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Introduction to Methods of Applied Mathematics or Advanced Mathematical Methods for Scientists and EngineersSean Mauch March 19, 2003

ContentsAnti-Copyright Preface 0.1 Advice to Teachers . . . . 0.2 Acknowledgments . . . . 0.3 Warnings and Disclaimers 0.4 Suggested Use . . . . . . 0.5 About the Title . . . . . xxiv xxv xxv xxv xxvi xxvii xxvii

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I

Algebraand Functions Sets . . . . . . . . . . . . . . . . . Single Valued Functions . . . . . . . Inverses and Multi-Valued Functions . Transforming Equations . . . . . . . Exercises . . . . . . . . . . . . . . . Hints . . . . . . . . . . . . . . . . . Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 2 4 6 9 12 15 17

1 Sets 1.1 1.2 1.3 1.4 1.5 1.6 1.7

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2 Vectors 2.1 Vectors . . . . . . . . . . . . . . . . . . 2.1.1 Scalars and Vectors . . . . . . . 2.1.2 The Kronecker Delta and Einstein 2.1.3 The Dot and Cross Product . . . 2.2 Sets of Vectors in n Dimensions . . . . . 2.3 Exercises . . . . . . . . . . . . . . . . . 2.4 Hints . . . . . . . . . . . . . . . . . . . 2.5 Solutions . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . Summation Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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23 23 23 26 27 34 37 39 41

II

Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4849 49 54 57 62 64 67 69 74 76 82 82 82 83 85 85 85

3 Dierential Calculus 3.1 Limits of Functions . . . . . . . . . . . . . . . 3.2 Continuous Functions . . . . . . . . . . . . . 3.3 The Derivative . . . . . . . . . . . . . . . . . 3.4 Implicit Dierentiation . . . . . . . . . . . . . 3.5 Maxima and Minima . . . . . . . . . . . . . . 3.6 Mean Value Theorems . . . . . . . . . . . . . 3.6.1 Application: Using Taylors Theorem to 3.6.2 Application: Finite Dierence Schemes 3.7 LHospitals Rule . . . . . . . . . . . . . . . . 3.8 Exercises . . . . . . . . . . . . . . . . . . . . 3.8.1 Limits of Functions . . . . . . . . . . 3.8.2 Continuous Functions . . . . . . . . . 3.8.3 The Derivative . . . . . . . . . . . . . 3.8.4 Implicit Dierentiation . . . . . . . . . 3.8.5 Maxima and Minima . . . . . . . . . . 3.8.6 Mean Value Theorems . . . . . . . . .

ii

3.8.7 LHospitals Rule 3.9 Hints . . . . . . . . . . 3.10 Solutions . . . . . . . . 3.11 Quiz . . . . . . . . . . 3.12 Quiz Solutions . . . . .

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. 86 . 88 . 93 . 113 . 114 116 116 122 122 123 125 127 127 130 134 134 134 136 136 137 138 141 150 151 154 154 155 163

4 Integral Calculus 4.1 The Indenite Integral . . . . . . . . . . . . . . 4.2 The Denite Integral . . . . . . . . . . . . . . . 4.2.1 Denition . . . . . . . . . . . . . . . . 4.2.2 Properties . . . . . . . . . . . . . . . . 4.3 The Fundamental Theorem of Integral Calculus . 4.4 Techniques of Integration . . . . . . . . . . . . 4.4.1 Partial Fractions . . . . . . . . . . . . . 4.5 Improper Integrals . . . . . . . . . . . . . . . . 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . 4.6.1 The Indenite Integral . . . . . . . . . . 4.6.2 The Denite Integral . . . . . . . . . . . 4.6.3 The Fundamental Theorem of Integration 4.6.4 Techniques of Integration . . . . . . . . 4.6.5 Improper Integrals . . . . . . . . . . . . 4.7 Hints . . . . . . . . . . . . . . . . . . . . . . . 4.8 Solutions . . . . . . . . . . . . . . . . . . . . . 4.9 Quiz . . . . . . . . . . . . . . . . . . . . . . . 4.10 Quiz Solutions . . . . . . . . . . . . . . . . . .

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5 Vector Calculus 5.1 Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Gradient, Divergence and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.4 Hints . . . . . 5.5 Solutions . . . 5.6 Quiz . . . . . 5.7 Quiz Solutions

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166 168 177 178

III

Functions of a Complex Variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179180 180 184 188 193 195 197 201 208 211 239 239 242 246 248 251 257 267 269 285 296

6 Complex Numbers 6.1 Complex Numbers . . . 6.2 The Complex Plane . . 6.3 Polar Form . . . . . . . 6.4 Arithmetic and Vectors 6.5 Integer Exponents . . . 6.6 Rational Exponents . . 6.7 Exercises . . . . . . . . 6.8 Hints . . . . . . . . . . 6.9 Solutions . . . . . . . .

7 Functions of a Complex Variable 7.1 Curves and Regions . . . . . . . . . . . . 7.2 The Point at Innity and the Stereographic 7.3 Cartesian and Modulus-Argument Form . . 7.4 Graphing Functions of a Complex Variable 7.5 Trigonometric Functions . . . . . . . . . . 7.6 Inverse Trigonometric Functions . . . . . . 7.7 Riemann Surfaces . . . . . . . . . . . . . 7.8 Branch Points . . . . . . . . . . . . . . . 7.9 Exercises . . . . . . . . . . . . . . . . . . 7.10 Hints . . . . . . . . . . . . . . . . . . . .

. . . . . . Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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iv

7.11 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 8 Analytic Functions 8.1 Complex Derivatives . . . . . . . . . . . . . . 8.2 Cauchy-Riemann Equations . . . . . . . . . . 8.3 Harmonic Functions . . . . . . . . . . . . . . 8.4 Singularities . . . . . . . . . . . . . . . . . . 8.4.1 Categorization of Singularities . . . . . 8.4.2 Isolated and Non-Isolated Singularities 8.5 Application: Potential Flow . . . . . . . . . . 8.6 Exercises . . . . . . . . . . . . . . . . . . . . 8.7 Hints . . . . . . . . . . . . . . . . . . . . . . 8.8 Solutions . . . . . . . . . . . . . . . . . . . . 359 359 366 371 376 376 380 382 387 395 398 436 436 439 441 446 449 453 455 456 461 461 463 466 467 469

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9 Analytic Continuation 9.1 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Analytic Continuation of Sums . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Analytic Functions Dened in Terms of Real Variables . . . . . . . . . . . . 9.3.1 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Analytic Functions Dened in Terms of Their Real or Imaginary Parts 9.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Contour Integration and the Cauchy-Goursat Theorem 10.1 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . 10.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . 10.2.1 Maximum Modulus Integral Bound . . . . . . . 10.3 The Cauchy-Goursat Theorem . . . . . . . . . . . . . . 10.4 Contour Deformation . . . . . . . . . . . . . . . . . .

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v

10.5 Moreras Theorem. . . . . . . . . . . 10.6 Indenite Integrals . . . . . . . . . . 10.7 Fundamental Theorem of Calculus via 10.7.1 Line Integrals and Primitives . 10.7.2 Contour Integrals . . . . . . 10.8 Fundamental Theorem of Calculus via 10.9 Exercises . . . . . . . . . . . . . . . 10.10Hints . . . . . . . . . . . . . . . . . 10.11Solutions . . . . . . . . . . . . . . . 11 Cauchys Integral Formula 11.1 Cauchys Integral Formula 11.2 The Argument Theorem . 11.3 Rouches Theorem . . . . 11.4 Exercises . . . . . . . . . 11.5 Hints . . . . . . . . . . . 11.6 Solutions . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . Primitives . . . . . . . . . . . . . . . . . . . . . . . . Complex Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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470 472 473 473 473 474 477 481 482 492 493 500 501 504 508 510 524 524 524 526 528 535 536 538 539 546 549 552

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12 Series and Convergence 12.1 Series of Constants . . . . . . . . . . . . . . . . . . . . 12.1.1 Denitions . . . . . . . . . . . . . . . . . . . . 12.1.2 Special Series . . . . . . . . . . . . . . . . . . 12.1.3 Convergence Tests . . . . . . . . . . . . . . . . 12.2 Uniform Convergence . . . . . . . . . . . . . . . . . . 12.2.1 Tests for Uniform Convergence . . . . . . . . . 12.2.2 Uniform Convergence and Continuous Functions. 12.3 Uniformly Convergent Power Series . . . . . . . . . . . 12.4 Integration and Dierentiation of Power Series . . . . . 12.5 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Newtons Binomial Formula. . . . . . . . . . . .

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vi

12.6 Laurent Series . . . . . . . . . . . . . . . . . . . . . 12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 12.7.1 Series of Constants . . . . . . . . . . . . . . 12.7.2 Uniform Convergence . . . . . . . . . . . . . 12.7.3 Uniformly Convergent Power Series . . . . . . 12.7.4 Integration and Dierentiation of Power Series 12.7.5 Taylor Series . . . . . . . . . . . . . . . . . . 12.7.6 Laurent Series . . . . . . . . . . . . . . . . . 12.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . 13 The Residue Theorem 13.1 The Residue Theorem . . . . . . . . . . . . 13.2 Cauchy Principal Value for Real Integrals . . 13.2.1 The Cauchy Principal Value . . . . . 13.3 Cauchy Principal Value for Contour Integrals 13.4 Integrals on the Real Axis . . . . . . . . . . 13.5 Fourier Integrals . . . . . . . . . . . . . . . 13.6 Fourier Cosine and Sine Integrals . . . . . . 13.7 Contour Integration and Branch Cuts . . . . 13.8 Exploiting Symmetry . . . . . . . . . . . . . 13.8.1 Wedge Contours . . . . . . . . . . . 13.8.2 Box Contours . . . . . . . . . . . . 13.9 Denite Integrals Involving Sine and Cosine . 13.10Innite Sums . . . . . . . . . . . . . . . . . 13.11Exercises . . . . . . . . . . . . . . . . . . . 13.12Hints . . . . . . . . . . . . . . . . . . . . . 13.13Solutions . . . . . . . . . . . . . . . . . . .

