S. Mauch
Table of contents :
Anti-Copyright……Page 24
Acknowledgments……Page 25
Warnings and Disclaimers……Page 26
About the Title……Page 27
I Algebra……Page 28
Sets……Page 29
Single Valued Functions……Page 31
Inverses and Multi-Valued Functions……Page 32
Transforming Equations……Page 36
Exercises……Page 38
Hints……Page 42
Solutions……Page 43
Scalars and Vectors……Page 49
The Kronecker Delta and Einstein Summation Convention……Page 52
The Dot and Cross Product……Page 53
Sets of Vectors in n Dimensions……Page 60
Exercises……Page 63
Hints……Page 65
Solutions……Page 67
II Calculus……Page 73
Limits of Functions……Page 74
Continuous Functions……Page 79
The Derivative……Page 81
Implicit Differentiation……Page 86
Maxima and Minima……Page 88
Mean Value Theorems……Page 91
Application: Using Taylor’s Theorem to Approximate Functions…….Page 93
Application: Finite Difference Schemes……Page 98
L’Hospital’s Rule……Page 100
Exercises……Page 106
Hints……Page 112
Solutions……Page 118
The Indefinite Integral……Page 138
Definition……Page 144
Properties……Page 145
The Fundamental Theorem of Integral Calculus……Page 147
Partial Fractions……Page 149
Improper Integrals……Page 152
Exercises……Page 156
Hints……Page 160
Solutions……Page 164
Vector Functions……Page 174
Gradient, Divergence and Curl……Page 175
Exercises……Page 183
Hints……Page 186
Solutions……Page 188
III Functions of a Complex Variable……Page 197
Complex Numbers……Page 198
The Complex Plane……Page 201
Polar Form……Page 206
Arithmetic and Vectors……Page 210
Integer Exponents……Page 212
Rational Exponents……Page 214
Exercises……Page 218
Hints……Page 225
Solutions……Page 228
Curves and Regions……Page 255
The Point at Infinity and the Stereographic Projection……Page 258
Cartesian and Modulus-Argument Form……Page 260
Graphing Functions of a Complex Variable……Page 264
Trigonometric Functions……Page 266
Inverse Trigonometric Functions……Page 272
Riemann Surfaces……Page 281
Branch Points……Page 283
Exercises……Page 300
Hints……Page 311
Solutions……Page 316
Complex Derivatives……Page 373
Cauchy-Riemann Equations……Page 380
Harmonic Functions……Page 385
Categorization of Singularities……Page 390
Isolated and Non-Isolated Singularities……Page 394
Application: Potential Flow……Page 396
Exercises……Page 401
Hints……Page 407
Solutions……Page 410
Analytic Continuation……Page 446
Analytic Continuation of Sums……Page 449
Analytic Functions Defined in Terms of Real Variables……Page 451
Polar Coordinates……Page 456
Analytic Functions Defined in Terms of Their Real or Imaginary Parts……Page 459
Exercises……Page 463
Hints……Page 465
Solutions……Page 466
Line Integrals……Page 471
Contour Integrals……Page 473
Maximum Modulus Integral Bound……Page 476
The Cauchy-Goursat Theorem……Page 477
Contour Deformation……Page 479
Morera’s Theorem…….Page 480
Indefinite Integrals……Page 482
Contour Integrals……Page 483
Fundamental Theorem of Calculus via Complex Calculus……Page 484
Exercises……Page 487
Hints……Page 491
Solutions……Page 492
Cauchy’s Integral Formula……Page 502
Cauchy’s Integral Formula……Page 503
The Argument Theorem……Page 510
Rouche’s Theorem……Page 511
Exercises……Page 514
Hints……Page 518
Solutions……Page 520
Definitions……Page 535
Special Series……Page 537
Convergence Tests……Page 539
Uniform Convergence……Page 546
Tests for Uniform Convergence……Page 547
Uniform Convergence and Continuous Functions…….Page 549
Uniformly Convergent Power Series……Page 550
Integration and Differentiation of Power Series……Page 557
Taylor Series……Page 560
Newton’s Binomial Formula…….Page 563
Laurent Series……Page 565
Exercises……Page 570
Hints……Page 585
Solutions……Page 594
The Residue Theorem……Page 641
The Cauchy Principal Value……Page 649
Cauchy Principal Value for Contour Integrals……Page 654
Integrals on the Real Axis……Page 658
Fourier Integrals……Page 662
Fourier Cosine and Sine Integrals……Page 664
Contour Integration and Branch Cuts……Page 667
Wedge Contours……Page 670
Box Contours……Page 673
Definite Integrals Involving Sine and Cosine……Page 674
