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Introduction to Methods of Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers Sean Mauch April 8, 2002 Contents Anti-Copyright xxiii Preface xxiv 0.1 Advice to Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv 0.2 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv 0.3 Warnings and Disclaimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv 0.4 Suggested Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi 0.5 About the Title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi I Algebra 1 1 Sets and Functions 2 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Single Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Inverses and Multi-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Transforming Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 i 2 Vectors 22 2.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.1 Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.2 The Kronecker Delta and Einstein Summation Convention . . . . . . . . . . . . . . . . . . . . 25 2.1.3 The Dot and Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Sets of Vectors in n Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 II Calculus 46 3 Differential Calculus 47 3.1 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.6 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.6.1 Application: Using Taylor’s Theorem to Approximate Functions. . . . . . . . . . . . . . . . . . 66 3.6.2 Application: Finite Difference Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.7 L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.10 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4 Integral Calculus 111 4.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 ii 4.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.3 The Fundamental Theorem of Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.4 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.4.1 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5 Vector Calculus 147 5.1 Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.2 Gradient, Divergence and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 III Functions of a Complex Variable 170 6 Complex Numbers 171 6.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.2 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.3 Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.4 Arithmetic and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.5 Integer Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.6 Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 iii 7 Functions of a Complex Variable 228 7.1 Curves and Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 7.2 The Point at Infinity and the Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.3 Cartesian and Modulus-Argument Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.4 Graphing Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 7.5 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 7.6 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 7.7 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 7.8 Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 256 7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 7.10 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 7.11 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 8 Analytic Functions 346 8.1 Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 8.2 Cauchy-Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 8.3 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 8.4 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 8.4.1 Categorization of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 8.4.2 Isolated and Non-Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 8.5 Application: Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 8.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 8.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 9 Analytic Continuation 419 9.1 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 9.2 Analytic Continuation of Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 9.3 Analytic Functions Defined in Terms of Real Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 424 9.3.1 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 iv 9.3.2 Analytic Functions Defined in Terms of Their Real or Imaginary Parts . . . . . . . . . . . . . . 432 9.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 9.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 9.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 10 Contour Integration and the Cauchy-Goursat Theorem 444 10.1 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 10.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 10.2.1 Maximum Modulus Integral Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 10.3 The Cauchy-Goursat Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 10.4 Contour Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 10.5 Morera’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 10.6 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 10.7 Fundamental Theorem of Calculus via Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 10.7.1 Line Integrals and Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 10.7.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 10.8 Fundamental Theorem of Calculus via Complex Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 457 10.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 10.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 10.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 11 Cauchy’s Integral Formula 475 11.1 Cauchy’s Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 11.2 The Argument Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 11.3 Rouche’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 11.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 11.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 v 12 Series and Convergence 508 12.1 Series of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 12.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 12.1.2 Special Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 12.1.3 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 12.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 12.2.1 Tests for Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 12.2.2 Uniform Convergence and Continuous Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 522 12.3 Uniformly Convergent Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 12.4 Integration and Differentiation of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 12.5 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 12.5.1 Newton’s Binomial Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 12.6 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 12.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 12.