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554 559 559 565 565 567 568 570 573 581 625 625 633 633 638 642 646 648 651 654 654 657 658 661 666 680 686

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vii

IV

Ordinary Dierential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

772773 773 775 775 777 780 780 782 786 791 791 792 795 796 797 801 803 803 807 812 814 817 820 823 844 845

14 First Order Dierential Equations 14.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Growth and Decay . . . . . . . . . . . . . . . . . . . . 14.3 One Parameter Families of Functions . . . . . . . . . . . . . . 14.4 Integrable Forms . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 Separable Equations . . . . . . . . . . . . . . . . . . . 14.4.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . 14.4.3 Homogeneous Coecient Equations . . . . . . . . . . . 14.5 The First Order, Linear Dierential Equation . . . . . . . . . . 14.5.1 Homogeneous Equations . . . . . . . . . . . . . . . . . 14.5.2 Inhomogeneous Equations . . . . . . . . . . . . . . . . 14.5.3 Variation of Parameters. . . . . . . . . . . . . . . . . . 14.6 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.1 Piecewise Continuous Coecients and Inhomogeneities . 14.7 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . 14.8 Equations in the Complex Plane . . . . . . . . . . . . . . . . . 14.8.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . 14.8.2 Regular Singular Points . . . . . . . . . . . . . . . . . 14.8.3 Irregular Singular Points . . . . . . . . . . . . . . . . . 14.8.4 The Point at Innity . . . . . . . . . . . . . . . . . . . 14.9 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . 14.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.12Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.13Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

15 First Order Linear Systems of Dierential Equations 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Using Eigenvalues and Eigenvectors to nd Homogeneous Solutions 15.3 Matrices and Jordan Canonical Form . . . . . . . . . . . . . . . . 15.4 Using the Matrix Exponential . . . . . . . . . . . . . . . . . . . . 15.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Theory of Linear Ordinary Dierential Equations 16.1 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Nature of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Transformation to a First Order System . . . . . . . . . . . . . . 16.4 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 Derivative of a Determinant. . . . . . . . . . . . . . . . 16.4.2 The Wronskian of a Set of Functions. . . . . . . . . . . . 16.4.3 The Wronskian of the Solutions to a Dierential Equation 16.5 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . 16.6 The Fundamental Set of Solutions . . . . . . . . . . . . . . . . . 16.7 Adjoint Equations . . . . . . . . . . . . . . . . . . . . . . . . . 16.8 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . 16.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.11Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.12Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Techniques for Linear Dierential 17.1 Constant Coecient Equations 17.1.1 Second Order Equations 17.1.2 Real-Valued Solutions .

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847 847 848 853 861 866 871 873 901 901 902 906 907 907 908 910 912 914 917 920 921 923 929 930

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Equations 931 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936

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17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9

17.1.3 Higher Order Equations . . . . . . . . Euler Equations . . . . . . . . . . . . . . . . 17.2.1 Real-Valued Solutions . . . . . . . . . Exact Equations . . . . . . . . . . . . . . . . Equations Without Explicit Dependence on y . Reduction of Order . . . . . . . . . . . . . . . *Reduction of Order and the Adjoint Equation Additional Exercises . . . . . . . . . . . . . . Hints . . . . . . . . . . . . . . . . . . . . . . Solutions . . . . . . . . . . . . . . . . . . . .

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938 941 943 946 947 948 949 952 958 961 985 985 987 991 993 996 998 1001 1002 1005 1007

18 Techniques for Nonlinear Dierential Equations 18.1 Bernoulli Equations . . . . . . . . . . . . . . . . . . 18.2 Riccati Equations . . . . . . . . . . . . . . . . . . . 18.3 Exchanging the Dependent and Independent Variables 18.4 Autonomous Equations . . . . . . . . . . . . . . . . 18.5 *Equidimensional-in-x Equations . . . . . . . . . . . . 18.6 *Equidimensional-in-y Equations . . . . . . . . . . . . 18.7 *Scale-Invariant Equations . . . . . . . . . . . . . . . 18.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 18.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . 18.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . 19 Transformations and Canonical Forms 19.1 The Constant Coecient Equation . . . . . . . . . 19.2 Normal Form . . . . . . . . . . . . . . . . . . . . . 19.2.1 Second Order Equations . . . . . . . . . . . 19.2.2 Higher Order Dierential Equations . . . . . 19.3 Transformations of the Independent Variable . . . . 19.3.1 Transformation to the form u + a(x) u = 0

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19.4

19.5 19.6 19.7 20 The 20.1 20.2 20.3 20.4 20.5 20.6 20.7

19.3.2 Transformation to a Constant Coecient Equation Integral Equations . . . . . . . . . . . . . . . . . . . . . 19.4.1 Initial Value Problems . . . . . . . . . . . . . . . 19.4.2 Boundary Value Problems . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . Dirac Delta Function Derivative of the Heaviside Function The Delta Function as a Limit . . . . Higher Dimensions . . . . . . . . . . Non-Rectangular Coordinate Systems Exercises . . . . . . . . . . . . . . . Hints . . . . . . . . . . . . . . . . . Solutions . . . . . . . . . . . . . . .

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1026 1028 1028 1030 1033 1035 1036

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1042 . 1042 . 1044 . 1046 . 1047 . 1049 . 1051 . 1053 1060 . 1060 . 1062 . 1066 . 1066 . 1069 . 1072 . 1075 . 1075 . 1077 . 1078 . 1080 . 1083

21 Inhomogeneous Dierential Equations 21.1 Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Method of Undetermined Coecients . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.1 Second Order Dierential Equations . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Higher Order Dierential Equations . . . . . . . . . . . . . . . . . . . . . . . 21.4 Piecewise Continuous Coecients and Inhomogeneities . . . . . . . . . . . . . . . . . 21.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5.1 Eliminating Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . 21.5.2 Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions 21.5.3 Existence of Solutions of Problems with Inhomogeneous Boundary Conditions . 21.6 Green Functions for First Order Equations . . . . . . . . . . . . . . . . . . . . . . . 21.7 Green Functions for Second Order Equations . . . . . . . . . . . . . . . . . . . . . .

