Introduction to Methods of Applied Mathematicsor

Advanced Mathematical Methods for Scientists and Engineers

Sean Mauch

April 8, 2002

Contents

Anti-Copyright xxiii

Preface xxiv0.1 Advice to Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv0.2 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv0.3 Warnings and Disclaimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv0.4 Suggested Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi0.5 About the Title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi

I Algebra 1

1 Sets and Functions 21.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Single Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Inverses and Multi-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Transforming Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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2 Vectors 222.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.1 Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.2 The Kronecker Delta and Einstein Summation Convention . . . . . . . . . . . . . . . . . . . . 252.1.3 The Dot and Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Sets of Vectors in n Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

II Calculus 46

3 Differential Calculus 473.1 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.6 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.6.1 Application: Using Taylors Theorem to Approximate Functions. . . . . . . . . . . . . . . . . . 663.6.2 Application: Finite Difference Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.7 LHospitals Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.10 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4 Integral Calculus 1114.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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4.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.3 The Fundamental Theorem of Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.4 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.4.1 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5 Vector Calculus 1475.1 Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.2 Gradient, Divergence and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1595.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

III Functions of a Complex Variable 170

6 Complex Numbers 1716.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1716.2 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1746.3 Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1796.4 Arithmetic and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.5 Integer Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1856.6 Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1876.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1986.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

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7 Functions of a Complex Variable 2287.1 Curves and Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2287.2 The Point at Infinity and the Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . . . . 2317.3 Cartesian and Modulus-Argument Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2337.4 Graphing Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2377.5 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2397.6 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2457.7 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2547.8 Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2567.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2737.10 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2847.11 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

8 Analytic Functions 3468.1 Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3468.2 Cauchy-Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3538.3 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3588.4 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

8.4.1 Categorization of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3638.4.2 Isolated and Non-Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

8.5 Application: Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3698.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3748.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3808.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

9 Analytic Continuation 4199.1 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4199.2 Analytic Continuation of Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4229.3 Analytic Functions Defined in Terms of Real Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 424

9.3.1 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

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9.3.2 Analytic Functions Defined in Terms of Their Real or Imaginary Parts . . . . . . . . . . . . . . 4329.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4369.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4389.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

10 Contour Integration and the Cauchy-Goursat Theorem 44410.1 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44410.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

10.2.1 Maximum Modulus Integral Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44910.3 The Cauchy-Goursat Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45010.4 Contour Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45210.5 Moreras Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45310.6 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45510.7 Fundamental Theorem of Calculus via Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

10.7.1 Line Integrals and Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45610.7.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

10.8 Fundamental Theorem of Calculus via Complex Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 45710.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46010.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46410.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

11 Cauchys Integral Formula 47511.1 Cauchys Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47611.2 The Argument Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48311.3 Rouches Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48411.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48711.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49111.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

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12 Series and Convergence 50812.1 Series of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

12.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50812.1.2 Special Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51012.1.3 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

12.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51912.2.1 Tests for Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52012.2.2 Uniform Convergence and Continuous Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 522

12.3 Uniformly Convergent Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52312.4 Integration and Differentiation of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53012.5 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

12.5.1 Newtons Binomial Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53612.6 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53812.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54312.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55812.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

13 The Residue Theorem 61413.1 The Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61413.2 Cauchy Principal Value for Real Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622

13.2.1 The Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62213.3 Cauchy Principal Value for Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62713.4 Integrals on the Real Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63113.5 Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63513.6 Fourier Cosine and Sine Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63713.7 Contour Integration and Branch Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64013.8 Exploiting Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

13.8.1 Wedge Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64313.8.2 Box Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646

13.9 Definite Integrals Involving Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647

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13.10Infinite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65013.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65513.12Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66913.13Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675

IV Ordinary Differential Equations 761

14 First Order Differential Equations 76214.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76214.2 One Parameter Families of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76414.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766

14.3.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77114.3.2 Homogeneous Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773

14.4 The First Order, Linear Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77714.4.1 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77714.4.2 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77914.4.3 Variation of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782

14.5 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78214.5.1 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . 783

14.6 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78814.7 Equations in the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791

