مقدمه‌ای بر روش‌های ریاضی کاربردی: روش‌های ریاضی پیشرفته برای دانشمندان و مهندسان

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