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