14.7.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79114.7.2 Regular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79414.7.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79914.7.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801

14.8 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80414.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80714.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810

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15 First Order Linear Systems of Differential Equations 83115.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83115.2 Using Eigenvalues and Eigenvectors to find Homogeneous Solutions . . . . . . . . . . . . . . . . . . . 83215.3 Matrices and Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83715.4 Using the Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84415.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85015.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85515.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857

16 Theory of Linear Ordinary Differential Equations 88516.1 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88516.2 Nature of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88616.3 Transformation to a First Order System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88916.4 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890

16.4.1 Derivative of a Determinant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89016.4.2 The Wronskian of a Set of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89116.4.3 The Wronskian of the Solutions to a Differential Equation . . . . . . . . . . . . . . . . . . . . 893

16.5 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89616.6 The Fundamental Set of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89816.7 Adjoint Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90016.8 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90416.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90516.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906

17 Techniques for Linear Differential Equations 91117.1 Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911

17.1.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91217.1.2 Higher Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91617.1.3 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917

17.2 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 921

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17.2.1 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92317.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92617.4 Equations Without Explicit Dependence on y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92717.5 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92817.6 *Reduction of Order and the Adjoint Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92917.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93217.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93817.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941

18 Techniques for Nonlinear Differential Equations 96518.1 Bernoulli Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96518.2 Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96718.3 Exchanging the Dependent and Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 97118.4 Autonomous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97318.5 *Equidimensional-in-x Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97618.6 *Equidimensional-in-y Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97818.7 *Scale-Invariant Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98118.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98218.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98518.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987

19 Transformations and Canonical Forms 99919.1 The Constant Coefficient Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99919.2 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002

19.2.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100219.2.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003

19.3 Transformations of the Independent Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100519.3.1 Transformation to the form u + a(x) u = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 100519.3.2 Transformation to a Constant Coefficient Equation . . . . . . . . . . . . . . . . . . . . . . . . 1006

19.4 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008

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19.4.1 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100819.4.2 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010

19.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101319.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101519.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016

20 The Dirac Delta Function 102220.1 Derivative of the Heaviside Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102220.2 The Delta Function as a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102420.3 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102620.4 Non-Rectangular Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102720.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102920.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103120.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033

21 Inhomogeneous Differential Equations 104021.1 Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104021.2 Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104221.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046

21.3.1 Second Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104621.3.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049

21.4 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . 105221.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055

21.5.1 Eliminating Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 105521.5.2 Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions . . . . . . . . . 105721.5.3 Existence of Solutions of Problems with Inhomogeneous Boundary Conditions . . . . . . . . . . 1058

21.6 Green Functions for First Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106021.7 Green Functions for Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063

21.7.1 Green Functions for Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 107321.7.2 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076

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21.7.3 Problems with Unmixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 107821.7.4 Problems with Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081

21.8 Green Functions for Higher Order Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108521.9 Fredholm Alternative Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109021.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109821.11Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110421.12Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107

22 Difference Equations 114522.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114522.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114722.3 Homogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114822.4 Inhomogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115022.5 Homogeneous Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115322.6 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115622.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115822.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115922.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1160

23 Series Solutions of Differential Equations 116323.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163

23.1.1 Taylor Series Expansion for a Second Order Differential Equation . . . . . . . . . . . . . . . . 116723.2 Regular Singular Points of Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177

23.2.1 Indicial Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118023.2.2 The Case: Double Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118223.2.3 The Case: Roots Differ by an Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186

23.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119623.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119623.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119923.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204

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23.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205

24 Asymptotic Expansions 122824.1 Asymptotic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122824.2 Leading Order Behavior of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123224.3 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124124.4 Asymptotic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124824.5 Asymptotic Expansions of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249

24.5.1 The Parabolic Cylinder Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249

25 Hilbert Spaces 125525.1 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125525.2 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125725.3 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125825.4 Linear Independence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126025.5 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126025.6 Gramm-Schmidt Orthogonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126125.7 Orthonormal Function Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126325.8 Sets Of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126525.9 Least Squares Fit to a Function and Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127225.10Closure Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127525.11Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128025.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128125.13Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128225.14Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283

