Introduction To Methods Of Applied Mathematics Or Advanced Mathematical Methods For Scientists And Engineers

Anti-Copyright......Page 23
Acknowledgments......Page 24
Warnings and Disclaimers......Page 25
About the Title......Page 26
Algebra......Page 27
Sets......Page 28
Single Valued Functions......Page 30
Inverses and Multi-Valued Functions......Page 32
Transforming Equations......Page 35
Scalars and Vectors......Page 37
The Kronecker Delta and Einstein Summation Convention......Page 40
The Dot and Cross Product......Page 41
Sets of Vectors in n Dimensions......Page 49
Exercises......Page 51
Hints......Page 53
Solutions......Page 55
Calculus......Page 62
Limits of Functions......Page 63
Continuous Functions......Page 68
The Derivative......Page 71
Implicit Differentiation......Page 76
Maxima and Minima......Page 78
Mean Value Theorems......Page 81
Application: Using Taylor's Theorem to Approximate Functions.......Page 83
Application: Finite Difference Schemes......Page 88
L'Hospital's Rule......Page 90
Exercises......Page 96
Hints......Page 101
Solutions......Page 107
The Indefinite Integral......Page 126
Definition......Page 132
Properties......Page 133
The Fundamental Theorem of Integral Calculus......Page 135
Partial Fractions......Page 137
Improper Integrals......Page 140
Exercises......Page 144
Hints......Page 147
Solutions......Page 151
Vector Functions......Page 160
Gradient, Divergence and Curl......Page 161
Exercises......Page 168
Hints......Page 170
Solutions......Page 171
Functions of a Complex Variable......Page 176
Complex Numbers......Page 177
The Complex Plane......Page 180
Polar Form......Page 184
Arithmetic and Vectors......Page 189
Integer Exponents......Page 190
Rational Exponents......Page 192
Exercises......Page 196
Hints......Page 202
Solutions......Page 205
Curves and Regions......Page 228
Cartesian and Modulus-Argument Form......Page 232
Graphing Functions of a Complex Variable......Page 234
Trigonometric Functions......Page 238
Inverse Trigonometric Functions......Page 243
Branch Points......Page 252
Exercises......Page 269
Hints......Page 279
Solutions......Page 284
Complex Derivatives......Page 329
Cauchy-Riemann Equations......Page 336
Harmonic Functions......Page 341
Singularities......Page 346
Categorization of Singularities......Page 347
Isolated and Non-Isolated Singularities......Page 351
Exercises......Page 353
Hints......Page 358
Solutions......Page 360
Analytic Continuation......Page 382
Analytic Continuation of Sums......Page 385
Analytic Functions Defined in Terms of Real Variables......Page 386
Polar Coordinates......Page 392
Analytic Functions Defined in Terms of Their Real or Imaginary Parts......Page 395
Exercises......Page 399
Hints......Page 401
Solutions......Page 402
Line Integrals......Page 407
Under Construction......Page 412
Cauchy's Theorem......Page 414
Indefinite Integrals......Page 416
Contour Integrals......Page 417
Exercises......Page 421
Hints......Page 423
Solutions......Page 424
Cauchy's Integral Formula......Page 429
Cauchy's Integral Formula......Page 430
The Argument Theorem......Page 437
Rouche's Theorem......Page 439
Exercises......Page 441
Hints......Page 443
Solutions......Page 444
Definitions......Page 448
Special Series......Page 451
Convergence Tests......Page 452
Uniform Convergence......Page 458
Tests for Uniform Convergence......Page 459
Uniform Convergence and Continuous Functions.......Page 461
Uniformly Convergent Power Series......Page 462
Integration and Differentiation of Power Series......Page 469
Taylor Series......Page 472
Newton's Binomial Formula.......Page 475
Laurent Series......Page 478
Exercises......Page 481
Hints......Page 488
Solutions......Page 491
The Residue Theorem......Page 516
The Cauchy Principal Value......Page 524
Cauchy Principal Value for Contour Integrals......Page 529
Integrals on the Real Axis......Page 533
Fourier Integrals......Page 538
Fourier Cosine and Sine Integrals......Page 541
Contour Integration and Branch Cuts......Page 543
Wedge Contours......Page 547
Box Contours......Page 550
Definite Integrals Involving Sine and Cosine......Page 551
Infinite Sums......Page 554
Exercises......Page 558
Hints......Page 572