Infinite Sums……Page 677
Exercises……Page 682
Hints……Page 696
Solutions……Page 702
IV Ordinary Differential Equations……Page 788
Notation……Page 789
One Parameter Families of Functions……Page 791
Exact Equations……Page 793
Separable Equations……Page 798
Homogeneous Coefficient Equations……Page 800
Homogeneous Equations……Page 804
Inhomogeneous Equations……Page 806
Initial Conditions……Page 809
Piecewise Continuous Coefficients and Inhomogeneities……Page 810
Well-Posed Problems……Page 815
Ordinary Points……Page 818
Regular Singular Points……Page 821
Irregular Singular Points……Page 826
The Point at Infinity……Page 828
Additional Exercises……Page 831
Hints……Page 834
Solutions……Page 837
Introduction……Page 858
Using Eigenvalues and Eigenvectors to find Homogeneous Solutions……Page 859
Matrices and Jordan Canonical Form……Page 864
Using the Matrix Exponential……Page 871
Exercises……Page 877
Hints……Page 882
Solutions……Page 884
Exact Equations……Page 912
Nature of Solutions……Page 913
Transformation to a First Order System……Page 916
Derivative of a Determinant…….Page 917
The Wronskian of a Set of Functions…….Page 918
The Wronskian of the Solutions to a Differential Equation……Page 920
Well-Posed Problems……Page 923
The Fundamental Set of Solutions……Page 925
Adjoint Equations……Page 927
Additional Exercises……Page 931
Hints……Page 932
Solutions……Page 933
Constant Coefficient Equations……Page 938
Second Order Equations……Page 939
Higher Order Equations……Page 943
Real-Valued Solutions……Page 944
Euler Equations……Page 948
Real-Valued Solutions……Page 950
Exact Equations……Page 953
Equations Without Explicit Dependence on y……Page 954
Reduction of Order……Page 955
*Reduction of Order and the Adjoint Equation……Page 956
Exercises……Page 959
Hints……Page 965
Solutions……Page 968
Bernoulli Equations……Page 992
Riccati Equations……Page 994
Exchanging the Dependent and Independent Variables……Page 998
Autonomous Equations……Page 1000
*Equidimensional-in-x Equations……Page 1003
*Equidimensional-in-y Equations……Page 1005
*Scale-Invariant Equations……Page 1008
Exercises……Page 1009
Hints……Page 1012
Solutions……Page 1014
The Constant Coefficient Equation……Page 1026
Second Order Equations……Page 1029
Higher Order Differential Equations……Page 1030
Transformation to the form u” + a(x) u = 0……Page 1032
Transformation to a Constant Coefficient Equation……Page 1033
Initial Value Problems……Page 1035
Boundary Value Problems……Page 1037
Exercises……Page 1040
Hints……Page 1042
Solutions……Page 1043
Derivative of the Heaviside Function……Page 1049
The Delta Function as a Limit……Page 1051
Higher Dimensions……Page 1053
Non-Rectangular Coordinate Systems……Page 1054
Exercises……Page 1056
Hints……Page 1058
Solutions……Page 1060
Particular Solutions……Page 1067
Method of Undetermined Coefficients……Page 1069
Second Order Differential Equations……Page 1073
Higher Order Differential Equations……Page 1076
Piecewise Continuous Coefficients and Inhomogeneities……Page 1079
Eliminating Inhomogeneous Boundary Conditions……Page 1082
Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions……Page 1084
Existence of Solutions of Problems with Inhomogeneous Boundary Conditions……Page 1085
Green Functions for First Order Equations……Page 1087
Green Functions for Second Order Equations……Page 1090
Green Functions for Sturm-Liouville Problems……Page 1100
Initial Value Problems……Page 1103
Problems with Unmixed Boundary Conditions……Page 1105
Problems with Mixed Boundary Conditions……Page 1108
Green Functions for Higher Order Problems……Page 1112
Fredholm Alternative Theorem……Page 1117
Exercises……Page 1125
Hints……Page 1131
Solutions……Page 1134
Introduction……Page 1172
Exact Equations……Page 1174
Homogeneous First Order……Page 1175
Inhomogeneous First Order……Page 1177
Homogeneous Constant Coefficient Equations……Page 1180
Reduction of Order……Page 1183
Exercises……Page 1185
Hints……Page 1186
Solutions……Page 1187
Ordinary Points……Page 1190