26 Self Adjoint Linear Operators 128526.1 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128526.2 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128626.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289

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26.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129026.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1291

27 Self-Adjoint Boundary Value Problems 129227.1 Summary of Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129227.2 Formally Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129327.3 Self-Adjoint Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129627.4 Self-Adjoint Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129627.5 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130127.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130427.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130527.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306

28 Fourier Series 130828.1 An Eigenvalue Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130828.2 Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131128.3 Least Squares Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131528.4 Fourier Series for Functions Defined on Arbitrary Ranges . . . . . . . . . . . . . . . . . . . . . . . . . 131928.5 Fourier Cosine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132228.6 Fourier Sine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132328.7 Complex Fourier Series and Parsevals Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132428.8 Behavior of Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132728.9 Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133628.10Integrating and Differentiating Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133628.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134128.12Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134928.13Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1351

29 Regular Sturm-Liouville Problems 139829.1 Derivation of the Sturm-Liouville Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398

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29.2 Properties of Regular Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140029.3 Solving Differential Equations With Eigenfunction Expansions . . . . . . . . . . . . . . . . . . . . . . 141129.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141729.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142129.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1423

30 Integrals and Convergence 144830.1 Uniform Convergence of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144830.2 The Riemann-Lebesgue Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144930.3 Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1450

30.3.1 Integrals on an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145030.3.2 Singular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1451

31 The Laplace Transform 145331.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145331.2 The Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455

31.2.1 f(s) with Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145831.2.2 f(s) with Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146331.2.3 Asymptotic Behavior of f(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466

31.3 Properties of the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146831.4 Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147131.5 Systems of Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 147331.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147631.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148331.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486

32 The Fourier Transform 151832.1 Derivation from a Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151832.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1520

32.2.1 A Word of Caution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523

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32.3 Evaluating Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152432.3.1 Integrals that Converge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152432.3.2 Cauchy Principal Value and Integrals that are Not Absolutely Convergent. . . . . . . . . . . . . 152732.3.3 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1529

32.4 Properties of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153132.4.1 Closure Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153132.4.2 Fourier Transform of a Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153232.4.3 Fourier Convolution Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153432.4.4 Parsevals Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153732.4.5 Shift Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153932.4.6 Fourier Transform of x f(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1539

32.5 Solving Differential Equations with the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 154032.6 The Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1542

32.6.1 The Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154232.6.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1543

32.7 Properties of the Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 154432.7.1 Transforms of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154432.7.2 Convolution Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154632.7.3 Cosine and Sine Transform in Terms of the Fourier Transform . . . . . . . . . . . . . . . . . . 1548

32.8 Solving Differential Equations with the Fourier Cosine and Sine Transforms . . . . . . . . . . . . . . . 154932.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155132.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155832.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1561

33 The Gamma Function 158533.1 Eulers Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158533.2 Hankels Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158733.3 Gauss Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158933.4 Weierstrass Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159133.5 Stirlings Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593

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33.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159833.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159933.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1600

34 Bessel Functions 160234.1 Bessels Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160234.2 Frobeneius Series Solution about z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1603

34.2.1 Behavior at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160634.3 Bessel Functions of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608

34.3.1 The Bessel Function Satisfies Bessels Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 160934.3.2 Series Expansion of the Bessel Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161034.3.3 Bessel Functions of Non-Integer Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161334.3.4 Recursion Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161634.3.5 Bessel Functions of Half-Integer Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1619

34.4 Neumann Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162034.5 Bessel Functions of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162434.6 Hankel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162634.7 The Modified Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162634.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163034.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163534.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637

V Partial Differential Equations 1660

35 Transforming Equations 166135.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166235.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166335.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664

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36 Classification of Partial Differential Equations 166536.1 Classification of Second Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1665

36.1.1 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166636.1.2 Parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167136.1.3 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1672

36.2 Equilibrium Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167436.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167636.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167736.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1678

37 Separation of Variables 168437.1 Eigensolutions of Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168437.2 Homogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . 168437.3 Time-Independent Sources and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 168637.4 Inhomogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 168937.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169037.6 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169337.7 General Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169637.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169837.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171437.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1719