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 13 The Residue Theorem 614 13.1 The Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 13.2 Cauchy Principal Value for Real Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 13.2.1 The Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 13.3 Cauchy Principal Value for Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 13.4 Integrals on the Real Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 13.5 Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 13.6 Fourier Cosine and Sine Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 13.7 Contour Integration and Branch Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 13.8 Exploiting Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 13.8.1 Wedge Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 13.8.2 Box Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 646 13.9 Definite Integrals Involving Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 vi 13.10Infinite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 13.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 13.12Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 13.13Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 IV Ordinary Differential Equations 761 14 First Order Differential Equations 762 14.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762 14.2 One Parameter Families of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 14.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766 14.3.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 14.3.2 Homogeneous Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 14.4 The First Order, Linear Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 14.4.1 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 14.4.2 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779 14.4.3 Variation of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782 14.5 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782 14.5.1 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . 783 14.6 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788 14.7 Equations in the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 14.7.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 14.7.2 Regular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794 14.7.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799 14.7.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801 14.8 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804 14.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807 14.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810 vii 15 First Order Linear Systems of Differential Equations 831 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831 15.2 Using Eigenvalues and Eigenvectors to find Homogeneous Solutions . . . . . . . . . . . . . . . . . . . 832 15.3 Matrices and Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837 15.4 Using the Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844 15.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850 15.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 15.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 16 Theory of Linear Ordinary Differential Equations 885 16.1 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885 16.2 Nature of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886 16.3 Transformation to a First Order System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889 16.4 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890 16.4.1 Derivative of a Determinant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890 16.4.2 The Wronskian of a Set of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891 16.4.3 The Wronskian of the Solutions to a Differential Equation . . . . . . . . . . . . . . . . . . . . 893 16.5 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 16.6 The Fundamental Set of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898 16.7 Adjoint Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 900 16.8 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904 16.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 16.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906 17 Techniques for Linear Differential Equations 911 17.1 Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911 17.1.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912 17.1.2 Higher Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916 17.1.3 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917 17.2 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 921 viii 17.2.1 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923 17.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926 17.4 Equations Without Explicit Dependence on y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927 17.5 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928 17.6 *Reduction of Order and the Adjoint Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929 17.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932 17.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938 17.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941 18 Techniques for Nonlinear Differential Equations 965 18.1 Bernoulli Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965 18.2 Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 18.3 Exchanging the Dependent and Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 971 18.4 Autonomous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973 18.5 *Equidimensional-in-x Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976 18.6 *Equidimensional-in-y Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978 18.7 *Scale-Invariant Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981 18.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 982 18.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985 18.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 19 Transformations and Canonical Forms 999 19.1 The Constant Coefficient Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999 19.2 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002 19.2.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002 19.2.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003 19.3 Transformations of the Independent Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005 19.3.1 Transformation to the form u” + a(x) u = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005 19.3.2 Transformation to a Constant Coefficient Equation . . . . . . . . . . . . . . . . . . . . . . . . 1006 19.4 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008 ix 19.4.1 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008 19.4.2 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 19.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013 19.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015 19.