Solutions......Page 579
Ordinary Differential Equations......Page 660
Notation......Page 661
One Parameter Families of Functions......Page 663
Exact Equations......Page 665
Separable Equations......Page 669
Homogeneous Coefficient Equations......Page 671
Homogeneous Equations......Page 675
Inhomogeneous Equations......Page 676
Variation of Parameters.......Page 677
Initial Conditions......Page 679
Piecewise Continuous Coefficients and Inhomogeneities......Page 680
Well-Posed Problems......Page 685
Ordinary Points......Page 687
Regular Singular Points......Page 690
Irregular Singular Points......Page 695
The Point at Infinity......Page 697
Exercises......Page 700
Hints......Page 706
Solutions......Page 709
Matrices and Jordan Canonical Form......Page 731
Systems of Differential Equations......Page 739
Exercises......Page 745
Hints......Page 751
Solutions......Page 753
Theory of Linear Ordinary Differential Equations......Page 783
Nature of Solutions......Page 784
Transformation to a First Order System......Page 787
Derivative of a Determinant.......Page 788
The Wronskian of a Set of Functions.......Page 789
The Wronskian of the Solutions to a Differential Equation......Page 791
Well-Posed Problems......Page 794
The Fundamental Set of Solutions......Page 796
Adjoint Equations......Page 799
Exercises......Page 802
Hints......Page 804
Solutions......Page 806
Constant Coefficient Equations......Page 812
Second Order Equations......Page 813
Higher Order Equations......Page 817
Real-Valued Solutions......Page 819
Euler Equations......Page 821
Real-Valued Solutions......Page 824
Exact Equations......Page 827
Equations Without Explicit Dependence on y......Page 828
Reduction of Order......Page 829
*Reduction of Order and the Adjoint Equation......Page 830
Exercises......Page 833
Hints......Page 840
Solutions......Page 843
Bernoulli Equations......Page 868
Riccati Equations......Page 870
Exchanging the Dependent and Independent Variables......Page 874
Autonomous Equations......Page 876
*Equidimensional-in-x Equations......Page 880
*Equidimensional-in-y Equations......Page 882
*Scale-Invariant Equations......Page 885
Exercises......Page 886
Hints......Page 890
Solutions......Page 892
The Constant Coefficient Equation......Page 904
Second Order Equations......Page 907
Higher Order Differential Equations......Page 909
Transformation to the form u'' + a(x) u = 0......Page 911
Transformation to a Constant Coefficient Equation......Page 912
Integral Equations......Page 914
Initial Value Problems......Page 915
Boundary Value Problems......Page 917
Exercises......Page 920
Hints......Page 922
Solutions......Page 923
Derivative of the Heaviside Function......Page 930
The Delta Function as a Limit......Page 932
Non-Rectangular Coordinate Systems......Page 934
Exercises......Page 936
Hints......Page 937
Solutions......Page 938
Particular Solutions......Page 941
Method of Undetermined Coefficients......Page 943
Second Order Differential Equations......Page 947
Higher Order Differential Equations......Page 951
Piecewise Continuous Coefficients and Inhomogeneities......Page 953
Eliminating Inhomogeneous Boundary Conditions......Page 957
Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions......Page 959
Existence of Solutions of Problems with Inhomogeneous Boundary Conditions......Page 960
Green Functions for First Order Equations......Page 962
Green Functions for Second Order Equations......Page 965
Green Functions for Sturm-Liouville Problems......Page 975
Initial Value Problems......Page 979
Problems with Unmixed Boundary Conditions......Page 982
Problems with Mixed Boundary Conditions......Page 984
Green Functions for Higher Order Problems......Page 988
Fredholm Alternative Theorem......Page 994
Exercises......Page 1001
Hints......Page 1008
Solutions......Page 1012
Introduction......Page 1053
Exact Equations......Page 1055
Homogeneous First Order......Page 1056
Inhomogeneous First Order......Page 1058
Homogeneous Constant Coefficient Equations......Page 1061
Reduction of Order......Page 1064
Exercises......Page 1066
Hints......Page 1067
Solutions......Page 1068
Ordinary Points......Page 1072