38 Finite Transforms 180138.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180538.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180638.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1807

39 The Diffusion Equation 181139.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181239.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1814

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39.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1815

40 Laplaces Equation 182140.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182140.2 Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1821

40.2.1 Two Dimensional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182240.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182340.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182640.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827

41 Waves 183941.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184041.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184641.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1848

42 Similarity Methods 186842.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187342.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187442.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1875

43 Method of Characteristics 187843.1 First Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187843.2 First Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187943.3 The Method of Characteristics and the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 188143.4 The Wave Equation for an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188243.5 The Wave Equation for a Semi-Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188343.6 The Wave Equation for a Finite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188543.7 Envelopes of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188643.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188943.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1891

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43.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1892

44 Transform Methods 189944.1 Fourier Transform for Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189944.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190144.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190144.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190344.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190744.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1909

45 Green Functions 193145.1 Inhomogeneous Equations and Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 193145.2 Homogeneous Equations and Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 193245.3 Eigenfunction Expansions for Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193445.4 The Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193945.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194145.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195245.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1955

46 Conformal Mapping 201546.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201646.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201946.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2020

47 Non-Cartesian Coordinates 203247.1 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203247.2 Laplaces Equation in a Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203347.3 Laplaces Equation in an Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2036

xix

VI Calculus of Variations 2040

48 Calculus of Variations 204148.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204248.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205648.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2060

VII Nonlinear Differential Equations 2147

49 Nonlinear Ordinary Differential Equations 214849.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214949.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215449.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2155

50 Nonlinear Partial Differential Equations 217750.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217850.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218150.3 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 2222

I Table of Laplace Transforms 2225I.1 Properties of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2225I.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 2246Q.1 Circular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2246Q.2 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2248

R Bessel Functions 2251R.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 2260W.1 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2260W.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 ordesecrated 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 norpolished. I have a Warnings and Disclaimers section below that is a little amusing, and an appendix on probabilitythat I feel concisesly captures the essence of the subject. However, all the material in between is in some stage ofdevelopment. 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 newversions 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 providingthe funding for me to write the first edition of this book.

xxiv

http://www.its.caltech.edu/~sean

0.3 Warnings and Disclaimers

This book is a work in progress. It contains quite a few mistakes and typos. I would greatly appreciate yourconstructive 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 contentscan 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 chequerfrom thyme too thyme, sew their should knot bee two many misspellings. Though I aint so sure the grammarstoo 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 ofobtaining 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 thatis 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 tolearn about some mathematical subject, look for books with Introduction or Elementary in the title. If it is anIntermediate text it will be incomprehensible. If it is Advanced then not only will it be incomprehensible, it willhave low production qualities, i.e. a crappy typewriter font, no graphics and no examples. There is an exception to thisrule: When the title also contains the word Scientists or Engineers the advanced book may be quite suitable foractually 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 elementsbetween braces. For example: {e, , , 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 thanenumerated. This is read as the set of elements x such that x satisfies . . . . x S is the notation for x is anelement 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. 11Guess what R is for.

2

C = {a + b|a, b R, 2 = 1} is the set of complex numbers. is the square root of 1. (If you havent seencomplex 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 finite sets, the cardinality is the number of elements in theset. 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 aproper subset of T , S T , if S T and S 6= T . For example: The empty set is a subset of every set, S. Therational 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 union of nsets,

nj=1Sj S1 S2 Snis the set whose elements are in any of the sets Sj. The intersection of two sets, S T , is the set whose elements arein both of the two sets. In other words, the intersection of two sets in the set of elements that the two sets have incommon. 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 setsare 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 Xinto elements y Y . This is expressed as f : X Y or X f Y . If such a function is well-defined, then for eachx X there exists a unique element of y such that f(x) = y. The set X is the domain of the function, Y is thecodomain, (not to be confused with the range, which we introduce shortly). To denote the value of a function on aparticular 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 onX 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:

f1(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 numbersto real numbers.

