Mauch S.
Table of contents :
Anti-Copyright……Page 25
Acknowledgments……Page 26
Warnings and Disclaimers……Page 27
About the Title……Page 28
I Algebra……Page 29
Sets……Page 30
Single Valued Functions……Page 32
Inverses and Multi-Valued Functions……Page 34
Transforming Equations……Page 37
Exercises……Page 39
Hints……Page 42
Solutions……Page 44
Scalars and Vectors……Page 50
The Kronecker Delta and Einstein Summation Convention……Page 53
The Dot and Cross Product……Page 54
Sets of Vectors in n Dimensions……Page 61
Exercises……Page 64
Hints……Page 66
Solutions……Page 68
II Calculus……Page 75
Limits of Functions……Page 76
Continuous Functions……Page 81
The Derivative……Page 84
Implicit Differentiation……Page 89
Maxima and Minima……Page 90
Mean Value Theorems……Page 94
Application: Using Taylor’s Theorem to Approximate Functions…….Page 96
Application: Finite Difference Schemes……Page 101
L’Hospital’s Rule……Page 103
Continuous Functions……Page 109
The Derivative……Page 110
Maxima and Minima……Page 112
L’Hospital’s Rule……Page 113
Hints……Page 115
Solutions……Page 121
Quiz……Page 141
Quiz Solutions……Page 142
The Indefinite Integral……Page 144
Definition……Page 150
Properties……Page 151
The Fundamental Theorem of Integral Calculus……Page 153
Partial Fractions……Page 155
Improper Integrals……Page 158
The Definite Integral……Page 162
Techniques of Integration……Page 164
Improper Integrals……Page 165
Hints……Page 166
Solutions……Page 169
Quiz……Page 178
Quiz Solutions……Page 179
Vector Functions……Page 182
Gradient, Divergence and Curl……Page 183
Exercises……Page 191
Hints……Page 194
Solutions……Page 196
Quiz……Page 205
Quiz Solutions……Page 206
III Functions of a Complex Variable……Page 207
Complex Numbers……Page 208
The Complex Plane……Page 212
Polar Form……Page 216
Arithmetic and Vectors……Page 221
Integer Exponents……Page 223
Rational Exponents……Page 225
Exercises……Page 229
Hints……Page 236
Solutions……Page 239
Curves and Regions……Page 267
The Point at Infinity and the Stereographic Projection……Page 270
Cartesian and Modulus-Argument Form……Page 274
Graphing Functions of a Complex Variable……Page 277
Trigonometric Functions……Page 280
Inverse Trigonometric Functions……Page 287
Riemann Surfaces……Page 296
Branch Points……Page 298
Exercises……Page 314
Hints……Page 325
Solutions……Page 330
Complex Derivatives……Page 388
Cauchy-Riemann Equations……Page 395
Harmonic Functions……Page 400
Categorization of Singularities……Page 405
Isolated and Non-Isolated Singularities……Page 409
Application: Potential Flow……Page 411
Exercises……Page 416
Hints……Page 424
Solutions……Page 427
Analytic Continuation……Page 465
Analytic Continuation of Sums……Page 468
Analytic Functions Defined in Terms of Real Variables……Page 470
Polar Coordinates……Page 474
Analytic Functions Defined in Terms of Their Real or Imaginary Parts……Page 478
Exercises……Page 482
Hints……Page 484
Solutions……Page 485
Line Integrals……Page 490
Contour Integrals……Page 492
Maximum Modulus Integral Bound……Page 494
The Cauchy-Goursat Theorem……Page 495
Contour Deformation……Page 497
Morera’s Theorem…….Page 499
Indefinite Integrals……Page 501
Contour Integrals……Page 502
Fundamental Theorem of Calculus via Complex Calculus……Page 503
Exercises……Page 506
Hints……Page 510
Solutions……Page 511
Cauchy’s Integral Formula……Page 521
Cauchy’s Integral Formula……Page 522
The Argument Theorem……Page 529
Rouche’s Theorem……Page 530
Exercises……Page 533
Hints……Page 537
Solutions……Page 539
Definitions……Page 553
Special Series……Page 555
Convergence Tests……Page 557
Uniform Convergence……Page 564
Tests for Uniform Convergence……Page 565
Uniformly Convergent Power Series……Page 567
Integration and Differentiation of Power Series……Page 575
Taylor Series……Page 578
Newton’s Binomial Formula…….Page 581
Laurent Series……Page 583
Series of Constants……Page 588
Uniformly Convergent Power Series……Page 594
Integration and Differentiation of Power Series……Page 596
Taylor Series……Page 597
Laurent Series……Page 599
Hints……Page 602
Solutions……Page 610
The Residue Theorem……Page 654
The Cauchy Principal Value……Page 662
Cauchy Principal Value for Contour Integrals……Page 667
Integrals on the Real Axis……Page 671