Taylor Series Expansion for a Second Order Differential Equation......Page 1077
Regular Singular Points of Second Order Equations......Page 1086
Indicial Equation......Page 1089
The Case: Double Root......Page 1091
The Case: Roots Differ by an Integer......Page 1095
The Point at Infinity......Page 1105
Exercises......Page 1108
Hints......Page 1113
Solutions......Page 1115
Asymptotic Relations......Page 1139
Leading Order Behavior of Differential Equations......Page 1143
Integration by Parts......Page 1152
Asymptotic Series......Page 1159
The Parabolic Cylinder Equation.......Page 1161
Linear Spaces......Page 1167
Inner Products......Page 1169
Norms......Page 1170
Gramm-Schmidt Orthogonalization......Page 1173
Orthonormal Function Expansion......Page 1177
Sets Of Functions......Page 1178
Least Squares Fit to a Function and Completeness......Page 1184
Closure Relation......Page 1187
Linear Operators......Page 1192
Exercises......Page 1193
Hints......Page 1194
Solutions......Page 1195
Adjoint Operators......Page 1197
Self-Adjoint Operators......Page 1198
Exercises......Page 1201
Hints......Page 1202
Solutions......Page 1203
Summary of Adjoint Operators......Page 1204
Formally Self-Adjoint Operators......Page 1205
Self-Adjoint Problems......Page 1208
Self-Adjoint Eigenvalue Problems......Page 1209
Inhomogeneous Equations......Page 1214
Exercises......Page 1217
Hints......Page 1218
Solutions......Page 1219
An Eigenvalue Problem.......Page 1221
Fourier Series.......Page 1224
Least Squares Fit......Page 1230
Fourier Series for Functions Defined on Arbitrary Ranges......Page 1233
Fourier Cosine Series......Page 1236
Fourier Sine Series......Page 1237
Complex Fourier Series and Parseval's Theorem......Page 1238
Behavior of Fourier Coefficients......Page 1241
Integrating and Differentiating Fourier Series......Page 1250
Exercises......Page 1255
Hints......Page 1264
Solutions......Page 1267
Derivation of the Sturm-Liouville Form......Page 1317
Properties of Regular Sturm-Liouville Problems......Page 1320
Solving Differential Equations With Eigenfunction Expansions......Page 1331
Exercises......Page 1337
Hints......Page 1341
Solutions......Page 1343
Uniform Convergence of Integrals......Page 1368
The Riemann-Lebesgue Lemma......Page 1369
Singular Functions......Page 1371
The Laplace Transform......Page 1373
The Inverse Laplace Transform......Page 1375
F(s) with Poles......Page 1378
mathaccent "705E f(s) with Branch Points......Page 1383
Asymptotic Behavior of F(s)......Page 1386
Properties of the Laplace Transform......Page 1388
Constant Coefficient Differential Equations......Page 1392
Systems of Constant Coefficient Differential Equations......Page 1394
Exercises......Page 1396
Hints......Page 1404
Solutions......Page 1408
Derivation from a Fourier Series......Page 1441
The Fourier Transform......Page 1443
A Word of Caution......Page 1446
Integrals that Converge......Page 1447
Cauchy Principal Value and Integrals that are Not Absolutely Convergent.......Page 1450
Analytic Continuation......Page 1452
Properties of the Fourier Transform......Page 1454
Closure Relation.......Page 1455
Fourier Transform of a Derivative.......Page 1456
Fourier Convolution Theorem.......Page 1457
Parseval's Theorem.......Page 1461
Shift Property.......Page 1462
Solving Differential Equations with the Fourier Transform......Page 1463
The Fourier Cosine Transform......Page 1466
The Fourier Sine Transform......Page 1467
Transforms of Derivatives......Page 1468
Convolution Theorems......Page 1470
Cosine and Sine Transform in Terms of the Fourier Transform......Page 1472
Solving Differential Equations with the Fourier Cosine and Sine Transforms......Page 1473
Exercises......Page 1475
Hints......Page 1481
Solutions......Page 1484
Euler's Formula......Page 1510
Hankel's Formula......Page 1512
Gauss' Formula......Page 1514
Weierstrass' Formula......Page 1516
Stirling's Approximation......Page 1518
Exercises......Page 1523
Hints......Page 1524
Solutions......Page 1525
Bessel's Equation......Page 1527
Frobeneius Series Solution about z = 0......Page 1528
Behavior at Infinity......Page 1531
Bessel Functions of the First Kind......Page 1533
The Bessel Function Satisfies Bessel's Equation......Page 1534