The greatest integer function, bc, 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 domainthere 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 setof 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, thereare 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 ofx such that y = sin x.

1.3 Inverses and Multi-Valued Functions

If y = f(x), then we can write x = f1(y) where f1 is the inverse of f . If y = f(x) is a one-to-one function, thenf1(y) is also a one-to-one function. In this case, x = f1(f(x)) = f(f1(x)) for values of x where both f(x) andf1(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 = f1(y) is a one-to-many function. f1(y) is a multi-valued function.We have x = f(f1(x)) for values of x where f1(x) is defined, however x 6= f1(f(x)). There are diagrams showingone-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 ofx 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 positivebranch 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 aninfinite 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 branchesof y = arcsinx.

Example 1.3.2 arcsinx has an infinite number of branches. We will denote the principal branch by Arcsin x whichmaps [1, 1] to

[

2,

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 = /3 in the range x [, ]. Since cos(x+ ) = cosx,

arccos(1/2) = {/3 + n}.

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. Theequations 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 andy = sin x to the equation x =

2results in x2 =

2

4and sin x = 1. The former equation has the two solutions x =

2;

the latter has the infinite number of solutions x = 2

+ 2n, n Z.

We do not generally apply a one-to-many function to both sides of an equation as this rarely is useful. Consider theequation

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 off(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 = /2 + 2n and x = arcsin(1) has the solutions x = /2 + 2n. Thus

x =

2+ n, n Z.

Note that we cannot just apply arcsin to both sides of the equation as arcsin(sinx) 6= x.

10

1.5 ExercisesExercise 1.1The area of a circle is directly proportional to the square of its diameter. What is the constant of proportionality?Hint, Solution

Exercise 1.2Consider the equation

x+ 1

y 2=x2 1y2 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 1.3Consider the function of a real variable,

f(x) =1

x2 + 2.

What is the domain and range of the function?Hint, Solution

Exercise 1.4The 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 degrees Celsius as a functionof degrees Fahrenheit.

2 Originally, it was called degrees Centigrade. centi because there are 100 degrees between the two calibration points. It is nowcalled 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 tobe 0. Later, the calibration points became the freezing point of water, 32, and body temperature, 96. With this method, there are64 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 1.5Consider the function graphed in Figure 1.9. Sketch graphs of f(x), f(x + 3), f(3 x) + 2, and f1(x). You mayuse the blank grids in Figure 1.10.

Figure 1.9: Graph of the function.

Hint, Solution

Exercise 1.6A culture of bacteria grows at the rate of 10% per minute. At 6:00 pm there are 1 billion bacteria. How many bacteriaare there at 7:00 pm? How many were there at 3:00 pm?

Hint, Solution

Exercise 1.7The graph in Figure 1.11 shows an even function f(x) = p(x)/q(x) where p(x) and q(x) are rational quadraticpolynomials. Give possible formulas for p(x) and q(x).

Hint, Solution

12

Figure 1.10: Blank grids.

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

Exercise 1.9Hint, 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 1.10Hint, Solution

Exercise 1.11Hint, Solution

Exercise 1.12Hint, Solution

Exercise 1.13Hint, Solution

Exercise 1.14Hint, Solution

Exercise 1.15Hint, Solution

Exercise 1.16Hint, Solution

14

1.6 HintsHint 1.1area = constant diameter2.

Hint 1.2A pair (x, y) is a solution of the equation if it make the equation an identity.

Hint 1.3The domain is the subset of R on which the function is defined.

Hint 1.4Find the slope and x-intercept of the line.

Hint 1.5The inverse of the function is the reflection of the function across the line y = x.

Hint 1.6The formula for geometric growth/decay is x(t) = x0r

t, where r is the rate.

Hint 1.7Note that p(x) and q(x) appear as a ratio, they are determined only up to a multiplicative constant. We may take theleading coefficient of q(x) to be unity.

f(x) =p(x)

q(x)=ax2 + bx+ c

x2 + x+

Use the properties of the function to solve for the unknown parameters.