Taylor Series Expansion for a Second Order Differential Equation……Page 1194
Regular Singular Points of Second Order Equations……Page 1204
Indicial Equation……Page 1207
The Case: Double Root……Page 1209
The Case: Roots Differ by an Integer……Page 1213
The Point at Infinity……Page 1223
Exercises……Page 1226
Hints……Page 1231
Solutions……Page 1232
Asymptotic Relations……Page 1255
Leading Order Behavior of Differential Equations……Page 1259
Integration by Parts……Page 1268
Asymptotic Series……Page 1275
The Parabolic Cylinder Equation…….Page 1276
Linear Spaces……Page 1282
Inner Products……Page 1284
Norms……Page 1285
Orthogonality……Page 1287
Gramm-Schmidt Orthogonalization……Page 1288
Orthonormal Function Expansion……Page 1290
Sets Of Functions……Page 1292
Least Squares Fit to a Function and Completeness……Page 1299
Closure Relation……Page 1302
Linear Operators……Page 1307
Exercises……Page 1308
Hints……Page 1309
Solutions……Page 1310
Adjoint Operators……Page 1312
Self-Adjoint Operators……Page 1313
Exercises……Page 1316
Hints……Page 1317
Solutions……Page 1318
Summary of Adjoint Operators……Page 1319
Formally Self-Adjoint Operators……Page 1320
Self-Adjoint Eigenvalue Problems……Page 1323
Inhomogeneous Equations……Page 1328
Exercises……Page 1331
Hints……Page 1332
Solutions……Page 1333
An Eigenvalue Problem…….Page 1335
Fourier Series…….Page 1338
Least Squares Fit……Page 1342
Fourier Series for Functions Defined on Arbitrary Ranges……Page 1346
Fourier Cosine Series……Page 1349
Fourier Sine Series……Page 1350
Complex Fourier Series and Parseval’s Theorem……Page 1351
Behavior of Fourier Coefficients……Page 1354
Integrating and Differentiating Fourier Series……Page 1363
Exercises……Page 1368
Hints……Page 1376
Solutions……Page 1378
Derivation of the Sturm-Liouville Form……Page 1425
Properties of Regular Sturm-Liouville Problems……Page 1427
Solving Differential Equations With Eigenfunction Expansions……Page 1438
Exercises……Page 1444
Hints……Page 1448
Solutions……Page 1450
Uniform Convergence of Integrals……Page 1475
The Riemann-Lebesgue Lemma……Page 1476
Integrals on an Infinite Domain……Page 1477
Singular Functions……Page 1478
The Laplace Transform……Page 1480
The Inverse Laplace Transform……Page 1482
(s) with Poles……Page 1485
(s) with Branch Points……Page 1490
Asymptotic Behavior of (s)……Page 1493
Properties of the Laplace Transform……Page 1495
Constant Coefficient Differential Equations……Page 1498
Systems of Constant Coefficient Differential Equations……Page 1500
Exercises……Page 1503
Hints……Page 1510
Solutions……Page 1513
Derivation from a Fourier Series……Page 1545
The Fourier Transform……Page 1547
A Word of Caution……Page 1550
Integrals that Converge……Page 1551
Cauchy Principal Value and Integrals that are Not Absolutely Convergent…….Page 1554
Analytic Continuation……Page 1556
Closure Relation…….Page 1558
Fourier Transform of a Derivative…….Page 1559
Fourier Convolution Theorem…….Page 1561
Parseval’s Theorem…….Page 1564
Fourier Transform of x f(x)…….Page 1566
Solving Differential Equations with the Fourier Transform……Page 1567
The Fourier Cosine Transform……Page 1569
The Fourier Sine Transform……Page 1570
Transforms of Derivatives……Page 1571
Convolution Theorems……Page 1573
Cosine and Sine Transform in Terms of the Fourier Transform……Page 1575
Solving Differential Equations with the Fourier Cosine and Sine Transforms……Page 1576
Exercises……Page 1578
Hints……Page 1585
Solutions……Page 1588
Euler’s Formula……Page 1612
Hankel’s Formula……Page 1614
Gauss’ Formula……Page 1616
Weierstrass’ Formula……Page 1618
Stirling’s Approximation……Page 1620
Exercises……Page 1625
Hints……Page 1626
Solutions……Page 1627
Bessel’s Equation……Page 1629
Frobeneius Series Solution about z = 0……Page 1630
Behavior at Infinity……Page 1633
Bessel Functions of the First Kind……Page 1635
The Bessel Function Satisfies Bessel’s Equation……Page 1636
Series Expansion of the Bessel Function……Page 1637
Bessel Functions of Non-Integer Order……Page 1640
Recursion Formulas……Page 1643