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xi

21.7.1 Green Functions for Sturm-Liouville Problems . 21.7.2 Initial Value Problems . . . . . . . . . . . . . 21.7.3 Problems with Unmixed Boundary Conditions . 21.7.4 Problems with Mixed Boundary Conditions . . 21.8 Green Functions for Higher Order Problems . . . . . . 21.9 Fredholm Alternative Theorem . . . . . . . . . . . . 21.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . 21.11Hints . . . . . . . . . . . . . . . . . . . . . . . . . . 21.12Solutions . . . . . . . . . . . . . . . . . . . . . . . . 21.13Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . 21.14Quiz Solutions . . . . . . . . . . . . . . . . . . . . . 22 Dierence Equations 22.1 Introduction . . . . . . . . . . . . . . . . . . 22.2 Exact Equations . . . . . . . . . . . . . . . . 22.3 Homogeneous First Order . . . . . . . . . . . 22.4 Inhomogeneous First Order . . . . . . . . . . 22.5 Homogeneous Constant Coecient Equations . 22.6 Reduction of Order . . . . . . . . . . . . . . . 22.7 Exercises . . . . . . . . . . . . . . . . . . . . 22.8 Hints . . . . . . . . . . . . . . . . . . . . . . 22.9 Solutions . . . . . . . . . . . . . . . . . . . .

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1093 1096 1099 1101 1105 1110 1118 1124 1127 1165 1166

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1167 . 1167 . 1169 . 1170 . 1172 . 1175 . 1178 . 1180 . 1181 . 1182 1185 . 1185 . 1189 . 1199 . 1202 . 1204 . 1208

23 Series Solutions of Dierential Equations 23.1 Ordinary Points . . . . . . . . . . . . . . . . . . 23.1.1 Taylor Series Expansion for a Second Order 23.2 Regular Singular Points of Second Order Equations 23.2.1 Indicial Equation . . . . . . . . . . . . . . 23.2.2 The Case: Double Root . . . . . . . . . . 23.2.3 The Case: Roots Dier by an Integer . . .

. . . . . . . . . . . . Dierential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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xii

23.3 23.4 23.5 23.6 23.7 23.8 23.9

Irregular Singular Points The Point at Innity . . Exercises . . . . . . . . Hints . . . . . . . . . . Solutions . . . . . . . . Quiz . . . . . . . . . . Quiz Solutions . . . . .

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1218 1218 1221 1226 1227 1250 1251

24 Asymptotic Expansions 24.1 Asymptotic Relations . . . . . . . . . . . . . . 24.2 Leading Order Behavior of Dierential Equations 24.3 Integration by Parts . . . . . . . . . . . . . . . 24.4 Asymptotic Series . . . . . . . . . . . . . . . . 24.5 Asymptotic Expansions of Dierential Equations 24.5.1 The Parabolic Cylinder Equation. . . . . 25 Hilbert Spaces 25.1 Linear Spaces . . . . . . . . . . . 25.2 Inner Products . . . . . . . . . . . 25.3 Norms . . . . . . . . . . . . . . . 25.4 Linear Independence. . . . . . . . . 25.5 Orthogonality . . . . . . . . . . . 25.6 Gramm-Schmidt Orthogonalization 25.7 Orthonormal Function Expansion . 25.8 Sets Of Functions . . . . . . . . . 25.9 Least Squares Fit to a Function and 25.10Closure Relation . . . . . . . . . . 25.11Linear Operators . . . . . . . . . . 25.12Exercises . . . . . . . . . . . . . . 25.13Hints . . . . . . . . . . . . . . . .

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1253 . 1253 . 1257 . 1266 . 1273 . 1274 . 1274 1280 . 1280 . 1282 . 1283 . 1285 . 1285 . 1286 . 1288 . 1290 . 1297 . 1300 . 1305 . 1306 . 1307

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xiii

25.14Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1308 26 Self 26.1 26.2 26.3 26.4 26.5 Adjoint Linear Operators Adjoint Operators . . . . . Self-Adjoint Operators . . . Exercises . . . . . . . . . . Hints . . . . . . . . . . . . Solutions . . . . . . . . . . 1310 . 1310 . 1311 . 1314 . 1315 . 1316 1317 . 1317 . 1318 . 1321 . 1321 . 1326 . 1329 . 1330 . 1331 1333 . 1333 . 1336 . 1340 . 1344 . 1347 . 1348 . 1349 . 1352 . 1361 . 1361

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27 Self-Adjoint Boundary Value Problems 27.1 Summary of Adjoint Operators . . . 27.2 Formally Self-Adjoint Operators . . . 27.3 Self-Adjoint Problems . . . . . . . . 27.4 Self-Adjoint Eigenvalue Problems . . 27.5 Inhomogeneous Equations . . . . . . 27.6 Exercises . . . . . . . . . . . . . . . 27.7 Hints . . . . . . . . . . . . . . . . . 27.8 Solutions . . . . . . . . . . . . . . .