Series Expansion of the Bessel Function......Page 1535
Bessel Functions of Non-Integral Order......Page 1538
Recursion Formulas......Page 1541
Bessel Functions of Half-Integral Order......Page 1544
Neumann Expansions......Page 1545
Bessel Functions of the Second Kind......Page 1549
The Modified Bessel Equation......Page 1551
Exercises......Page 1555
Hints......Page 1560
Solutions......Page 1562
Partial Differential Equations......Page 1585
Transforming Equations......Page 1586
Exercises......Page 1587
Hints......Page 1588
Solutions......Page 1589
Classification of Second Order Quasi-Linear Equations......Page 1590
Hyperbolic Equations......Page 1591
Elliptic Equations......Page 1596
Equilibrium Solutions......Page 1598
Exercises......Page 1600
Hints......Page 1601
Solutions......Page 1602
Homogeneous Equations with Homogeneous Boundary Conditions......Page 1606
Time-Independent Sources and Boundary Conditions......Page 1608
Inhomogeneous Equations with Homogeneous Boundary Conditions......Page 1611
Inhomogeneous Boundary Conditions......Page 1613
The Wave Equation......Page 1615
General Method......Page 1619
Exercises......Page 1620
Hints......Page 1634
Solutions......Page 1639
Finite Transforms......Page 1716
Exercises......Page 1720
Hints......Page 1721
Solutions......Page 1722
Waves......Page 1727
Exercises......Page 1728
Hints......Page 1734
Solutions......Page 1736
The Diffusion Equation......Page 1753
Exercises......Page 1754
Hints......Page 1756
Solutions......Page 1757
Similarity Methods......Page 1760
Exercises......Page 1765
Hints......Page 1766
Solutions......Page 1767
The Method of Characteristics and the Wave Equation......Page 1769
The Method of Characteristics for an Infinite Domain......Page 1771
The Method of Characteristics for a Semi-Infinite Domain......Page 1772
Envelopes of Curves......Page 1773
Exercises......Page 1776
Hints......Page 1778
Solutions......Page 1779
Fourier Transform for Partial Differential Equations......Page 1785
The Fourier Sine Transform......Page 1787
Fourier Transform......Page 1788
Exercises......Page 1789
Hints......Page 1794
Solutions......Page 1797
Inhomogeneous Equations and Homogeneous Boundary Conditions......Page 1820
Homogeneous Equations and Inhomogeneous Boundary Conditions......Page 1821
Eigenfunction Expansions for Elliptic Equations......Page 1823
The Method of Images......Page 1828
Exercises......Page 1829
Hints......Page 1834
Solutions......Page 1836
Conformal Mapping......Page 1877
Exercises......Page 1878
Hints......Page 1881
Solutions......Page 1882
Spherical Coordinates......Page 1890
Laplace's Equation in a Disk......Page 1891
Laplace's Equation in an Annulus......Page 1894
Calculus of Variations......Page 1898
Calculus of Variations......Page 1899
Exercises......Page 1900
Hints......Page 1917
Solutions......Page 1923
Nonlinear Differential Equations......Page 2016
Nonlinear Ordinary Differential Equations......Page 2017
Exercises......Page 2018
Hints......Page 2023
Solutions......Page 2025
Nonlinear Partial Differential Equations......Page 2047
Exercises......Page 2048
Hints......Page 2051
Solutions......Page 2052
Appendices......Page 2070
Greek Letters......Page 2071
Notation......Page 2073
Formulas from Complex Variables......Page 2075
Table of Derivatives......Page 2078
Table of Integrals......Page 2082
Definite Integrals......Page 2086
Table of Sums......Page 2089
Table of Taylor Series......Page 2092
Table of Laplace Transforms......Page 2095
Table of Fourier Transforms......Page 2100
Table of Fourier Transforms in n Dimensions......Page 2103
Table of Fourier Cosine Transforms......Page 2104
Table of Fourier Sine Transforms......Page 2106
Table of Wronskians......Page 2108
Sturm-Liouville Eigenvalue Problems......Page 2110
Green Functions for Ordinary Differential Equations......Page 2112
Circular Functions......Page 2115
Hyperbolic Functions......Page 2117
Definite Integrals......Page 2120
Formulas from Linear Algebra......Page 2121
Vector Analysis......Page 2123
Partial Fractions......Page 2125
Finite Math......Page 2129
Independent Events......Page 2130
Playing the Odds......Page 2131
Economics......Page 2132
Glossary......Page 2133