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Table of Contents

1 Anti-Copyright 22

2 Preface 23
2.1 Advice to Teachers ~ 23
2.2 Acknowledgments ~ 23
2.3 Warnings and Disclaimers ~ 24
2.4 Suggested Use ~ 25
2.5 About the Title ~ 25

I Algebra 1

3 Sets and Functions 2
3.1 Sets ~ 2
3.2 Single Valued Functions ~ 4
3.3 Inverses and Multi-Valued Functions ~ 6
3.4 Transforming Equations ~ 9

4 Vectors 11
4.1 Vectors ~ 11
4.1.1 Scalars and Vectors ~ 11
4.1.2 The Kronecker Delta and Einstein Summation Convention ~ 14
4.1.3 The Dot and Cross Product ~ 15
4.2 Sets of Vectors in n Dimensions ~ 23
4.3 Exercises ~ 25
4.4 Hints ~ 27
4.5 Solutions ~ 29

II Calculus 36

5 Differential Calculus 37
5.1 Limits of Functions ~ 37
5.2 Continuous Functions ~ 42
5.3 The Derivative ~ 45
5.4 Implicit Differentiation ~ 50
5.5 Maxima and Minima ~ 52
5.6 Mean Value Theorems ~ 55
5.6.1 Application: Using Taylor's Theorem to Approximate Functions ~ 57
5.6.2 Application: Finite Difference Schemes ~ 62
5.7 L'Hospital's Rule ~ 64
5.8 Exercises ~ 70
5.9 Hints ~ 75
5.10 Solutions ~ 81

6 Integral Calculus 100
6.1 The Indefi nite Integral ~ 100
6.2 The Defi nite Integral ~ 106
6.2.1 Defi nition ~ 106
6.2.2 Properties ~ 107
6.3 The Fundamental Theorem of Integral Calculus ~ 109
6.4 Techniques of Integration ~ 111
6.4.1 Partial Fractions ~ 111
6.5 Improper Integrals ~ 114
6.6 Exercises ~ 118
6.7 Hints ~ 121
6.8 Solutions ~ 125

7 Vector Calculus 134
7.1 Vector Functions ~ 134
7.2 Gradient, Divergence and Curl ~ 135
7.3 Exercises ~ 142
7.4 Hints ~ 144
7.5 Solutions ~ 145

III Functions of a Complex Variable 150

8 Complex Numbers 151
8.1 Complex Numbers ~ 151
8.2 The Complex Plane ~ 154
8.3 Polar Form ~ 158
8.4 Arithmetic and Vectors ~ 163
8.5 Integer Exponents ~ 164
8.6 Rational Exponents ~ 166
8.7 Exercises ~ 170
8.8 Hints ~ 176
8.9 Solutions ~ 179

9 Functions of a Complex Variable 202
9.1 Curves and Regions ~ 202
9.2 Cartesian and Modulus-Argument Form ~ 206
9.3 Graphing Functions of a Complex Variable ~ 208
9.4 Trigonometric Functions ~ 212
9.5 Inverse Trigonometric Functions ~ 217
9.6 Branch Points ~ 226
9.7 Exercises ~ 243
9.8 Hints ~ 253
9.9 Solutions ~ 258

10 Analytic Functions 303
10.1 Complex Derivatives ~ 303
10.2 Cauchy-Riemann Equations ~ 310
10.3 Harmonic Functions ~ 315
10.4 Singularities ~ 320
10.4.1 Categorization of Singularities ~ 321
10.4.2 Isolated and Non-Isolated Singularities ~ 325
10.5 Exercises ~ 327
10.6 Hints ~ 332
10.7 Solutions ~ 334

11 Analytic Continuation 356
11.1 Analytic Continuation ~ 356
11.2 Analytic Continuation of Sums ~ 359
11.3 Analytic Functions Defi ned in Terms of Real Variables ~ 360
11.3.1 Polar Coordinates ~ 366
11.3.2 Analytic Functions Defi ned in Terms of Their Real or Imaginary Parts ~ 369
11.4 Exercises ~ 373
11.5 Hints ~ 375
11.6 Solutions ~ 376