Bessel Functions of Half-Integer Order……Page 1646
Neumann Expansions……Page 1647
Bessel Functions of the Second Kind……Page 1651
The Modified Bessel Equation……Page 1653
Exercises……Page 1657
Hints……Page 1662
Solutions……Page 1664
V Partial Differential Equations……Page 1687
Transforming Equations……Page 1688
Exercises……Page 1689
Hints……Page 1690
Solutions……Page 1691
Classification of Second Order Quasi-Linear Equations……Page 1692
Hyperbolic Equations……Page 1693
Parabolic equations……Page 1698
Elliptic Equations……Page 1699
Equilibrium Solutions……Page 1701
Exercises……Page 1703
Hints……Page 1704
Solutions……Page 1705
Homogeneous Equations with Homogeneous Boundary Conditions……Page 1711
Time-Independent Sources and Boundary Conditions……Page 1713
Inhomogeneous Equations with Homogeneous Boundary Conditions……Page 1716
Inhomogeneous Boundary Conditions……Page 1717
The Wave Equation……Page 1720
General Method……Page 1723
Exercises……Page 1725
Hints……Page 1741
Solutions……Page 1746
Finite Transforms……Page 1828
Exercises……Page 1832
Hints……Page 1833
Solutions……Page 1834
The Diffusion Equation……Page 1838
Exercises……Page 1839
Hints……Page 1841
Solutions……Page 1842
Fundamental Solution……Page 1848
Two Dimensional Space……Page 1849
Exercises……Page 1850
Hints……Page 1853
Solutions……Page 1854
Waves……Page 1866
Exercises……Page 1867
Hints……Page 1873
Solutions……Page 1875
Similarity Methods……Page 1895
Exercises……Page 1900
Hints……Page 1901
Solutions……Page 1902
First Order Linear Equations……Page 1905
First Order Quasi-Linear Equations……Page 1906
The Method of Characteristics and the Wave Equation……Page 1908
The Wave Equation for an Infinite Domain……Page 1909
The Wave Equation for a Semi-Infinite Domain……Page 1910
The Wave Equation for a Finite Domain……Page 1912
Envelopes of Curves……Page 1913
Exercises……Page 1916
Hints……Page 1918
Solutions……Page 1919
Fourier Transform for Partial Differential Equations……Page 1926
Fourier Transform……Page 1928
Exercises……Page 1930
Hints……Page 1934
Solutions……Page 1936
Inhomogeneous Equations and Homogeneous Boundary Conditions……Page 1958
Homogeneous Equations and Inhomogeneous Boundary Conditions……Page 1959
Eigenfunction Expansions for Elliptic Equations……Page 1961
The Method of Images……Page 1966
Exercises……Page 1968
Hints……Page 1979
Solutions……Page 1982
Conformal Mapping……Page 2042
Exercises……Page 2043
Hints……Page 2046
Solutions……Page 2047
Spherical Coordinates……Page 2059
Laplace’s Equation in a Disk……Page 2060
Laplace’s Equation in an Annulus……Page 2063
VI Calculus of Variations……Page 2067
Calculus of Variations……Page 2068
Exercises……Page 2069
Hints……Page 2083
Solutions……Page 2087
VII Nonlinear Differential Equations……Page 2174
Nonlinear Ordinary Differential Equations……Page 2175
Exercises……Page 2176
Hints……Page 2181
Solutions……Page 2182
Nonlinear Partial Differential Equations……Page 2204
Exercises……Page 2205
Hints……Page 2208
Solutions……Page 2209
VIII Appendices……Page 2228
Greek Letters……Page 2229
Notation……Page 2231
Formulas from Complex Variables……Page 2233
Table of Derivatives……Page 2236
Table of Integrals……Page 2240
Definite Integrals……Page 2244
Table of Sums……Page 2246
Table of Taylor Series……Page 2249
Properties of Laplace Transforms……Page 2252
Table of Laplace Transforms……Page 2254
Table of Fourier Transforms……Page 2258
Table of Fourier Transforms in n Dimensions……Page 2261
Table of Fourier Cosine Transforms……Page 2262
Table of Fourier Sine Transforms……Page 2264
Table of Wronskians……Page 2266
Sturm-Liouville Eigenvalue Problems……Page 2268
Green Functions for Ordinary Differential Equations……Page 2270
Circular Functions……Page 2273
Hyperbolic Functions……Page 2275
Definite Integrals……Page 2278
Formulas from Linear Algebra……Page 2279
Vector Analysis……Page 2280
Partial Fractions……Page 2282
Finite Math……Page 2286
Independent Events……Page 2287
Playing the Odds……Page 2288
Economics……Page 2289
Glossary……Page 2290
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