Hint 1.8Write the polynomial in factored form.

15

1.7 SolutionsSolution 1.1

area = radius2

area =

4 diameter2

The constant of proportionality is 4

.

Solution 1.21. 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 areno solutions for these two values of y and we assume from this point that y 6= 2. We multiply by (y2)(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 obtainthe equation of a line.

y + 2 = x 1y = 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+1y2 =

x21y24 .

Solution 1.3The denominator is nonzero for all x R. Since we dont have any division by zero problems, the domain of thefunction is R. For x R,

0 0 there exists a > 0 such that |y(x) | < for all0 < |x | < . That is, there is an interval surrounding the point x = for which the function is within of . SeeFigure 3.1. Note that the interval surrounding x = is a deleted neighborhood, that is it does not contain the pointx = . Thus the value function at x = need not be equal to for the limit to exist. Indeed the function need noteven be defined at x = .

To prove that a function has a limit at a point we first bound |y(x) | in terms of for values of x satisfying0 < |x | < . Denote this upper bound by u(). Then for an arbitrary > 0, we determine a > 0 such that thethe upper bound u() and hence |y(x) | is less than .

47

x

y

+

+

Figure 3.1: The neighborhood of x = such that |y(x) | < .

Example 3.1.1 Show thatlimx1

x2 = 1.

Consider any > 0. We need to show that there exists a > 0 such that |x2 1| < for all |x 1| < . First weobtain a bound on |x2 1|.

|x2 1| = |(x 1)(x+ 1)|= |x 1||x+ 1|< |x+ 1|= |(x 1) + 2|< ( + 2)

Now we choose a positive such that,( + 2) = .

We see that =

1 + 1,

is positive and satisfies the criterion that |x2 1| < for all 0 < |x 1| < . Thus the limit exists.

48

Note that the value of the function y() need not be equal to limx y(x). This is illustrated in Example 3.1.2.

Example 3.1.2 Consider the function

y(x) =

{1 for x Z,0 for x 6 Z.

For what values of does limx y(x) exist?First consider 6 Z. Then there exists an open neighborhood a < < b around such that y(x) is identically zero

for x (a, b). Then trivially, limx y(x) = 0.Now consider Z. Consider any > 0. Then if |x | < 1 then |y(x) 0| = 0 < . Thus we see that

limx y(x) = 0.Thus, regardless of the value of , limx y(x) = 0.

Left and Right Limits. With the notation limx+ y(x) we denote the right limit of y(x). This is the limit as xapproaches from above. Mathematically: limx+ exists if for any > 0 there exists a > 0 such that |y(x)| < for all 0 < x < . The left limit limx y(x) is defined analogously.

Example 3.1.3 Consider the function, sinx|x| , defined for x 6= 0. (See Figure 3.2.) The left and right limits exist as xapproaches zero.

limx0+

sin x

|x|= 1, lim

x0

sin x

|x|= 1

However the limit,

limx0

sinx

|x|,

does not exist.

Properties of Limits. Let limx u(x) and limx v(x) exist.

limx (au(x) + bv(x)) = a limx u(x) + b limx v(x).

49

Figure 3.2: Plot of sin(x)/|x|.

limx (u(x)v(x)) = (limx u(x)) (limx v(x)).

limx(u(x)v(x)

)=

limx u(x)

limx v(x)if limx v(x) 6= 0.

Example 3.1.4 Prove that if limx u(x) = and limx v(x) = exist then

limx

(u(x)v(x)) =

(limx

u(x)

)(limx

v(x)

).

Assume that and are nonzero. (The cases where one or both are zero are similar and simpler.)

|u(x)v(x) | = |uv (u+ u)|= |u(v ) + (u )|= |u||v |+ |u |||

A sufficient condition for |u(x)v(x) | < is

|u | < 2||

and |v | <

2(||+

2||

) .Since the two right sides of the inequalities are positive, there exists 1 > 0 and 2 > 0 such that the first inequality issatisfied for all |x | < 1 and the second inequality is satisfied for all |x | < 2. By choosing to be the smaller

50

of 1 and 2 we see that

|u(x)v(x) | < for all |x | < .