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28 Fourier Series 28.1 An Eigenvalue Problem. . . . . . . . . . . . . . . . . . 28.2 Fourier Series. . . . . . . . . . . . . . . . . . . . . . . 28.3 Least Squares Fit . . . . . . . . . . . . . . . . . . . . 28.4 Fourier Series for Functions Dened on Arbitrary Ranges 28.5 Fourier Cosine Series . . . . . . . . . . . . . . . . . . . 28.6 Fourier Sine Series . . . . . . . . . . . . . . . . . . . . 28.7 Complex Fourier Series and Parsevals Theorem . . . . . 28.8 Behavior of Fourier Coecients . . . . . . . . . . . . . 28.9 Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . 28.10Integrating and Dierentiating Fourier Series . . . . . .

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xiv

28.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366 28.12Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374 28.13Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376 29 Regular Sturm-Liouville Problems 29.1 Derivation of the Sturm-Liouville Form . . . . . . . . . . . . 29.2 Properties of Regular Sturm-Liouville Problems . . . . . . . . 29.3 Solving Dierential Equations With Eigenfunction Expansions 29.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Integrals and Convergence 30.1 Uniform Convergence of Integrals . . . 30.2 The Riemann-Lebesgue Lemma . . . . 30.3 Cauchy Principal Value . . . . . . . . 30.3.1 Integrals on an Innite Domain 30.3.2 Singular Functions . . . . . . . 1423 . 1423 . 1425 . 1436 . 1442 . 1446 . 1448 1473 . 1473 . 1474 . 1475 . 1475 . 1476 1478 . 1478 . 1480 . 1483 . 1488 . 1491 . 1493 . 1496 . 1498 . 1501 . 1508

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31 The Laplace Transform 31.1 The Laplace Transform . . . . . . . . . . . . . . . . 31.2 The Inverse Laplace Transform . . . . . . . . . . . . 31.2.1 f (s) with Poles . . . . . . . . . . . . . . . . 31.2.2 f (s) with Branch Points . . . . . . . . . . . . 31.2.3 Asymptotic Behavior of f (s) . . . . . . . . . 31.3 Properties of the Laplace Transform . . . . . . . . . . 31.4 Constant Coecient Dierential Equations . . . . . . 31.5 Systems of Constant Coecient Dierential Equations 31.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 31.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . .

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xv

31.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1511 32 The Fourier Transform 32.1 Derivation from a Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2.1 A Word of Caution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Evaluating Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.1 Integrals that Converge . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.2 Cauchy Principal Value and Integrals that are Not Absolutely Convergent. . 32.3.3 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Properties of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 32.4.1 Closure Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4.2 Fourier Transform of a Derivative. . . . . . . . . . . . . . . . . . . . . . . 32.4.3 Fourier Convolution Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 32.4.4 Parsevals Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4.5 Shift Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4.6 Fourier Transform of x f(x). . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 Solving Dierential Equations with the Fourier Transform . . . . . . . . . . . . . 32.6 The Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . 32.6.1 The Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . 32.6.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . 32.7 Properties of the Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . 32.7.1 Transforms of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 32.7.2 Convolution Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.7.3 Cosine and Sine Transform in Terms of the Fourier Transform . . . . . . . 32.8 Solving Dierential Equations with the Fourier Cosine and Sine Transforms . . . . 32.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1543 . 1543 . 1545 . 1548 . 1549 . 1549 . 1552 . 1554 . 1556 . 1556 . 1557 . 1559 . 1562 . 1564 . 1564 . 1565 . 1567 . 1567 . 1568 . 1569 . 1569 . 1571 . 1573 . 1574 . 1576 . 1583 . 1586

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xvi

33 The 33.1 33.2 33.3 33.4 33.5 33.6 33.7 33.8

Gamma Function Eulers Formula . . . . Hankels Formula . . . . Gauss Formula . . . . . Weierstrass Formula . . Stirlings Approximation Exercises . . . . . . . . Hints . . . . . . . . . . Solutions . . . . . . . .

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1610 . 1610 . 1612 . 1614 . 1616 . 1618 . 1623 . 1624 . 1625 1627 . 1627 . 1628 . 1631 . 1633 . 1634 . 1635 . 1638 . 1641 . 1644 . 1645 . 1649 . 1651 . 1651 . 1655 . 1660 . 1662

34 Bessel Functions 34.1 Bessels Equation . . . . . . . . . . . . . . . . . . . . 34.2 Frobeneius Series Solution about z = 0 . . . . . . . . . 34.2.1 Behavior at Innity . . . . . . . . . . . . . . . . 34.3 Bessel Functions of the First Kind . . . . . . . . . . . . 34.3.1 The Bessel Function Satises Bessels Equation . 34.3.2 Series Expansion of the Bessel Function . . . . . 34.3.3 Bessel Functions of Non-Integer Order . . . . . 34.3.4 Recursion Formulas . . . . . . . . . . . . . . . 34.3.5 Bessel Functions of Half-Integer Order . . . . . 34.4 Neumann Expansions . . . . . . . . . . . . . . . . . . 34.5 Bessel Functions of the Second Kind . . . . . . . . . . 34.6 Hankel Functions . . . . . . . . . . . . . . . . . . . . . 34.7 The Modied Bessel Equation . . . . . . . . . . . . . . 34.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 34.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . .