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016 20 The Dirac Delta Function 1022 20.1 Derivative of the Heaviside Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022 20.2 The Delta Function as a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 20.3 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026 20.4 Non-Rectangular Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027 20.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029 20.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 20.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033 21 Inhomogeneous Differential Equations 1040 21.1 Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1040 21.2 Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042 21.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 21.3.1 Second Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 21.3.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 21.4 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . 1052 21.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 21.5.1 Eliminating Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 1055 21.5.2 Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions . . . . . . . . . 1057 21.5.3 Existence of Solutions of Problems with Inhomogeneous Boundary Conditions . . . . . . . . . . 1058 21.6 Green Functions for First Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060 21.7 Green Functions for Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063 21.7.1 Green Functions for Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073 21.7.2 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076 x 21.7.3 Problems with Unmixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078 21.7.4 Problems with Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 21.8 Green Functions for Higher Order Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085 21.9 Fredholm Alternative Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1090 21.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 21.11Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104 21.12Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107 22 Difference Equations 1145 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145 22.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147 22.3 Homogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1148 22.4 Inhomogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150 22.5 Homogeneous Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153 22.6 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156 22.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1158 22.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159 22.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1160 23 Series Solutions of Differential Equations 1163 23.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163 23.1.1 Taylor Series Expansion for a Second Order Differential Equation . . . . . . . . . . . . . . . . 1167 23.2 Regular Singular Points of Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177 23.2.1 Indicial Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1180 23.2.2 The Case: Double Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182 23.2.3 The Case: Roots Differ by an Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186 23.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196 23.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196 23.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199 23.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204 xi 23.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205 24 Asymptotic Expansions 1228 24.1 Asymptotic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228 24.2 Leading Order Behavior of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232 24.3 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 1241 24.4 Asymptotic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248 24.5 Asymptotic Expansions of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249 24.5.1 The Parabolic Cylinder Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249 25 Hilbert Spaces 1255 25.1 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255 25.2 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257 25.3 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258 25.4 Linear Independence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1260 25.5 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1260 25.6 Gramm-Schmidt Orthogonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1261 25.7 Orthonormal Function Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263 25.8 Sets Of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265 25.9 Least Squares Fit to a Function and Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272 25.10Closure Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275 25.11Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1280 25.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1281 25.13Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1282 25.14Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283 26 Self Adjoint Linear Operators 1285 26.1 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285 26.2 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286 26.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289 xii 26.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1290 26.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1291 27 Self-Adjoint Boundary Value Problems 1292 27.1 Summary of Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1292 27.2 Formally Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293 27.3 Self-Adjoint Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296 27.4 Self-Adjoint Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296 27.5 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301 27.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304 27.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305 27.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306 28 Fourier Series 1308 28.1 An Eigenvalue Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1308 28.2 Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1311 28.3 Least Squares Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315 28.4 Fourier Series for Functions Defined on Arbitrary Ranges . . . . . . . . . . . . . . . . . . . . . . . . . 1319 28.5 Fourier Cosine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1322 28.6 Fourier Sine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323 28.7 Complex Fourier Series and Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324 28.8 Behavior of Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327 28.9 Gibb’s Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336 28.10Integrating and Differentiating Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336 28.