12 Contour Integration and Cauchy's Theorem 381
12.1 Line Integrals ~ 381
12.2 Under Construction ~ 386
12.3 Cauchy's Theorem ~ 388
12.4 Indefi nite Integrals ~ 390
12.5 Contour Integrals ~ 391
12.6 Exercises ~ 395
12.7 Hints ~ 397
12.8 Solutions ~ 398

13 Cauchy's Integral Formula 403
13.1 Cauchy's Integral Formula ~ 404
13.2 The Argument Theorem ~ 411
13.3 Rouche's Theorem ~ 413
13.4 Exercises ~ 415
13.5 Hints ~ 417
13.6 Solutions ~ 418

14 Series and Convergence 422
14.1 Series of Constants ~ 422
14.1.1 Defi nitions ~ 422
14.1.2 Special Series ~ 425
14.1.3 Convergence Tests ~ 426
14.2 Uniform Convergence ~ 432
14.2.1 Tests for Uniform Convergence ~ 433
14.2.2 Uniform Convergence and Continuous Functions ~ 435
14.3 Uniformly Convergent Power Series ~ 436
14.4 Integration and Differentiation of Power Series ~ 443
14.5 Taylor Series ~ 446
14.5.1 Newton's Binomial Formula ~ 449
14.6 Laurent Series ~ 452
14.7 Exercises ~ 455
14.8 Hints ~ 462
14.9 Solutions ~ 465

15 The Residue Theorem 490
15.1 The Residue Theorem ~ 490
15.2 Cauchy Principal Value for Real Integrals ~ 498
15.2.1 The Cauchy Principal Value ~ 498
15.3 Cauchy Principal Value for Contour Integrals ~ 503
15.4 Integrals on the Real Axis ~ 507
15.5 Fourier Integrals ~ 512
15.6 Fourier Cosine and Sine Integrals ~ 515
15.7 Contour Integration and Branch Cuts ~ 517
15.8 Exploiting Symmetry ~ 521
15.8.1 Wedge Contours ~ 521
15.8.2 Box Contours ~ 524
15.9 Defi nite Integrals Involving Sine and Cosine ~ 525
15.10Infi nite Sums ~ 528
15.11Exercises ~ 532
15.12Hints ~ 546
15.13Solutions ~ 553

IV Ordinary Differential Equations 634

16 First Order Differential Equations 635
16.1 Notation ~ 635
16.2 One Parameter Families of Functions ~ 637
16.3 Exact Equations ~ 639
16.3.1 Separable Equations ~ 643
16.3.2 Homogeneous Coeficient Equations ~ 645
16.4 The First Order, Linear Differential Equation ~ 649
16.4.1 Homogeneous Equations ~ 649
16.4.2 Inhomogeneous Equations ~ 650
16.4.3 Variation of Parameters ~ 651
16.5 Initial Conditions ~ 653
16.5.1 Piecewise Continuous Coeficients and Inhomogeneities ~ 654
16.6 Well-Posed Problems ~ 659
16.7 Equations in the Complex Plane ~ 661
16.7.1 Ordinary Points ~ 661
16.7.2 Regular Singular Points ~ 664
16.7.3 Irregular Singular Points ~ 669
16.7.4 The Point at Infi nity ~ 671
16.8 Exercises ~ 674
16.9 Hints ~ 680
16.10Solutions ~ 683

17 First Order Systems of Differential Equations 705
17.1 Matrices and Jordan Canonical Form ~ 705
17.2 Systems of Differential Equations ~ 713
17.3 Exercises ~ 719
17.4 Hints ~ 725
17.5 Solutions ~ 727

18 Theory of Linear Ordinary Differential Equations 757
18.1 Nature of Solutions ~ 758
18.2 Transformation to a First Order System ~ 761
18.3 The Wronskian ~ 762
18.3.1 Derivative of a Determinant ~ 762
18.3.2 The Wronskian of a Set of Functions ~ 763
18.3.3 The Wronskian of the Solutions to a Differential Equation ~ 765
18.4 Well-Posed Problems ~ 768
18.5 The Fundamental Set of Solutions ~ 770
18.6 Adjoint Equations ~ 773
18.7 Exercises ~ 776
18.8 Hints ~ 778
18.9 Solutions ~ 780