Fourier Integrals……Page 675
Fourier Cosine and Sine Integrals……Page 677
Contour Integration and Branch Cuts……Page 680
Wedge Contours……Page 683
Box Contours……Page 686
Definite Integrals Involving Sine and Cosine……Page 687
Infinite Sums……Page 690
Exercises……Page 694
Hints……Page 708
Solutions……Page 714
IV Ordinary Differential Equations……Page 800
Notation……Page 801
Growth and Decay……Page 803
One Parameter Families of Functions……Page 805
Integrable Forms……Page 807
Separable Equations……Page 808
Exact Equations……Page 810
Homogeneous Coefficient Equations……Page 814
Homogeneous Equations……Page 819
Inhomogeneous Equations……Page 820
Variation of Parameters…….Page 823
Initial Conditions……Page 824
Piecewise Continuous Coefficients and Inhomogeneities……Page 825
Well-Posed Problems……Page 829
Ordinary Points……Page 831
Regular Singular Points……Page 834
Irregular Singular Points……Page 840
The Point at Infinity……Page 842
Additional Exercises……Page 844
Hints……Page 847
Solutions……Page 850
Quiz……Page 871
Quiz Solutions……Page 872
Introduction……Page 874
Using Eigenvalues and Eigenvectors to find Homogeneous Solutions……Page 875
Matrices and Jordan Canonical Form……Page 880
Using the Matrix Exponential……Page 888
Exercises……Page 893
Hints……Page 898
Solutions……Page 900
Exact Equations……Page 928
Nature of Solutions……Page 929
Derivative of a Determinant…….Page 933
The Wronskian of a Set of Functions…….Page 934
The Wronskian of the Solutions to a Differential Equation……Page 936
Well-Posed Problems……Page 939
The Fundamental Set of Solutions……Page 941
Adjoint Equations……Page 943
Additional Exercises……Page 947
Hints……Page 948
Solutions……Page 950
Quiz……Page 956
Quiz Solutions……Page 957
Constant Coefficient Equations……Page 958
Second Order Equations……Page 959
Real-Valued Solutions……Page 963
Higher Order Equations……Page 965
Euler Equations……Page 968
Real-Valued Solutions……Page 970
Exact Equations……Page 973
Equations Without Explicit Dependence on y……Page 974
Reduction of Order……Page 975
*Reduction of Order and the Adjoint Equation……Page 976
Additional Exercises……Page 979
Hints……Page 985
Solutions……Page 988
Bernoulli Equations……Page 1012
Riccati Equations……Page 1014
Exchanging the Dependent and Independent Variables……Page 1018
Autonomous Equations……Page 1020
*Equidimensional-in-x Equations……Page 1023
*Equidimensional-in-y Equations……Page 1025
*Scale-Invariant Equations……Page 1028
Exercises……Page 1029
Hints……Page 1032
Solutions……Page 1034
The Constant Coefficient Equation……Page 1046
Second Order Equations……Page 1049
Higher Order Differential Equations……Page 1050
Transformation to the form u” + a(x) u = 0……Page 1052
Transformation to a Constant Coefficient Equation……Page 1053
Initial Value Problems……Page 1055
Boundary Value Problems……Page 1057
Exercises……Page 1060
Hints……Page 1062
Solutions……Page 1063
Derivative of the Heaviside Function……Page 1069
The Delta Function as a Limit……Page 1071
Higher Dimensions……Page 1073
Non-Rectangular Coordinate Systems……Page 1074
Exercises……Page 1076
Hints……Page 1078
Solutions……Page 1080
Particular Solutions……Page 1087
Method of Undetermined Coefficients……Page 1089
Second Order Differential Equations……Page 1093
Higher Order Differential Equations……Page 1096
Piecewise Continuous Coefficients and Inhomogeneities……Page 1099
Eliminating Inhomogeneous Boundary Conditions……Page 1102
Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions……Page 1104
Existence of Solutions of Problems with Inhomogeneous Boundary Conditions……Page 1105
Green Functions for First Order Equations……Page 1107
Green Functions for Second Order Equations……Page 1110
Green Functions for Sturm-Liouville Problems……Page 1120
Initial Value Problems……Page 1123
Problems with Unmixed Boundary Conditions……Page 1126
Problems with Mixed Boundary Conditions……Page 1128
Green Functions for Higher Order Problems……Page 1132
Fredholm Alternative Theorem……Page 1137
Exercises……Page 1145
Hints……Page 1151
Solutions……Page 1154
Quiz……Page 1192
Quiz Solutions……Page 1193
Introduction……Page 1194
Exact Equations……Page 1196
Homogeneous First Order……Page 1197