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1341 28.12Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349 28.13Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1351 29 Regular Sturm-Liouville Problems 1398 29.1 Derivation of the Sturm-Liouville Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398 xiii 29.2 Properties of Regular Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1400 29.3 Solving Differential Equations With Eigenfunction Expansions . . . . . . . . . . . . . . . . . . . . . . 1411 29.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417 29.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1421 29.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1423 30 Integrals and Convergence 1448 30.1 Uniform Convergence of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1448 30.2 The Riemann-Lebesgue Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1449 30.3 Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1450 30.3.1 Integrals on an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1450 30.3.2 Singular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1451 31 The Laplace Transform 1453 31.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453 31.2 The Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455 31.2.1 fˆ(s) with Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458 31.2.2 fˆ(s) with Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1463 31.2.3 Asymptotic Behavior of fˆ(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466 31.3 Properties of the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1468 31.4 Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1471 31.5 Systems of Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1473 31.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476 31.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1483 31.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486 32 The Fourier Transform 151832.1 Derivation from a Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518 32.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1520 32.2.1 A Word of Caution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523 xiv 32.3 Evaluating Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1524 32.3.1 Integrals that Converge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1524 32.3.2 Cauchy Principal Value and Integrals that are Not Absolutely Convergent. . . . . . . . . . . . . 1527 32.3.3 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1529 32.4 Properties of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1531 32.4.1 Closure Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1531 32.4.2 Fourier Transform of a Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1532 32.4.3 Fourier Convolution Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534 32.4.4 Parseval’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537 32.4.5 Shift Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1539 32.4.6 Fourier Transform of x f(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1539 32.5 Solving Differential Equations with the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 1540 32.6 The Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1542 32.6.1 The Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1542 32.6.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1543 32.7 Properties of the Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544 32.7.1 Transforms of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544 32.7.2 Convolution Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546 32.7.3 Cosine and Sine Transform in Terms of the Fourier Transform . . . . . . . . . . . . . . . . . . 1548 32.8 Solving Differential Equations with the Fourier Cosine and Sine Transforms . . . . . . . . . . . . . . . 1549 32.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1551 32.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558 32.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1561 33 The Gamma Function 1585 33.1 Euler’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1585 33.2 Hankel’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587 33.3 Gauss’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1589 33.4 Weierstrass’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1591 33.5 Stirling’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593 xv 33.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1598 33.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1599 33.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1600 34 Bessel Functions 1602 34.1 Bessel’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1602 34.2 Frobeneius Series Solution about z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1603 34.2.1 Behavior at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606 34.3 Bessel Functions of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608 34.3.1 The Bessel Function Satisfies Bessel’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 1609 34.3.2 Series Expansion of the Bessel Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1610 34.3.3 Bessel Functions of Non-Integer Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613 34.3.4 Recursion Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1616 34.3.5 Bessel Functions of Half-Integer Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1619 34.4 Neumann Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1620 34.5 Bessel Functions of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1624 34.6 Hankel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626 34.7 The Modified Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626 34.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1630 34.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1635 34.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637 V Partial Differential Equations 1660 35 Transforming Equations 1661 35.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1662 35.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1663 35.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664 xvi 36 Classification of Partial Differential Equations 1665 36.1 Classification of Second Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1665 36.1.1 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1666 36.1.2 Parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1671 36.1.3 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1672 36.2 Equilibrium Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1674 36.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1676 36.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1677 36.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1678 37 Separation of Variables 1684 37.1 Eigensolutions of Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1684 37.2 Homogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . 1684 37.3 Time-Independent Sources and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686 37.4 Inhomogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 1689 37.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1690 37.6 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 1693 37.7 General Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1696 37.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698 37.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714 37.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1719 38 Finite Transforms 1801 38.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1805 38.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806 38.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1807 39 The Diffusion Equation 1811 39.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1812 39.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1814 xvii 39.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1815 40 Laplace’s Equation 1821 40.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1821 40.2 Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1821 40.2.1 Two Dimensional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1822 40.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1823 40.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826 40.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827 41 Waves 1839 41.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1840 41.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1846 41.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1848 42 Similarity Methods 1868 42.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1873 42.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1874 42.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1875 43 Method of Characteristics 1878 43.1 First Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1878 43.2 First Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1879 43.3 The Method of Characteristics and the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 1881 43.4 The Wave Equation for an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1882 43.5 The Wave Equation for a Semi-Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1883 43.6 The Wave Equation for a Finite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1885 43.7 Envelopes of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1886 43.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1889 43.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1891 xviii 43.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1892 44 Transform Methods 1899 44.1 Fourier Transform for Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1899 44.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1901 44.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1901 44.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1903 44.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1907 44.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1909 45 Green Functions 1931 45.1 Inhomogeneous Equations and Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 1931 45.2 Homogeneous Equations and Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 1932 45.3 Eigenfunction Expansions for Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1934 45.4 The Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1939 45.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1941 45.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1952 45.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1955 46 Conformal Mapping 2015 46.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2016 46.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2019 46.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2020 47 Non-Cartesian Coordinates 2032 47.1 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2032 47.2 Laplace’s Equation in a Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2033 47.3 Laplace’s Equation in an Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2036 xix VI Calculus of Variations 2040 48 Calculus of Variations 2041 48.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2042 48.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2056 48.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2060 VII Nonlinear Differential Equations 2147 49 Nonlinear Ordinary Differential Equations 2148 49.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2149 49.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2154 49.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2155 50 Nonlinear Partial Differential Equations 2177 50.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2178 50.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2181 50.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2182 VIII Appendices 2201 A Greek Letters 2202 B Notation 2204 C Formulas from Complex Variables 2206 D Table of Derivatives 2209 xx E Table of Integrals 2213 F Definite Integrals 2217 G Table of Sums 2219 H Table of Taylor Series 2222I Table of Laplace Transforms 2225 I.1 Properties of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2225 I.2 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2227 J Table of Fourier Transforms 2231 K Table of Fourier Transforms in n Dimensions 2234 L Table of Fourier Cosine Transforms 2235 M Table of Fourier Sine Transforms 2237 N Table of Wronskians 2239 O Sturm-Liouville Eigenvalue Problems 2241 P Green Functions for Ordinary Differential Equations 2243 Q Trigonometric Identities 2246 Q.1 Circular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2246 Q.2 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2248 R Bessel Functions 2251 R.1 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2251 xxi S Formulas from Linear Algebra 2252 T Vector Analysis 2253 U Partial Fractions 2255 V Finite Math 2259 W Probability 2260 W.1 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2260 W.2 Playing the Odds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2261 X Economics 2262 Y Glossary 2263 xxii Anti-Copyright Anti-Copyright @ 1995-2001 by Mauch Publishing Company, un-Incorporated. No rights reserved. Any part of this publication may be reproduced, stored in a retrieval system, transmitted or desecrated without permission. xxiii Preface During the summer before my final undergraduate year at Caltech I set out to write a math text unlike any other, namely, one written by me. In that respect I have succeeded beautifully. Unfortunately, the text is neither complete nor polished. I have a “Warnings and Disclaimers” section below that is a little amusing, and an appendix on probability that I feel concisesly captures the essence of the subject. However, all the material in between is in some stage of development. I am currently working to improve and expand this text. This text is freely available from my web set. Currently I’m at http://www.its.caltech.edu/˜sean. I post