19 Techniques for Linear Differential Equations 786
19.1 Constant Coeficient Equations ~ 786
19.1.1 Second Order Equations ~ 787
19.1.2 Higher Order Equations ~ 791
19.1.3 Real-Valued Solutions ~ 793
19.2 Euler Equations ~ 795
19.2.1 Real-Valued Solutions ~ 798
19.3 Exact Equations ~ 801
19.4 Equations Without Explicit Dependence on y ~ 802
19.5 Reduction of Order ~ 803
19.6 *Reduction of Order and the Adjoint Equation ~ 804
19.7 Exercises ~ 807
19.8 Hints ~ 814
19.9 Solutions ~ 817

20 Techniques for Nonlinear Differential Equations 842
20.1 Bernoulli Equations ~ 842
20.2 Riccati Equations ~ 844
20.3 Exchanging the Dependent and Independent Variables ~ 848
20.4 Autonomous Equations ~ 850
20.5 *Equidimensional-in-x Equations ~ 854
20.6 *Equidimensional-in-y Equations ~ 856
20.7 *Scale-Invariant Equations ~ 859
20.8 Exercises ~ 860
20.9 Hints ~ 864
20.10Solutions ~ 866

21 Transformations and Canonical Forms 878
21.1 The Constant Coeficient Equation ~ 878
21.2 Normal Form ~ 881
21.2.1 Second Order Equations ~ 881
21.2.2 Higher Order Differential Equations ~ 883
21.3 Transformations of the Independent Variable ~ 885
21.3.1 Transformation to the form u" + a(x) u = 0 ~ 885
21.3.2 Transformation to a Constant Coeficient Equation ~ 886
21.4 Integral Equations ~ 888
21.4.1 Initial Value Problems ~ 889
21.4.2 Boundary Value Problems ~ 891
21.5 Exercises ~ 894
21.6 Hints ~ 896
21.7 Solutions ~ 897

22 The Dirac Delta Function 904
22.1 Derivative of the Heaviside Function ~ 904
22.2 The Delta Function as a Limit ~ 906
22.3 Higher Dimensions ~ 908
22.4 Non-Rectangular Coordinate Systems ~ 908
22.5 Exercises ~ 910
22.6 Hints ~ 911
22.7 Solutions ~ 912

23 Inhomogeneous Differential Equations 915
23.1 Particular Solutions ~ 915
23.2 Method of Undetermined Coeficients ~ 917
23.3 Variation of Parameters ~ 921
23.3.1 Second Order Differential Equations ~ 921
23.3.2 Higher Order Differential Equations ~ 925
23.4 Piecewise Continuous Coeficients and Inhomogeneities ~ 927
23.5 Inhomogeneous Boundary Conditions ~ 931
23.5.1 Eliminating Inhomogeneous Boundary Conditions ~ 931
23.5.2 Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions ~ 933
23.5.3 Existence of Solutions of Problems with Inhomogeneous Boundary Conditions ~ 934
23.6 Green Functions for First Order Equations ~ 936
23.7 Green Functions for Second Order Equations ~ 939
23.7.1 Green Functions for Sturm-Liouville Problems ~ 949
23.7.2 Initial Value Problems ~ 953
23.7.3 Problems with Unmixed Boundary Conditions ~ 956
23.7.4 Problems with Mixed Boundary Conditions ~ 958
23.8 Green Functions for Higher Order Problems ~ 962
23.9 Fredholm Alternative Theorem ~ 968
23.10Exercises ~ 975
23.11Hints ~ 982
23.12Solutions ~ 986

24 Difference Equations 1027
24.1 Introduction ~ 1027
24.2 Exact Equations ~ 1029
24.3 Homogeneous First Order ~ 1030
24.4 Inhomogeneous First Order ~ 1032
24.5 Homogeneous Constant Coeficient Equations ~ 1035
24.6 Reduction of Order ~ 1038
24.7 Exercises ~ 1040
24.8 Hints ~ 1041
24.9 Solutions ~ 1042