Thus

limx

(u(x)v(x)) =

(limx

u(x)

)(limx

v(x)

)= .

Result 3.1.1 Definition of a Limit. The statement:

limx

y(x) =

means that y(x) gets arbitrarily close to as x approaches . For any > 0 there exists a > 0 such that |y(x) | < for all x in the neighborhood 0 < |x | < . The left andright limits,

limx

y(x) = and limx+

y(x) =

denote the limiting value as x approaches respectively from below and above. The neigh-borhoods are respectively < x < 0 and 0 < x < .Properties of Limits. Let limx u(x) and limx v(x) exist.

limx (au(x) + bv(x)) = a limx u(x) + b limx v(x).

limx (u(x)v(x)) = (limx u(x)) (limx v(x)).

limx(u(x)v(x)

)=

limx u(x)limx v(x)

if limx v(x) 6= 0.

51

3.2 Continuous Functions

Definition of Continuity. A function y(x) is said to be continuous at x = if the value of the function isequal to its limit, that is, limx y(x) = y(). Note that this one condition is actually the three conditions: y() isdefined, limx y(x) exists and limx y(x) = y(). A function is continuous if it is continuous at each point in itsdomain. A function is continuous on the closed interval [a, b] if the function is continuous for each point x (a, b) andlimxa+ y(x) = y(a) and limxb y(x) = y(b).

Discontinuous Functions. If a function is not continuous at a point it is called discontinuous at that point. Iflimx y(x) exists but is not equal to y(), then the function has a removable discontinuity. It is thus named becausewe could define a continuous function

z(x) =

{y(x) for x 6= ,limx y(x) for x = ,

to remove the discontinuity. If both the left and right limit of a function at a point exist, but are not equal, then thefunction has a jump discontinuity at that point. If either the left or right limit of a function does not exist, then thefunction is said to have an infinite discontinuity at that point.

Example 3.2.1 sinxx

has a removable discontinuity at x = 0. The Heaviside function,

H(x) =

0 for x < 0,

1/2 for x = 0,

1 for x > 0,

has a jump discontinuity at x = 0. 1x

has an infinite discontinuity at x = 0. See Figure 3.3.

Properties of Continuous Functions.

52

Figure 3.3: A Removable discontinuity, a Jump Discontinuity and an Infinite Discontinuity

Arithmetic. If u(x) and v(x) are continuous at x = then u(x) v(x) and u(x)v(x) are continuous at x = . u(x)v(x)

is continuous at x = if v() 6= 0.

Function Composition. If u(x) is continuous at x = and v(x) is continuous at x = = u() then u(v(x)) iscontinuous at x = . The composition of continuous functions is a continuous function.

Boundedness. A function which is continuous on a closed interval is bounded in that closed interval.

Nonzero in a Neighborhood. If y() 6= 0 then there exists a neighborhood ( , + ), > 0 of the point suchthat y(x) 6= 0 for x ( , + ).

Intermediate Value Theorem. Let u(x) be continuous on [a, b]. If u(a) u(b) then there exists [a, b] suchthat u() = . This is known as the intermediate value theorem. A corollary of this is that if u(a) and u(b) areof opposite sign then u(x) has at least one zero on the interval (a, b).

Maxima and Minima. If u(x) is continuous on [a, b] then u(x) has a maximum and a minimum on [a, b]. That is, thereis at least one point [a, b] such that u() u(x) for all x [a, b] and there is at least one point [a, b]such that u() u(x) for all x [a, b].

Piecewise Continuous Functions. A function is piecewise continuous on an interval if the function is bounded onthe interval and the interval can be divided into a finite number of intervals on each of which the function is continuous.

53

For example, the greatest integer function, bxc, is piecewise continuous. (bxc is defined to the the greatest integer lessthan or equal to x.) See Figure 3.4 for graphs of two piecewise continuous functions.

