosine and Sine Transforms 1569
32.9 Exercises 1571
32.10Hints 1578
32.11Solutions 1581
Trang 1833 The Gamma Function 1605
33.1 Euler’s Formula 1605
33.2 Hankel’s Formula 1607
33.3 Gauss’ Formula 1609
33.4 Weierstrass’ Formula 1611
33.5 Stirling’s Approximation 1613
33.6 Exercises 1618
33.7 Hints 1619
33.8 Solutions 1620
34 Bessel Functions 1622 34.1 Bessel’s Equation 1622
34.2 Frobeneius Series Solution about z = 0 1623
34.2.1 Behavior at Infinity 1626
34.3 Bessel Functions of the First Kind 1628
34.3.1 The Bessel Function Satisfies Bessel’s Equation 1629
34.3.2 Series Expansion of the Bessel Function 1630
34.3.3 Bessel Functions of Non-Integer Order 1633
34.3.4 Recursion Formulas 1636
34.3.5 Bessel Functions of Half-Integer Order 1639
34.4 Neumann Expansions 1640
34.5 Bessel Functions of the Second Kind 1644
34.6 Hankel Functions 1646
34.7 The Modified Bessel Equation 1646
34.8 Exercises 1650
34.9 Hints 1655
34.10Solutions 1657
Trang 19V Partial Differential Equations 1680
35.1 Exercises 1682
35.2 Hints 1683
35.3 Solutions 1684
36 Classification of Partial Differential Equations 1685 36.1 Classification of Second Order Quasi-Linear Equations 1685
36.1.1 Hyperbolic Equations 1686
36.1.2 Parabolic equations 1691
36.1.3 Elliptic Equations 1692
36.2 Equilibrium Solutions 1694
36.3 Exercises 1696
36.4 Hints 1697
36.5 Solutions 1698
37 Separation of Variables 1704 37.1 Eigensolutions of Homogeneous Equations 1704
37.2 Homogeneous Equations with Homogeneous Boundary Conditions 1704
37.3 Time-Independent Sources and Boundary Conditions 1706
37.4 Inhomogeneous Equations with Homogeneous Boundary Conditions 1709
37.5 Inhomogeneous Boundary Conditions 1710
37.6 The Wave Equation 1713
37.7 General Method 1716
37.8 Exercises 1718
37.9 Hints 1734
37.10Solutions 1739
Trang 2038 Finite Transforms 1821
38.1 Exercises 1825
38.2 Hints 1826
38.3 Solutions 1827
39 The Diffusion Equation 1831 39.1 Exercises 1832
39.2 Hints 1834
39.3 Solutions 1835
40 Laplace’s Equation 1841 40.1 Introduction 1841
40.2 Fundamental Solution 1841
40.2.1 Two Dimensional Space 1842
40.3 Exercises 1843
40.4 Hints 1846
40.5 Solutions 1847
41 Waves 1859 41.1 Exercises 1860
41.2 Hints 1866
41.3 Solutions 1868
42 Similarity Methods 1888 42.1 Exercises 1892
42.2 Hints 1893
42.3 Solutions 1894
43 Method of Characteristics 1897 43.1 First Order Linear Equations 1897
43.2 First Order Quasi-Linear Equations 1898