25 Series Solutions of Differential Equations 1046
25.1 Ordinary Points ~ 1046
25.1.1 Taylor Series Expansion for a Second Order Differential Equation ~ 1051
25.2 Regular Singular Points of Second Order Equations ~ 1060
25.2.1 Indicial Equation ~ 1063
25.2.2 The Case: Double Root ~ 1065
25.2.3 The Case: Roots Differ by an Integer ~ 1069
25.3 Irregular Singular Points ~ 1079
25.4 The Point at Infi nity ~ 1079
25.5 Exercises ~ 1082
25.6 Hints ~ 1087
25.7 Solutions ~ 1089

26 Asymptotic Expansions 1113
26.1 Asymptotic Relations ~ 1113
26.2 Leading Order Behavior of Differential Equations ~ 1117
26.3 Integration by Parts ~ 1126
26.4 Asymptotic Series ~ 1133
26.5 Asymptotic Expansions of Differential Equations ~ 1135
26.5.1 The Parabolic Cylinder Equation ~ 1135

27 Hilbert Spaces 1141
27.1 Linear Spaces ~ 1141
27.2 Inner Products ~ 1143
27.3 Norms ~ 1144
27.4 Linear Independence ~ 1147
27.5 Orthogonality ~ 1147
27.6 Gramm-Schmidt Orthogonalization ~ 1147
27.7 Orthonormal Function Expansion ~ 1151
27.8 Sets Of Functions ~ 1152
27.9 Least Squares Fit to a Function and Completeness ~ 1158
27.10Closure Relation ~ 1161
27.11Linear Operators ~ 1166
27.12Exercises ~ 1167
27.13Hints ~ 1168
27.14Solutions ~ 1169

28 Self Adjoint Linear Operators 1171
28.1 Adjoint Operators ~ 1171
28.2 Self-Adjoint Operators ~ 1172
28.3 Exercises ~ 1175
28.4 Hints ~ 1176
28.5 Solutions ~ 1177

29 Self-Adjoint Boundary Value Problems 1178
29.1 Summary of Adjoint Operators ~ 1178
29.2 Formally Self-Adjoint Operators ~ 1179
29.3 Self-Adjoint Problems ~ 1182
29.4 Self-Adjoint Eigenvalue Problems ~ 1183
29.5 Inhomogeneous Equations ~ 1188
29.6 Exercises ~ 1191
29.7 Hints ~ 1192
29.8 Solutions ~ 1193

30 Fourier Series 1195
30.1 An Eigenvalue Problem ~ 1195
30.2 Fourier Series ~ 1198
30.3 Least Squares Fit ~ 1204
30.4 Fourier Series for Functions Defi ned on Arbitrary Ranges ~ 1207
30.5 Fourier Cosine Series ~ 1210
30.6 Fourier Sine Series ~ 1211
30.7 Complex Fourier Series and Parseval's Theorem ~ 1212
30.8 Behavior of Fourier Coeficients ~ 1215
30.9 Gibb's Phenomenon ~ 1224
30.10Integrating and Differentiating Fourier Series ~ 1224
30.11Exercises ~ 1229
30.12Hints ~ 1238
30.13Solutions ~ 1241

31 Regular Sturm-Liouville Problems 1291
31.1 Derivation of the Sturm-Liouville Form ~ 1291
31.2 Properties of Regular Sturm-Liouville Problems ~ 1294
31.3 Solving Differential Equations With Eigenfunction Expansions ~ 1305
31.4 Exercises ~ 1311
31.5 Hints ~ 1315
31.6 Solutions ~ 1317

32 Integrals and Convergence 1342
32.1 Uniform Convergence of Integrals ~ 1342
32.2 The Riemann-Lebesgue Lemma ~ 1343
32.3 Cauchy Principal Value ~ 1345
32.3.1 Integrals on an Infi nite Domain ~ 1345
32.3.2 Singular Functions ~ 1345

33 The Laplace Transform 1347
33.1 The Laplace Transform ~ 1347
33.2 The Inverse Laplace Transform ~ 1349
33.2.1 F(s) with Poles ~ 1352
33.2.2 ^ f(s) with Branch Points ~ 1357
33.2.3 Asymptotic Behavior of F(s) ~ 1360
33.3 Properties of the Laplace Transform ~ 1362
33.4 Constant Coeficient Differential Equations ~ 1366
33.5 Systems of Constant Coeficient Differential Equations ~ 1368
33.6 Exercises ~ 1370
33.7 Hints ~ 1378
33.8 Solutions ~ 1382

