# 教科書

1. Unabridged Version of Sean's Applied Math Book (No rights reserved)
2. 專門為程式人寫的 — 微積分

# 參考書

1. Whiteman (CC.BY.NC.SA)
2. Calculus for Mathematicians, Computer Scientists, and Physicists An Introduction to Abstract Mathematics , Andrew D. Hwang (適合資工系）
3. Elementary Calculus An Approach Using Infinitesimals, Keisler H.J (CC.BY.NC.SA 讚！)
4. Calculus, Banjamin Crowell (CC.BY.SA)
5. 紙本：工程數學 材雲清

# 授課內容

``````I Algebra 1
○ 1 Sets and Functions 2
○ 2 Vectors 22
II Calculus 47
◎ 3 Differential Calculus 48
◎ 4 Integral Calculus 116
◎ 5 Vector Calculus 154
III Functions of a Complex Variable 179
◎ 6 Complex Numbers 180
○ 7 Functions of a Complex Variable 239
○ 8 Analytic Functions 360
○ 9 Analytic Continuation 437
○ 10 Contour Integration and the Cauchy-Goursat Theorem 462
○ 11 Cauchy’s Integral Formula 493
◎ 12 Series and Convergence 525
○ 13 The Residue Theorem 626
IV Ordinary Differential Equations 772
◎ 14 First Order Differential Equations 773
15 First Order Linear Systems of Differential Equations 846
16 Theory of Linear Ordinary Differential Equations 900
17 Techniques for Linear Differential Equations 930
18 Techniques for Nonlinear Differential Equations 984
19 Transformations and Canonical Forms 1018
20 The Dirac Delta Function 1041
21 Inhomogeneous Differential Equations 1059
◎ 22 Difference Equations 1166
23 Series Solutions of Differential Equations 1184
○ 24 Asymptotic Expansions 1251
25 Hilbert Spaces 1278
26 Self Adjoint Linear Operators 1307
27 Self-Adjoint Boundary Value Problems 1314
◎ 28 Fourier Series 1330
29 Regular Sturm-Liouville Problems 1420
○ 30 Integrals and Convergence 1470
◎ 31 The Laplace Transform 1475
◎ 32 The Fourier Transform 1539
33 The Gamma Function 1605
34 Bessel Functions 1622
V Partial Differential Equations 1680
35 Transforming Equations 1681
36 Classification of Partial Differential Equations 1685
37 Separation of Variables 1704
38 Finite Transforms 1821
39 The Diffusion Equation 1831
40 Laplace’s Equation 1841
41 Waves 1859
42 Similarity Methods 1888
43 Method of Characteristics 1897
◎ 44 Transform Methods 1918
45 Green Functions 1950
46 Conformal Mapping 2034
47 Non-Cartesian Coordinates 2051
VI Calculus of Variations 2059
48 Calculus of Variations 2060
VII Nonlinear Differential Equations 2166
49 Nonlinear Ordinary Differential Equations 2167
50 Nonlinear Partial Differential Equations 2196```
```

# 內容

1. Introduction to Methods of Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers

``````Contents
Preface xxv
0 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
2 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
3 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
5 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
7 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
10 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
12 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
IV Ordinary Differential Equations 772
14 First Order Differential Equations 773
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 10Hints                         819
14 11Solutions                     822
14 12Quiz                         843
14 13Quiz Solutions                       844
15 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 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
17 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 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
19 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
21 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
23 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
25 14Solutions                     1305
26 Self Adjoint Linear Operators 1307
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 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
28 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 ? f(s) with Poles                     1480
31 2 2 ? f(s) with Branch Points                     1484
31 2 3 Asymptotic Behavior of ? f(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
31 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
33 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
V Partial Differential Equations 1680
35 Transforming Equations 1681
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
38 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
43 3 The Method of Characteristics and the Wave Equation             1900
43 4 The Wave Equation for an Infinite Domain                 1901
43 5 The Wave Equation for a Semi-Infinite Domain             1902
43 6 The Wave Equation for a Finite Domain                   1904
43 7 Envelopes of Curves                         1905
43 8 Exercises                     1908
43 9 Hints                         1910
43 10Solutions                     1911
44 Transform Methods 1918
44 1 Fourier Transform for Partial Differential Equations                 1918
44 2 The Fourier Sine Transform                 1920
44 3 Fourier Transform                   1920
44 4 Exercises                     1922
44 5 Hints                         1926
44 6 Solutions                     1928
45 Green Functions 1950
45 1 Inhomogeneous Equations and Homogeneous Boundary Conditions               1950
45 2 Homogeneous Equations and Inhomogeneous Boundary Conditions               1951
45 3 Eigenfunction Expansions for Elliptic Equations             1953
45 4 The Method of Images                       1958
45 5 Exercises                     1960
45 6 Hints                         1971
45 7 Solutions                     1974
46 Conformal Mapping 2034
46 1 Exercises                     2035
46 2 Hints                         2038
46 3 Solutions                     2039
47 Non-Cartesian Coordinates 2051
47 1 Spherical Coordinates                       2051
47 2 Laplace’s Equation in a Disk                 2052
47 3 Laplace’s Equation in an Annulus                   2055
VI Calculus of Variations 2059
48 Calculus of Variations 2060
48 1 Exercises                     2061
48 2 Hints                         2075
48 3 Solutions                     2079
VII Nonlinear Differential Equations 2166
49 Nonlinear Ordinary Differential Equations 2167
49 1 Exercises                     2168
49 2 Hints                         2173
49 3 Solutions                     2174
50 Nonlinear Partial Differential Equations 2196
50 1 Exercises                     2197
50 2 Hints                         2200
50 3 Solutions                     2201
VIII Appendices 2220
A Greek Letters 2221
B Notation 2223
C Formulas from Complex Variables 2225
D Table of Derivatives 2228
E Table of Integrals 2232
F Definite Integrals 2236
G Table of Sums 2238
H Table of Taylor Series 2241
I Continuous Transforms 2244
I 1 Properties of Laplace Transforms                   2244
I 2 Table of Laplace Transforms                 2247
I 3 Table of Fourier Transforms                 2250
I 4 Table of Fourier Transforms in n Dimensions               2253
I 5 Table of Fourier Cosine Transforms                 2254
I 6 Table of Fourier Sine Transforms                   2255
J Table of Wronskians 2257
K Sturm-Liouville Eigenvalue Problems 2259
L Green Functions for Ordinary Differential Equations 2261
M Trigonometric Identities 2264
M 1 Circular Functions                   2264
M 2 Hyperbolic Functions                         2266
N Bessel Functions 2269
N 1 Definite Integrals                     2269
O Formulas from Linear Algebra 2270
P Vector Analysis 2271
Q Partial Fractions 2273
R Finite Math 2276
S Physics 2277
T Probability 2278
T 1 Independent Events                         2278
T 2 Playing the Odds                     2279
U Economics 2280
V Glossary 2281
W whoami 2283```
```