Inhomogeneous First Order……Page 1199
Homogeneous Constant Coefficient Equations……Page 1202
Reduction of Order……Page 1205
Exercises……Page 1207
Hints……Page 1208
Solutions……Page 1209
Ordinary Points……Page 1212
Taylor Series Expansion for a Second Order Differential Equation……Page 1216
Regular Singular Points of Second Order Equations……Page 1226
Indicial Equation……Page 1229
The Case: Double Root……Page 1231
The Case: Roots Differ by an Integer……Page 1234
The Point at Infinity……Page 1244
Exercises……Page 1247
Hints……Page 1252
Solutions……Page 1253
Quiz……Page 1276
Quiz Solutions……Page 1277
Asymptotic Relations……Page 1279
Leading Order Behavior of Differential Equations……Page 1283
Integration by Parts……Page 1291
Asymptotic Series……Page 1298
The Parabolic Cylinder Equation…….Page 1300
Linear Spaces……Page 1306
Inner Products……Page 1308
Norms……Page 1309
Orthogonality……Page 1311
Gramm-Schmidt Orthogonalization……Page 1312
Orthonormal Function Expansion……Page 1315
Sets Of Functions……Page 1316
Least Squares Fit to a Function and Completeness……Page 1322
Closure Relation……Page 1325
Linear Operators……Page 1330
Exercises……Page 1331
Hints……Page 1332
Solutions……Page 1333
Adjoint Operators……Page 1335
Self-Adjoint Operators……Page 1336
Exercises……Page 1339
Hints……Page 1340
Solutions……Page 1341
Summary of Adjoint Operators……Page 1342
Formally Self-Adjoint Operators……Page 1343
Self-Adjoint Eigenvalue Problems……Page 1346
Inhomogeneous Equations……Page 1351
Exercises……Page 1354
Hints……Page 1355
Solutions……Page 1356
An Eigenvalue Problem…….Page 1358
Fourier Series…….Page 1361
Least Squares Fit……Page 1365
Fourier Series for Functions Defined on Arbitrary Ranges……Page 1369
Fourier Cosine Series……Page 1372
Fourier Sine Series……Page 1373
Complex Fourier Series and Parseval’s Theorem……Page 1374
Behavior of Fourier Coefficients……Page 1377
Integrating and Differentiating Fourier Series……Page 1386
Exercises……Page 1391
Hints……Page 1399
Solutions……Page 1401
Derivation of the Sturm-Liouville Form……Page 1448
Properties of Regular Sturm-Liouville Problems……Page 1450
Solving Differential Equations With Eigenfunction Expansions……Page 1461
Exercises……Page 1467
Hints……Page 1471
Solutions……Page 1473
Uniform Convergence of Integrals……Page 1498
The Riemann-Lebesgue Lemma……Page 1499
Integrals on an Infinite Domain……Page 1500
Singular Functions……Page 1501
The Laplace Transform……Page 1503
The Inverse Laplace Transform……Page 1505
(s) with Poles……Page 1508
(s) with Branch Points……Page 1512
Asymptotic Behavior of (s)……Page 1516
Properties of the Laplace Transform……Page 1517
Constant Coefficient Differential Equations……Page 1520
Systems of Constant Coefficient Differential Equations……Page 1523
Exercises……Page 1525
Hints……Page 1532
Solutions……Page 1535
Derivation from a Fourier Series……Page 1567
The Fourier Transform……Page 1569
A Word of Caution……Page 1572
Integrals that Converge……Page 1573
Cauchy Principal Value and Integrals that are Not Absolutely Convergent…….Page 1576
Analytic Continuation……Page 1578
Closure Relation…….Page 1580
Fourier Transform of a Derivative…….Page 1581
Fourier Convolution Theorem…….Page 1582
Parseval’s Theorem…….Page 1585
Fourier Transform of x f(x)…….Page 1587
Solving Differential Equations with the Fourier Transform……Page 1588
The Fourier Cosine Transform……Page 1590
The Fourier Sine Transform……Page 1591
Transforms of Derivatives……Page 1592
Convolution Theorems……Page 1594
Cosine and Sine Transform in Terms of the Fourier Transform……Page 1596
Solving Differential Equations with the Fourier Cosine and Sine Transforms……Page 1597
Exercises……Page 1599
Hints……Page 1606
Solutions……Page 1609
Euler’s Formula……Page 1633
Hankel’s Formula……Page 1635
Gauss’ Formula……Page 1637
Weierstrass’ Formula……Page 1639
Stirling’s Approximation……Page 1641
Exercises……Page 1646
Hints……Page 1647
Solutions……Page 1648
Bessel’s Equation……Page 1650
Frobeneius Series Solution about z = 0……Page 1651
Behavior at Infinity……Page 1654
Bessel Functions of the First Kind……Page 1656
The Bessel Function Satisfies Bessel’s Equation……Page 1657