34 The Fourier Transform 1415
34.1 Derivation from a Fourier Series ~ 1415
34.2 The Fourier Transform ~ 1417
34.2.1 A Word of Caution ~ 1420
34.3 Evaluating Fourier Integrals ~ 1421
34.3.1 Integrals that Converge ~ 1421
34.3.2 Cauchy Principal Value and Integrals that are Not Absolutely Convergent ~ 1424
34.3.3 Analytic Continuation ~ 1426
34.4 Properties of the Fourier Transform ~ 1428
34.4.1 Closure Relation ~ 1429
34.4.2 Fourier Transform of a Derivative ~ 1430
34.4.3 Fourier Convolution Theorem ~ 1431
34.4.4 Parseval's Theorem ~ 1435
34.4.5 Shift Property ~ 1436
34.4.6 Fourier Transform of x f(x) ~ 1437
34.5 Solving Differential Equations with the Fourier Transform ~ 1437
34.6 The Fourier Cosine and Sine Transform ~ 1440
34.6.1 The Fourier Cosine Transform ~ 1440
34.6.2 The Fourier Sine Transform ~ 1441
34.7 Properties of the Fourier Cosine and Sine Transform ~ 1442
34.7.1 Transforms of Derivatives ~ 1442
34.7.2 Convolution Theorems ~ 1444
34.7.3 Cosine and Sine Transform in Terms of the Fourier Transform ~ 1446
34.8 Solving Differential Equations with the Fourier Cosine and Sine Transforms ~ 1447
34.9 Exercises ~ 1449
34.10Hints ~ 1455
34.11Solutions ~ 1458

35 The Gamma Function 1484
35.1 Euler's Formula ~ 1484
35.2 Hankel's Formula ~ 1486
35.3 Gauss' Formula ~ 1488
35.4 Weierstrass' Formula ~ 1490
35.5 Stirling's Approximation ~ 1492
35.6 Exercises ~ 1497
35.7 Hints ~ 1498
35.8 Solutions ~ 1499

36 Bessel Functions 1501
36.1 Bessel's Equation ~ 1501
36.2 Frobeneius Series Solution about z = 0 ~ 1502
36.2.1 Behavior at Infi nity ~ 1505
36.3 Bessel Functions of the First Kind ~ 1507
36.3.1 The Bessel Function Satisfi es Bessel's Equation ~ 1508
36.3.2 Series Expansion of the Bessel Function ~ 1509
36.3.3 Bessel Functions of Non-Integral Order ~ 1512
36.3.4 Recursion Formulas ~ 1515
36.3.5 Bessel Functions of Half-Integral Order ~ 1518
36.4 Neumann Expansions ~ 1519
36.5 Bessel Functions of the Second Kind ~ 1523
36.6 Hankel Functions ~ 1525
36.7 The Modifi ed Bessel Equation ~ 1525
36.8 Exercises ~ 1529
36.9 Hints ~ 1534
36.10Solutions ~ 1536

V Partial Differential Equations 1559

37 Transforming Equations 1560
37.1 Exercises ~ 1561
37.2 Hints ~ 1562
37.3 Solutions ~ 1563

38 Classifi cation of Partial Differential Equations 1564
38.1 Classifi cation of Second Order Quasi-Linear Equations ~ 1564
38.1.1 Hyperbolic Equations ~ 1565
38.1.2 Parabolic equations ~ 1570
38.1.3 Elliptic Equations ~ 1570
38.2 Equilibrium Solutions ~ 1572
38.3 Exercises ~ 1574
38.4 Hints ~ 1575
38.5 Solutions ~ 1576