new versions a couple of times a year. 0.1 Advice to Teachers If you have something worth saying, write it down. 0.2 Acknowledgments I would like to thank Professor Saffman for advising me on this project and the Caltech SURF program for providing the funding for me to write the first edition of this book. xxiv 0.3 Warnings and Disclaimers • This book is a work in progress. It contains quite a few mistakes and typos. I would greatly appreciate your constructive criticism. You can reach me at ‘sean@its.caltech.edu’. • Reading this book impairs your ability to drive a car or operate machinery. • This book has been found to cause drowsiness in laboratory animals. • This book contains twenty-three times the US RDA of fiber. • Caution: FLAMMABLE - Do not read while smoking or near a fire. • If infection, rash, or irritation develops, discontinue use and consult a physician. • Warning: For external use only. Use only as directed. Intentional misuse by deliberately concentrating contents can be harmful or fatal. KEEP OUT OF REACH OF CHILDREN. • In the unlikely event of a water landing do not use this book as a flotation device. • The material in this text is fiction; any resemblance to real theorems, living or dead, is purely coincidental. • This is by far the most amusing section of this book. • Finding the typos and mistakes in this book is left as an exercise for the reader. (Eye ewes a spelling chequer from thyme too thyme, sew their should knot bee two many misspellings. Though I ain’t so sure the grammar’s too good.) • The theorems and methods in this text are subject to change without notice. • This is a chain book. If you do not make seven copies and distribute them to your friends within ten days of obtaining this text you will suffer great misfortune and other nastiness. • The surgeon general has determined that excessive studying is detrimental to your social life. xxv • This text has been buffered for your protection and ribbed for your pleasure. • Stop reading this rubbish and get back to work! 0.4 Suggested Use This text is well suited to the student, professional or lay-person. It makes a superb gift. This text has a boquet that is light and fruity, with some earthy undertones. It is ideal with dinner or as an apertif. Bon apetit! 0.5 About the Title The title is only making light of naming conventions in the sciences and is not an insult to engineers. If you want to learn about some mathematical subject, look for books with “Introduction” or “Elementary” in the title. If it is an “Intermediate” text it will be incomprehensible. If it is “Advanced” then not only will it be incomprehensible, it will have low production qualities, i.e. a crappy typewriter font, no graphics and no examples. There is an exception to this rule: When the title also contains the word “Scientists” or “Engineers” the advanced book may be quite suitable for actually learning the material. xxvi Part I Algebra 1 Chapter 1 Sets and Functions 1.1 Sets Definition. A set is a collection of objects. We call the objects, elements. A set is denoted by listing the elements between braces. For example: {e, ı, pi, 1}. We use ellipses to indicate patterns. The set of positive integers is {1, 2, 3, . . .}. We also denote a sets with the notation {x|conditions on x} for sets that are more easily described than enumerated. This is read as “the set of elements x such that x satisfies . . . ”. x ∈ S is the notation for “x is an element of the set S.” To express the opposite we have x 6∈ S for “x is not an element of the set S.” Examples. We have notations for denoting some of the commonly encountered sets. • ∅ = {} is the empty set, the set containing no elements. • Z = {. . . ,−1, 0, 1 . . .} is the set of integers. (Z is for “Zahlen”, the German word for “number”.) • Q = {p/q|p, q ∈ Z, q 6= 0} is the set of rational numbers. (Q is for quotient.) • R = {x|x = a1a2 · · · an.b1b2 · · · } is the set of real numbers, i.e. the set of numbers with decimal expansions. 1 1Guess what R is for. 2 • C = {a + ıb|a, b ∈ R, ı2 = −1} is the set of complex numbers. ı is the square root of −1. (If you haven’t seen complex numbers before, don’t dismay. We’ll cover them later.) • Z+, Q+ and R+ are the sets of positive integers, rationals and reals, respectively. For example, Z+ = {1, 2, 3, . . .}. • Z0+, Q0+ and R0+ are the sets of non-negative integers, rationals and reals, respectively. For example, Z0+ = {0, 1, 2, . . .}. • (a . . . b) denotes an open interval on the real axis. (a . . . b) ≡ {x|x ∈ R, a < x < b} • We use brackets to denote the closed interval. [a . . . b] ≡ {x|x ∈ R, a ≤ x ≤ b} The cardinality or order of a set S is denoted |S|. For finite sets, the cardinality is the number of elements in the set. The Cartesian product of two sets is the set of ordered pairs: X × Y ≡ {(x, y)|x ∈ X, y ∈ Y }. The Cartesian product of n sets is the set of ordered n-tuples: X1 ×X2 × · · · ×Xn ≡ {(x1, x2, . . . , xn)|x1 ∈ X1, x2 ∈ X2, . . . , xn ∈ Xn}. Equality. Two sets S and T are equal if each element of S is an element of T and vice versa. This is denoted, S = T . Inequality is S 6= T , of course. S is a subset of T , S ⊆ T , if every element of S is an element of T . S is a proper subset of T , S ⊂ T , if S ⊆ T and S 6= T . For example: The empty set is a subset of every set, ∅ ⊆ S. The rational numbers are a proper subset of the real numbers, Q ⊂ R. Operations. The union of two sets, S ∪ T , is the set whose elements are in either of the two sets. The unionof n sets, ∪nj=1Sj ≡ S1 ∪ S2 ∪ · · · ∪ Sn is the set whose elements are in any of the sets Sj. The intersection of two sets, S ∩ T , is the set whose elements are in both of the two sets. In other words, the intersection of two sets in the set of elements that the two sets have in common. The intersection of n sets, ∩nj=1Sj ≡ S1 ∩ S2 ∩ · · · ∩ Sn 3 is the set whose elements are in all of the sets Sj. If two sets have no elements in common, S ∩ T = ∅, then the sets are disjoint. If T ⊆ S, then the difference between S and T , S \ T , is the set of elements in S which are not in T . S \ T ≡ {x|x ∈ S, x 6∈ T} The difference of sets is also denoted S − T . Properties. The following properties are easily verified from the above definitions. • S ∪ ∅ = S, S ∩ ∅ = ∅, S \ ∅ = S, S \ S = ∅. • Commutative. S ∪ T = T ∪ S, S ∩ T = T ∩ S. • Associative. (S ∪ T ) ∪ U = S ∪ (T ∪ U) = S ∪ T ∪ U , (S ∩ T ) ∩ U = S ∩ (T ∩ U) = S ∩ T ∩ U . • Distributive. S ∪ (T ∩ U) = (S ∪ T ) ∩ (S ∪ U), S ∩ (T ∪ U) = (S ∩ T ) ∪ (S ∩ U). 