Series Expansion of the Bessel Function……Page 1658
Bessel Functions of Non-Integer Order……Page 1661
Recursion Formulas……Page 1664
Bessel Functions of Half-Integer Order……Page 1667
Neumann Expansions……Page 1668
Bessel Functions of the Second Kind……Page 1672
The Modified Bessel Equation……Page 1674
Exercises……Page 1678
Hints……Page 1683
Solutions……Page 1685
V Partial Differential Equations……Page 1708
Transforming Equations……Page 1709
Exercises……Page 1710
Hints……Page 1711
Solutions……Page 1712
Classification of Second Order Quasi-Linear Equations……Page 1713
Hyperbolic Equations……Page 1714
Parabolic equations……Page 1719
Elliptic Equations……Page 1720
Equilibrium Solutions……Page 1722
Exercises……Page 1724
Hints……Page 1725
Solutions……Page 1726
Homogeneous Equations with Homogeneous Boundary Conditions……Page 1732
Time-Independent Sources and Boundary Conditions……Page 1734
Inhomogeneous Equations with Homogeneous Boundary Conditions……Page 1737
Inhomogeneous Boundary Conditions……Page 1738
The Wave Equation……Page 1741
General Method……Page 1744
Exercises……Page 1746
Hints……Page 1762
Solutions……Page 1767
Finite Transforms……Page 1849
Exercises……Page 1853
Hints……Page 1854
Solutions……Page 1855
The Diffusion Equation……Page 1859
Exercises……Page 1860
Hints……Page 1862
Solutions……Page 1863
Fundamental Solution……Page 1869
Two Dimensional Space……Page 1870
Exercises……Page 1871
Hints……Page 1874
Solutions……Page 1875
Waves……Page 1887
Exercises……Page 1888
Hints……Page 1894
Solutions……Page 1896
Similarity Methods……Page 1916
Exercises……Page 1920
Hints……Page 1921
Solutions……Page 1922
First Order Linear Equations……Page 1925
First Order Quasi-Linear Equations……Page 1926
The Method of Characteristics and the Wave Equation……Page 1928
The Wave Equation for an Infinite Domain……Page 1929
The Wave Equation for a Semi-Infinite Domain……Page 1930
The Wave Equation for a Finite Domain……Page 1932
Envelopes of Curves……Page 1933
Exercises……Page 1936
Hints……Page 1938
Solutions……Page 1939
Fourier Transform for Partial Differential Equations……Page 1946
Fourier Transform……Page 1948
Exercises……Page 1950
Hints……Page 1954
Solutions……Page 1956
Inhomogeneous Equations and Homogeneous Boundary Conditions……Page 1978
Homogeneous Equations and Inhomogeneous Boundary Conditions……Page 1979
Eigenfunction Expansions for Elliptic Equations……Page 1981
The Method of Images……Page 1986
Exercises……Page 1988
Hints……Page 1999
Solutions……Page 2002
Conformal Mapping……Page 2062
Exercises……Page 2063
Hints……Page 2066
Solutions……Page 2067
Spherical Coordinates……Page 2079
Laplace’s Equation in a Disk……Page 2080
Laplace’s Equation in an Annulus……Page 2083
VI Calculus of Variations……Page 2087
Calculus of Variations……Page 2088
Exercises……Page 2089
Hints……Page 2103
Solutions……Page 2107
VII Nonlinear Differential Equations……Page 2194
Nonlinear Ordinary Differential Equations……Page 2195
Exercises……Page 2196
Hints……Page 2201
Solutions……Page 2202
Nonlinear Partial Differential Equations……Page 2224
Exercises……Page 2225
Hints……Page 2228
Solutions……Page 2229
VIII Appendices……Page 2248
Greek Letters……Page 2249
Notation……Page 2251
Formulas from Complex Variables……Page 2253
Table of Derivatives……Page 2256
Table of Integrals……Page 2260
Definite Integrals……Page 2264
Table of Sums……Page 2266
Table of Taylor Series……Page 2269
Properties of Laplace Transforms……Page 2272
Table of Laplace Transforms……Page 2275
Table of Fourier Transforms……Page 2278
Table of Fourier Transforms in n Dimensions……Page 2281
Table of Fourier Cosine Transforms……Page 2282
Table of Fourier Sine Transforms……Page 2283
Table of Wronskians……Page 2285
Sturm-Liouville Eigenvalue Problems……Page 2287
Green Functions for Ordinary Differential Equations……Page 2289
Circular Functions……Page 2292
Hyperbolic Functions……Page 2294
Definite Integrals……Page 2297
Formulas from Linear Algebra……Page 2298
Vector Analysis……Page 2299
Partial Fractions……Page 2301
Finite Math……Page 2304
Physics……Page 2305
Independent Events……Page 2306
Playing the Odds……Page 2307
Economics……Page 2308
Glossary……Page 2309
whoami……Page 2311
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