39 Separation of Variables 1580
39.1 Eigensolutions of Homogeneous Equations ~ 1580
39.2 Homogeneous Equations with Homogeneous Boundary Conditions ~ 1580
39.3 Time-Independent Sources and Boundary Conditions ~ 1582
39.4 Inhomogeneous Equations with Homogeneous Boundary Conditions ~ 1585
39.5 Inhomogeneous Boundary Conditions ~ 1587
39.6 The Wave Equation ~ 1589
39.7 General Method ~ 1593
39.8 Exercises ~ 1594
39.9 Hints ~ 1608
39.10Solutions ~ 1613

40 Finite Transforms 1690
40.1 Exercises ~ 1694
40.2 Hints ~ 1695
40.3 Solutions ~ 1696
41 Waves 1701
41.1 Exercises ~ 1702
41.2 Hints ~ 1708

41.3 Solutions ~ 1710
42 The Diffusion Equation 1727
42.1 Exercises ~ 1728
42.2 Hints ~ 1730
42.3 Solutions ~ 1731

43 Similarity Methods 1734
43.1 Exercises ~ 1739
43.2 Hints ~ 1740
43.3 Solutions ~ 1741

44 Method of Characteristics 1743
44.1 The Method of Characteristics and the Wave Equation ~ 1743
44.2 The Method of Characteristics for an Infi nite Domain ~ 1745
44.3 The Method of Characteristics for a Semi-Infi nite Domain ~ 1746
44.4 Envelopes of Curves ~ 1747
44.5 Exercises ~ 1750
44.6 Hints ~ 1752
44.7 Solutions ~ 1753

45 Transform Methods 1759
45.1 Fourier Transform for Partial Differential Equations ~ 1759
45.2 The Fourier Sine Transform ~ 1761
45.3 Fourier Transform ~ 1762
45.4 Exercises ~ 1763
45.5 Hints ~ 1768
45.6 Solutions ~ 1771

46 Green Functions 1794
46.1 Inhomogeneous Equations and Homogeneous Boundary Conditions ~ 1794
46.2 Homogeneous Equations and Inhomogeneous Boundary Conditions ~ 1795
46.3 Eigenfunction Expansions for Elliptic Equations ~ 1797
46.4 The Method of Images ~ 1802
46.5 Exercises ~ 1803
46.6 Hints ~ 1808
46.7 Solutions ~ 1810

47 Conformal Mapping 1851
47.1 Exercises ~ 1852
47.2 Hints ~ 1855
47.3 Solutions ~ 1856

48 Non-Cartesian Coordinates 1864
48.1 Spherical Coordinates ~ 1864
48.2 Laplace's Equation in a Disk ~ 1865
48.3 Laplace's Equation in an Annulus ~ 1868

VI Calculus of Variations 1872

49 Calculus of Variations 1873
49.1 Exercises ~ 1874
49.2 Hints ~ 1891
49.3 Solutions ~ 1897

VII Nonlinear Differential Equations 1990

50 Nonlinear Ordinary Differential Equations 1991
50.1 Exercises ~ 1992
50.2 Hints ~ 1997
50.3 Solutions ~ 1999

51 Nonlinear Partial Differential Equations 2021
51.1 Exercises ~ 2022
51.2 Hints ~ 2025
51.3 Solutions ~ 2026

VIII Appendices 2044

A Greek Letters 2045
B Notation 2047
C Formulas from Complex Variables 2049
D Table of Derivatives 2052
E Table of Integrals 2056
F Defi nite Integrals 2060
G Table of Sums 2063
H Table of Taylor Series 2066
I Table of Laplace Transforms 2069
J Table of Fourier Transforms 2074
K Table of Fourier Transforms in n Dimensions 2077
L Table of Fourier Cosine Transforms 2078
M Table of Fourier Sine Transforms 2080
N Table of Wronskians 2082
O Sturm-Liouville Eigenvalue Problems 2084
P Green Functions for Ordinary Differential Equations 2086
Q Trigonometric Identities 2089
Q.1 Circular Functions ~ 2089
Q.2 Hyperbolic Functions ~ 2091
R Bessel Functions 2094
R.1 Defi nite Integrals ~ 2094
S Formulas from Linear Algebra 2095
T Vector Analysis 2097
U Partial Fractions 2099
V Finite Math 2103
W Probability 2104
W.1 Independent Events ~ 2104
W.2 Playing the Odds ~ 2105
X Economics 2106
Y Glossary 2107