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xvii

V

Partial Dierential Equations

1685

35 Transforming 35.1 Exercises 35.2 Hints . . 35.3 Solutions

Equations 1686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1689 1690 . 1690 . 1691 . 1696 . 1697 . 1699 . 1701 . 1702 . 1703 1709 . 1709 . 1709 . 1711 . 1714 . 1715 . 1718 . 1721 . 1723 . 1739 . 1744

36 Classication of Partial Dierential Equations 36.1 Classication of Second Order Quasi-Linear Equations 36.1.1 Hyperbolic Equations . . . . . . . . . . . . . 36.1.2 Parabolic equations . . . . . . . . . . . . . . 36.1.3 Elliptic Equations . . . . . . . . . . . . . . . 36.2 Equilibrium Solutions . . . . . . . . . . . . . . . . . 36.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 36.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . 36.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . .

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37 Separation of Variables 37.1 Eigensolutions of Homogeneous Equations . . . . . . . . . . . . . 37.2 Homogeneous Equations with Homogeneous Boundary Conditions . 37.3 Time-Independent Sources and Boundary Conditions . . . . . . . . 37.4 Inhomogeneous Equations with Homogeneous Boundary Conditions 37.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . 37.6 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . 37.7 General Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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38 Finite Transforms 1826 38.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1830 38.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1831 38.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1832 39 The 39.1 39.2 39.3 Diusion Equation 1836 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1837 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1839 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1840 1846 . 1846 . 1846 . 1847 . 1848 . 1851 . 1852

40 Laplaces Equation 40.1 Introduction . . . . . . . . . . 40.2 Fundamental Solution . . . . . 40.2.1 Two Dimensional Space 40.3 Exercises . . . . . . . . . . . . 40.4 Hints . . . . . . . . . . . . . . 40.5 Solutions . . . . . . . . . . . .

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41 Waves 1864 41.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865 41.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1871 41.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1873 42 Similarity Methods 42.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1893 . 1898 . 1899 . 1900

43 Method of Characteristics 1903 43.1 First Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1903 43.2 First Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1904 xix

43.3 The Method of Characteristics and the Wave Equation 43.4 The Wave Equation for an Innite Domain . . . . . . 43.5 The Wave Equation for a Semi-Innite Domain . . . . 43.6 The Wave Equation for a Finite Domain . . . . . . . 43.7 Envelopes of Curves . . . . . . . . . . . . . . . . . . 43.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 43.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . 43.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . 44 Transform Methods 44.1 Fourier Transform for Partial Dierential Equations 44.2 The Fourier Sine Transform . . . . . . . . . . . . 44.3 Fourier Transform . . . . . . . . . . . . . . . . . 44.4 Exercises . . . . . . . . . . . . . . . . . . . . . . 44.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . 44.6 Solutions . . . . . . . . . . . . . . . . . . . . . .

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45 Green Functions 45.1 Inhomogeneous Equations and Homogeneous Boundary Conditions 45.2 Homogeneous Equations and Inhomogeneous Boundary Conditions 45.3 Eigenfunction Expansions for Elliptic Equations . . . . . . . . . . . 45.4 The Method of Images . . . . . . . . . . . . . . . . . . . . . . . . 45.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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46 Conformal Mapping 46.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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47 Non-Cartesian Coordinates 47.1 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47.2 Laplaces Equation in a Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47.3 Laplaces Equation in an Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2057 . 2057 . 2058 . 2061

VI

Calculus of Variations

2065

48 Calculus of Variations 2066 48.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2067 48.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2081 48.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2085

VII

Nonlinear Dierential Equations

2172

49 Nonlinear Ordinary Dierential Equations 2173 49.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2174 49.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2179 49.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2180 50 Nonlinear Partial 50.1 Exercises . . 50.2 Hints . . . . 50.3 Solutions . . Dierential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2202 . 2203 . 2206 . 2207

VIII

Appendices

22262227

A Greek Letters

xxi

B Notation C Formulas from Complex Variables D Table of Derivatives E Table of Integrals F Denite Integrals G Table of Sums H Table of Taylor Series I

2229 2231 2234 2238 2242 2244 2247

Table of Laplace Transforms 2250 I.1 Properties of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2250 I.2 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2252 2256 2259 2260 2262 2264 2266 2268

J Table of Fourier Transforms K Table of Fourier Transforms in n Dimensions L Table of Fourier Cosine Transforms M Table of Fourier Sine Transforms N Table of Wronskians O Sturm-Liouville Eigenvalue Problems P Green Functions for Ordinary Dierential Equations

xxii

Q Trigonometric Identities 2271 Q.1 Circular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2271 Q.2 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2273 R Bessel Functions 2276 R.1 Denite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2276 S Formulas from Linear Algebra T Vector Analysis U Partial Fractions V Finite Math W Physics 2277 2278 2280 2284 2285

X Probability 2286 X.1 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2286 X.2 Playing the Odds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2287 Y Economics Z Glossary 2288 2289

xxiii

Anti-CopyrightAnti-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.

xxiv

PrefaceDuring the summer before my nal 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 Im 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 Saman for advising me on this project and the Caltech SURF program for providing the funding for me to write the rst edition of this book.

xxv

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 ber. Caution: FLAMMABLE - Do not read while smoking or near a re. 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 otation device. The material in this text is ction; 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 aint so sure the grammars 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 suer great misfortune and other nastiness. The surgeon general has determined that excessive studying is detrimental to your social life. xxvi

This text has been buered 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.