1.2 Single Valued Functions Single-Valued Functions. A single-valued function or single-valued mapping is a mapping of the elements x ∈ X into elements y ∈ Y . This is expressed as f : X → Y or X f→ Y . If such a function is well-defined, then for each x ∈ X there exists a unique element of y such that f(x) = y. The set X is the domain of the function, Y is the codomain, (not to be confused with the range, which we introduce shortly). To denote the value of a function on a particular element we can use any of the notations: f(x) = y, f : x 7→ y or simply x 7→ y. f is the identity map on X if f(x) = x for all x ∈ X. Let f : X → Y . The range or image of f is f(X) = {y|y = f(x) for some x ∈ X}. The range is a subset of the codomain. For each Z ⊆ Y , the inverse image of Z is defined: f−1(Z) ≡ {x ∈ X|f(x) = z for some z ∈ Z}. 4 Examples. • Finite polynomials and the exponential function are examples of single valued functions which map real numbers to real numbers. • The greatest integer function, b·c, is a mapping from R to Z. bxc in the greatest integer less than or equal to x. Likewise, the least integer function, dxe, is the least integer greater than or equal to x. The -jectives. A function is injective if for each x1 6= x2, f(x1) 6= f(x2). In other words, for each x in the domain there is a unique y = f(x) in the range. f is surjective if for each y in the codomain, there is an x such that y = f(x). If a function is both injective and surjective, then it is bijective. A bijective function is also called a one-to-one mapping. Examples. • The exponential function y = ex is a bijective function, (one-to-one mapping), that maps R to R+. (R is the set of real numbers; R+ is the set of positive real numbers.) • f(x) = x2 is a bijection from R+ to R+. f is not injective from R to R+. For each positive y in the range, there are two values of x such that y = x2. • f(x) = sinx is not injective from R to [−1..1]. For each y ∈ [−1, 1] there exists an infinite number of values of x such that y = sin x. 1.3 Inverses and Multi-Valued Functions If y = f(x), then we can write x = f−1(y) where f−1 is the inverse of f . If y = f(x) is a one-to-one function, then f−1(y) is also a one-to-one function. In this case, x = f−1(f(x)) = f(f−1(x)) for values of x where both f(x) and f−1(x) are defined. For example log x, which maps R+ to R is the inverse of ex. x = elog x = log(ex) for all x ∈ R+. (Note the x ∈ R+ ensures that log x is defined.) 5 Injective Surjective Bijective Figure 1.1: Depictions of Injective, Surjective and Bijective Functions If y = f(x) is a many-to-one function, then x = f−1(y) is a one-to-many function. f−1(y) is a multi-valued function. We have x = f(f−1(x)) for values of x where f−1(x) is defined, however x 6= f−1(f(x)). There are diagrams showing one-to-one, many-to-one and one-to-many functions in Figure 1.2. rangedomain rangedomain rangedomain one-to-one many-to-one one-to-many Figure 1.2: Diagrams of One-To-One, Many-To-One and One-To-Many Functions Example 1.3.1 y = x2, a many-to-one function has the inverse x = y1/2. For each positive y, there are two values of x such that x = y1/2. y = x2 and y = x1/2 are graphed in Figure 1.3. 6 Figure 1.3: y = x2 and y = x1/2 We say that there are two branches of y = x1/2: the positive and the negative branch. We denote the positive branch as y = √ x; the negative branch is y = −√x. We call √x the principal branch of x1/2. Note that √x is a one-to-one function. Finally, x = (x1/2)2 since (±√x)2 = x, but x 6= (x2)1/2 since (x2)1/2 = ±x. y = √x is graphed in Figure 1.4. Figure 1.4: y = √ x Now consider the many-to-one function y = sin x. The inverse is x = arcsin y. For each y ∈ [−1, 1] there are an infinite number of values x such that x = arcsin y. In Figure 1.5 is a graph of y = sin x and a graph of a few branches of y = arcsinx. Example 1.3.2 arcsinx has an infinite number of branches. We will denote the principal branch by Arcsin x which maps [−1, 1] to [−pi 2 , pi 2 ] . Note that x = sin(arcsinx), but x 6= arcsin(sinx). y = Arcsin x in Figure 1.6. 7 Figure 1.5: y = sin x and y = arcsinx Figure 1.6: y = Arcsin x Example 1.3.3 Consider 11/3. Since x3 is a one-to-one function, x1/3 is a single-valued function. (See Figure 1.7.) 11/3 = 1. Figure 1.7: y = x3 and y = x1/3 8 Example 1.3.4 Consider arccos(1/2). cosx and a few branches of arccosx are graphed in Figure 1.8. cosx = 1/2 Figure 1.8: y = cos x and y = arccosx has the two solutions x = ±pi/3 in the range x ∈ [−pi, pi]. Since cos(x+ pi) = − cosx, arccos(1/2) = {±pi/3 + npi}. 1.4 Transforming Equations We must take care in applying functions to equations. It is always safe to apply a one-to-one function to an equation, (provided it is defined for that domain). For example, we can apply y = x3 or y = ex to the equation x = 1. The equations x3 = 1 and ex = e have the unique solution x = 1. If we apply a many-to-one function to an equation, we may introduce spurious solutions. Applying y = x2 and y = sin x to the equation x = pi 2 results in x2 = pi 2 4 and sin x = 1. The former equation has the two solutions x = ±pi 2 ; the latter has the infinite number of solutions x = pi 2 + 2npi, n ∈ Z. We do not generally apply a one-to-many function to both sides of an equation as this rarely is useful. Consider the equation sin2 x = 1. 9 Applying the function f(x) = x1/2 to the equation would not get us anywhere (sin2 x)1/2 = 11/2. Since (sin2 x)1/2 6= sin x, we cannot simplify the left side of the equation. Instead we could use the definition of f(x) = x1/2 as the inverse of the x2 function to obtain sin x = 11/2 = ±1. Then we could use the definition of arcsin as the inverse of sin to get x = arcsin(±1). x = arcsin(1) has the solutions x = pi/2 + 2npi and x = arcsin(−1) has the solutions x = −pi/2 + 2npi. Thus x = pi 2 + npi, n ∈ Z. Note that we cannot just apply arcsin to both sides of the equation as arcsin(sinx) 6= x. 10 1.5 Exercises Exercise 1.1 The area of a circle is directly proportional to the square of its diameter. What is the constant of proportionality? Hint, Solution Exercise 1.2 Consider the equation x+ 1 y − 2 = x2 − 1 y2 − 4 . 1. Why might one think that this is the equation of a line? 2. Graph the solutions of the equation to demonstrate that it is not the equation of a line. Hint, Solution Exercise 1.3 Consider the function of a real variable, f(x) = 1 x2 + 2 . What is the domain and range of the function? Hint, Solution Exercise 1.4 The temperature measured in degrees Celsius 2 is linearly related to the temperature measured in degrees Fahrenheit 3. Water freezes at 0◦ C = 32◦ F and boils at 100◦ C = 212◦ F . Write the temperature in degreesCelsius as a function of degrees Fahrenheit. 2 Originally, it was called degrees Centigrade. centi because there are 100 degrees between the two calibration points. It is now called degrees Celsius in honor of the inventor. 3 The Fahrenheit scale, named for Daniel Fahrenheit, was originally calibrated with the freezing point of salt-saturated water to be 0◦. Later, the calibration points became the freezing point of water, 32◦, and body temperature, 96◦. With this method, there are 64 divisions between the calibration points. Finally, the upper calibration point was changed to the boiling point of water at 212◦. This gave 180 divisions, (the number of degrees in a half circle), between the two calibration points. 11 Hint, Solution Exercise 1.5 Consider the function graphed in Figure 1.9. Sketch graphs of f(−x), f(x + 3), f(3− x) + 2, and f−1(x). You may use the blank grids in Figure 1.10. Figure 1.9: Graph of the function. Hint, Solution Exercise 1.6 A culture of bacteria grows at the rate of 10% per minute. At 6:00 pm there are 1 billion bacteria. How many bacteria are there at 7:00 pm? How many were there at 3:00 pm? Hint, Solution Exercise 1.7 The graph in Figure 1.11 shows an even function f(x) = p(x)/q(x) where p(x) and q(x) are rational quadratic polynomials. Give possible formulas for p(x) and q(x). Hint, Solution 12 Figure 1.10: Blank grids. Exercise 1.8 Find a polynomial of degree 100 which is zero only at x = −2, 1, pi and is non-negative. Hint, Solution Exercise 1.9 Hint, Solution 13 1 2 1 2 2 4 6 8 10 1 2 Figure 1.11: Plots of f(x) = p(x)/q(x). Exercise 1.10 Hint, Solution Exercise 1.11 Hint, Solution Exercise 1.12 Hint, Solution Exercise 1.13 Hint, Solution Exercise 1.14 Hint, Solution Exercise 1.15 Hint, Solution Exercise 1.16 Hint, Solution 14 1.6 Hints Hint 1.1 area = constant× diameter2. Hint 1.2 A pair (x, y) is a solution of the equation if it make the equation an identity. Hint 1.3 The domain is the subset of R on which the function is defined. Hint 1.4 Find the slope and x-intercept of the line. Hint 1.5 The inverse of the function is the reflection of the function across the line y = x. Hint 1.6 The formula for geometric growth/decay is x(t) = x0r t, where r is the rate. Hint 1.7 Note that p(x) and q(x) appear as a ratio, they are determined only up to a multiplicative constant. We may take the leading coefficient of q(x) to be unity. f(x) = p(x) q(x) = ax2 + bx+ c x2 + βx+ χ Use the properties of the function to solve for the unknown parameters. Hint 1.8 Write the polynomial in factored form. 15 1.7 Solutions Solution 1.1 area = pi × radius2 area = pi 4 × diameter2 The constant of proportionality is pi 4 . Solution 1.2 1. If we multiply the equation by y2 − 4 and divide by x+ 1, we obtain the equation of a line. y + 2 = x− 1 2. We factor the quadratics on the right side of the equation. x+ 1 y − 2 = (x+ 1)(x− 1) (y − 2)(y + 2) . We note that one or both sides of the equation are undefined at y = ±2 because of division by zero. There are no solutions for these two values of y and we assume from this point that y 6= ±2. We multiply by (y−2)(y+2). (x+ 1)(y + 2) = (x+ 1)(x− 1) For x = −1, the equation becomes the identity 0 = 0. Now we consider x 6= −1. We divide by x + 1 to obtain the equation of a line. y + 2 = x− 1 y = x− 3 Now we collect the solutions we have found. {(−1, y) : y 6= ±2} ∪ {(x, x− 3) : x 6= 1, 5} The solutions are depicted in Figure /reffig not a line. 16 -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 Figure 1.12: The solutions of x+1 y−2 = x2−1 y2−4 . Solution 1.3 The denominator is nonzero for all x ∈ R. Since we don’t have any division by zero problems, the domain of the function is R. For x ∈ R, 0 < 1 x2 + 2 ≤ 2. Consider y = 1 x2 + 2 . (1.1) For any y ∈ (0 . . . 1/2], there is at least one value of x that satisfies Equation 1.1. x2 + 2 = 1 y x = ± √ 1 y − 2 Thus the range of the function is (0 . . . 1/2] 17 Solution 1.4 Let c denote degrees Celsius and f denote degrees Fahrenheit. The line passes through the points (f, c) = (32, 0) and (f, c) = (212, 100). The x-intercept is f = 32. We calculate the slope of the line. slope = 100− 0 212− 32 = 100 180 = 5 9 The relationship between fahrenheit and celcius is c = 5 9 (f − 32). Solution 1.5 We plot the various transformations of f(x). Solution 1.6 The formula for geometric growth/decay is x(t) = x0r t, where r is the rate. Let t = 0 coincide with 6:00 pm. We determine x0. x(0) = 109 = x0 ( 11 10 )0 = x0 x0 = 10 9 At 7:00 pm the number of bacteria is 109 ( 11 10 )60 = 1160 1051 ≈ 3.04× 1011 At 3:00 pm the number of bacteria was 109 ( 11 10 )−180 = 10189 11180 ≈ 35.4 18 Figure 1.13: Graphs of f(−x), f(x+ 3), f(3− x) + 2, and f−1(x). Solution 1.7 We write p(x) and q(x) as general quadratic polynomials. f(x) = p(x) q(x) = ax2 + bx+ c αx2 + βx+ χ We will use the properties of the function to solve for the unknown parameters. 19 Note that p(x) and q(x) appear as a ratio, they are determined only up to a multiplicative constant. We may take the leading coefficient of q(x) to be unity. f(x) = p(x) q(x) = ax2 + bx+ c x2 + βx+ χ f(x) has a second order zero at x = 0. This means that p(x) has a second order zero there and that χ 6= 0. f(x) = ax2 x2 + βx+ χ We note that f(x)→ 2 as x→∞. This determines the parameter a. lim x→∞ f(x) = lim x→∞ ax2 x2 + βx+ χ = lim x→∞ 2ax 2x+ β = lim x→∞ 2a 2 = a f(x) = 2x2 x2 + βx+ χ Now we use the fact that f(x) is even to conclude that q(x) is even and thus β = 0. f(x) = 2x2 x2 + χ Finally, we use that f(1) = 1 to determine χ. f(x) = 2x2 x2 + 1 20 Solution 1.8 Consider the polynomial p(x) = (x+ 2)40(x− 1)30(x− pi)30. It is of degree 100. Since the factors only vanish at x = −2, 1, pi, p(x) only vanishes there. Since factors are non- negative, the polynomial is non-negative. 21 Chapter 2 Vectors 2.1 Vectors 2.1.1 Scalars and Vectors A vector is a quantity having both a magnitude and a direction. Examples of vector quantities are velocity, force and position. One can represent a vector in n-dimensional space with an arrow whose initial point is at the origin, (Figure 2.1). The magnitude is the length of the vector. Typographically, variables representing vectors are often written in capital letters, bold face or with a vector over-line, A, a,~a. The magnitude of a vector is denoted |a|. A scalar has only a magnitude. Examples of scalar quantities are mass, time and speed. Vector Algebra. Two vectors are equal if they have the same magnitude and direction. The negative of a vector, denoted −a, is a vector of the same magnitude as a but in the opposite direction. We add two vectors a and b by placing the tail of b at the head of a and defining a + b to be the vector with tail at the origin and head at the head of b. (See Figure 2.2.) The difference, a− b, is defined as the sum of a and the negative of b, a + (−b). The result of multiplying a by a scalar α is a vector of magnitude |α| |a| with the same/opposite direction if α is positive/negative. (See Figure 2.2.) 22 x z y Figure 2.1: Graphical Representation of a Vector in Three Dimensions a+b a b -a a 2a Figure 2.2: Vector Arithmetic Here are the properties of adding vectors and multiplying them by a scalar. They are evident from geometric considerations. a + b = b + a αa = aα commutative laws (a + b) + c = a + (b + c) α(βa) = (αβ)a associative laws α(a + b) = αa + αb (α + β)a = αa +
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