xxvii

Part I Algebra

1

Chapter 1 Sets and Functions1.1 Sets

Denition. A set is a collection of objects. We call the objects, elements. A set is denoted by listing elements the between braces. For example: {e, , , 1} is the set of the integer 1, the pure imaginary number = 1 and the transcendental numbers e = 2.7182818 . . . and = 3.1415926 . . .. For elements of a set, we do not count multiplicities. We regard the set {1, 2, 2, 3, 3, 3} as identical to the set {1, 2, 3}. Order is not signicant in sets. The set {1, 2, 3} is equivalent to {3, 2, 1}. In enumerating the elements of a set, we use ellipses to indicate patterns. We denote the set of positive integers as {1, 2, 3, . . .}. We also denote 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 S is the notation for x is an element of the set S. To express the opposite we have x 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 = {. . . , 3, 2, 1, 0, 1, 2, 3 . . .} is the set of integers. (Z is for Zahlen, the German word for number.) 2

Q = {p/q|p, q Z, q = 0} is the set of rational numbers. (Q is for quotient.)

1 2

R = {x|x = a1 a2 an .b1 b2 } is the set of real numbers, i.e. the set of numbers with decimal expansions.

C = {a + b|a, b R, 2 = 1} is the set of complex numbers. is the square root of 1. (If you havent seen complex numbers before, dont dismay. Well 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 nite 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 = 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 = 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.Note that with this description, we enumerate each rational number an innite number of times. For example: 1/2 = 2/4 = 3/6 = (1)/(2) = . This does not pose a problem as we do not count multiplicities. 2 Guess what R is for.1

3

Operations. The union of two sets, S T , is the set whose elements are in either of the two sets. The union of n sets, n Sj S1 S2 Sn j=1 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, n Sj S1 S2 Sn j=1 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 dierence between S and T , S \ T , is the set of elements in S which are not in T . S \ T {x|x S, x T } The dierence of sets is also denoted S T . Properties. The following properties are easily veried from the above denitions. 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 f into elements y Y . This is expressed as f : X Y or X Y . If such a function is well-dened, 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 4

particular element we can use any of the notations: f (x) = y, f : x y or simply x 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 dened: f 1 (Z) {x X|f (x) = z for some z Z}. Examples. Finite polynomials, f (x) = n ak xk , ak R, and the exponential function, f (x) = ex , are examples of single k=0 valued functions which map real numbers to real numbers. The greatest integer function, f (x) = x , is a mapping from R to Z. x is dened as the greatest integer less than or equal to x. Likewise, the least integer function, f (x) = x , is the least integer greater than or equal to x. The -jectives. A function is injective if for each x1 = x2 , f (x1 ) = f (x2 ). In other words, distinct elements are mapped to distinct elements. 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 f (x) = ex , considered as a mapping from R to R+ , is bijective, (a one-to-one mapping). 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) = sin x is not injective from R to [1..1]. For each y [1..1] there exists an innite number of values of x such that y = sin x.

5

Injective

Surjective

Bijective

Figure 1.1: Depictions of Injective, Surjective and Bijective Functions

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 dened. For example ln x, which maps R+ to R is the inverse of ex . x = eln x = ln(ex ) for all x R+ . (Note the x R+ ensures that ln x is dened.) 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 dened, however x = f 1 (f (x)). There are diagrams showing one-to-one, many-to-one and one-to-many functions in Figure 1.2. Example 1.3.1 y = x2 , a many-to-one function has the inverse x = y 1/2 . For each positive y, there are two values of x such that x = y 1/2 . y = x2 and y = x1/2 are graphed in Figure 1.3.

We say that there are two branches of y = x1/2 : the positive and the negative branch. We denote the positive 1/2 branch as y = x; the negative branch is y = x. We call x the principal branch of x . Note that x is a 6

one-to-one

many-to-one

one-to-many

domain

range

domain

range

domain

range

Figure 1.2: Diagrams of One-To-One, Many-To-One and One-To-Many Functions

Figure 1.3: y = x2 and y = x1/2 one-to-one function. Finally, x = (x1/2 )2 since ( x)2 = x, but x = (x2 )1/2 since (x2 )1/2 = x. y = x is graphed in Figure 1.4. Now consider the many-to-one function y = sin x. The inverse is x = arcsin y. For each y [1..1] there are an innite 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 = arcsin x. Example 1.3.2 arcsin x has an innite number of branches. We will denote the principal branch by Arcsin x which maps [1..1] to .. . Note that x = sin(arcsin x), but x = arcsin(sin x). y = Arcsin x in Figure 1.6. 2 2

7

Figure 1.4: y =

x

Figure 1.5: y = sin x and y = arcsin x

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.

8

Figure 1.7: y = x3 and y = x1/3 Example 1.3.4 Consider arccos(1/2). cos x and a portion of arccos x are graphed in Figure 1.8. The equation cos x = 1/2 has the two solutions x = /3 in the range x (..]. We use the periodicity of the cosine, cos(x + 2) = cos x, to nd the remaining solutions. arccos(1/2) = {/3 + 2n}, n Z.

Figure 1.8: y = cos x and y = arccos x

1.4

Transforming Equations

Consider the equation g(x) = h(x) and the single-valued function f (x). A particular value of x is a solution of the equation if substituting that value into the equation results in an identity. In determining the solutions of an equation, 9

we often apply functions to each side of the equation in order to simplify its form. We apply the function to obtain a second equation, f (g(x)) = f (h(x)). If x = is a solution of the former equation, (let = g() = h()), then it is necessarily a solution of latter. This is because f (g()) = f (h()) reduces to the identity f () = f (). If f (x) is bijective, then the converse is true: any solution of the latter equation is a solution of the former equation. Suppose that x = is a solution of the latter, f (g()) = f (h()). That f (x) is a one-to-one mapping implies that g() = h(). Thus x = is a solution of the former equation. It is always safe to apply a one-to-one, (bijective), function to an equation, (provided it is dened for that domain). For example, we can apply f (x) = x3 or f (x) = ex , considered as mappings on R, to the equation x = 1. The equations x3 = 1 and ex = e each have the unique solution x = 1 for x R. In general, we must take care in applying functions to equations. If we apply a many-to-one function, we may 2 introduce spurious solutions. Applying f (x) = x2 to the equation x = results in x2 = 4 , which has the two solutions, 2 2 x = { }. Applying f (x) = sin x results in x2 = 4 , which has an innite number of solutions, x = { +2n | n Z}. 2 2 We do not generally apply a one-to-many, (multi-valued), function to both sides of an equation as this rarely is useful. Rather, we typically use the denition of the inverse function. Consider the equation sin2 x = 1. Applying the function f (x) = x1/2 to the equation would not get us anywhere. sin2 x1/2

= 11/2

Since (sin2 x)1/2 = sin x, we cannot simplify the left side of the equation. Instead we could use the denition of f (x) = x1/2 as the inverse of the x2 function to obtain sin x = 11/2 = 1. Now note that we should not just apply arcsin to both sides of the equation as arcsin(sin x) = x. Instead we use the denition of arcsin as the inverse of sin. x = arcsin(1) 10

x = arcsin(1) has the solutions x = /2 + 2n and x = arcsin(1) has the solutions x = /2 + 2n. We enumerate the solutions. + n | n Z x= 2

11

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 x2 1 = 2 . y2 y 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) = What is the domain and range of the function? Hint, Solution Exercise 1.4 The temperature measured in degrees Celsius 3 is linearly related to the temperature measured in degrees Fahrenheit 4 . Water freezes at 0 C = 32 F and boils at 100 C = 212 F . Write the temperature in degrees Celsius as a function of degrees Fahrenheit.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. 4 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.3

x2

1 . +2

12

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 13

Figure 1.10: Blank grids. Exercise 1.8 Find a polynomial of degree 100 which is zero only at x = 2, 1, and is non-negative. Hint, Solution

14

2

2

1

1

1

2

2

4

6

8

10

Figure 1.11: Plots of f (x) = p(x)/q(x).

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 dened. Hint 1.4 Find the slope and x-intercept of the line. Hint 1.5 The inverse of the function is the reection of the function across the line y = x. Hint 1.6 The formula for geometric growth/decay is x(t) = x0 rt , where r is the rate.

15

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 coecient of q(x) to be unity. p(x) ax2 + bx + c f (x) = = 2 q(x) x + x + Use the properties of the function to solve for the unknown parameters. Hint 1.8 Write the polynomial in factored form.

16

1.7

Solutions

Solution 1.1 area = radius2 area = diameter2 4 The constant of proportionality is . 4 Solution 1.2 1. If we multiply the equation by y 2 4 and divide by x + 1, we obtain the equation of a line. y+2=x1 2. We factor the quadratics on the right side of the equation. x+1 (x + 1)(x 1) = . y2 (y 2)(y + 2) We note that one or both sides of the equation are undened 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 = 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 = 1. We divide by x + 1 to obtain the equation of a line. y+2=x1 y =x3 Now we collect the solutions we have found. {(1, y) : y = 2} {(x, x 3) : x = 1, 5} The solutions are depicted in Figure /refg not a line. 17

6

4

2

-6

-4

-2 -2

2

4

6

-4

-6

Figure 1.12: The solutions of

x+1 y2

=

x2 1 . y 2 4

Solution 1.3 The denominator is nonzero for all x R. Since we dont have any division by zero problems, the domain of the function is R. For x R, 1 2. 0< 2 x +2 Consider 1 . (1.1) y= 2 x +2 For any y (0 . . . 1/2], there is at least one value of x that satises Equation 1.1. x2 + 2 = x= Thus the range of the function is (0 . . . 1/2] 18 1 y

1 2 y

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 = The relationship between fahrenheit and celcius is 5 c = (f 32). 9 Solution 1.5 We plot the various transformations of f (x). Solution 1.6 The formula for geometric growth/decay is x(t) = x0 rt , where r is the rate. Let t = 0 coincide with 6:00 pm. We determine x0 . x(0) = 109 = x0 11 100

100 5 100 0 = = 212 32 180 9

= x0

x0 = 109 At 7:00 pm the number of bacteria is 10 At 3:00 pm the number of bacteria was 109 11 10180 9

11 10

60

=

1160 3.04 1011 51 10

=

10189 35.4 11180

19

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. p(x) ax2 + bx + c f (x) = = q(x) x2 + x + We will use the properties of the function to solve for the unknown parameters. 20

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 coecient of q(x) to be unity. f (x) = p(x) ax2 + bx + c = 2 q(x) x + x +

f (x) has a second order zero at x = 0. This means that p(x) has a second order zero there and that = 0. f (x) = ax2 x2 + x +

We note that f (x) 2 as x . This determines the parameter a. lim f (x) = lim ax2 x x2 + x + 2ax = lim x 2x + 2a = lim x 2 =a 2x2 x2 + x +

x

f (x) =

Now we use the fact that f