Figure 3.4: Piecewise Continuous Functions

Uniform Continuity. Consider a function f(x) that is continuous on an interval. This means that for any point in the interval and any positive there exists a > 0 such that |f(x) f()| < for all 0 < |x | < . In general,this value of depends on both and . If can be chosen so it is a function of alone and independent of thenthe function is said to be uniformly continuous on the interval. A sufficient condition for uniform continuity is that thefunction is continuous on a closed interval.

3.3 The Derivative

Consider a function y(x) on the interval (x . . . x+ x) for some x > 0. We define the increment y = y(x+ x)y(x). The average rate of change, (average velocity), of the function on the interval is y

x. The average rate of change

is the slope of the secant line that passes through the points (x, y(x)) and (x+ x, y(x+ x)). See Figure 3.5.If the slope of the secant line has a limit as x approaches zero then we call this slope the derivative or instantaneous

rate of change of the function at the point x. We denote the derivative by dydx

, which is a nice notation as the derivative

is the limit of yx

as x 0.dy

dx lim

x0

y(x+ x) y(x)x

.

54

y

x

y

x

Figure 3.5: The increments x and y.

x may approach zero from below or above. It is common to denote the derivative dydx

by ddxy, y(x), y or Dy.

A function is said to be differentiable at a point if the derivative exists there. Note that differentiability impliescontinuity, but not vice versa.

Example 3.3.1 Consider the derivative of y(x) = x2 at the point x = 1.

y(1) limx0

y(1 + x) y(1)x

= limx0

(1 + x)2 1x

= limx0

(2 + x)

= 2

Figure 3.6 shows the secant lines approaching the tangent line as x approaches zero from above and below.

55

0.5 1 1.5 2

0.5

1

1.5

2

2.5

3

3.5

4

0.5 1 1.5 2

0.5

1

1.5

2

2.5

3

3.5

4

Figure 3.6: Secant lines and the tangent to x2 at x = 1.

Example 3.3.2 We can compute the derivative of y(x) = x2 at an arbitrary point x.

d

dx

[x2]

= limx0

(x+ x)2 x2

x= lim

x0(2x+ x)

= 2x

56

Properties. Let u(x) and v(x) be differentiable. Let a and b be constants. Some fundamental properties ofderivatives are:

d

dx(au+ bv) = a

du

dx+ b

dv

dxLinearity

d

dx(uv) =

du

dxv + u

dv

dxProduct Rule

d

dx

(uv

)=v du

dx udv

dx

v2Quotient Rule

d

dx(ua) = aua1

du

dxPower Rule

d

dx(u(v(x))) =

du

dv

dv

dx= u(v(x))v(x) Chain Rule

These can be proved by using the definition of differentiation.

Example 3.3.3 Prove the quotient rule for derivatives.

d

dx

(uv

)= lim

x0

u(x+x)v(x+x)

u(x)v(x)

x

= limx0

u(x+ x)v(x) u(x)v(x+ x)xv(x)v(x+ x)

= limx0

u(x+ x)v(x) u(x)v(x) u(x)v(x+ x) + u(x)v(x)xv(x)v(x)

= limx0

(u(x+ x) u(x))v(x) u(x)(v(x+ x) v(x))xv2(x)

=limx0

u(x+x)u(x)x

v(x) u(x) limx0 v(x+x)v(x)xv2(x)

=v du

dx udv

dx

v2

57

Trigonometric Functions. Some derivatives of trigonometric functions are:

d

dxsin x = cos x

d

dxarcsinx =

1

(1 x2)1/2d

dxcosx = sinx d

dxarccosx = 1

(1 x2)1/2d

dxtanx =

1

cos2 x

d

dxarctanx =

1

1 + x2

d

dxex = ex

d

dxlnx =

1

xd

dxsinh x = cosh x

d

dxarcsinhx =

1

(x2 + 1)1/2

d

dxcoshx = sinh x

d

dxarccoshx =

1

(x2 1)1/2d

dxtanhx =

1

cosh2 x

d

dxarctanhx =

1

1 x2

Example 3.3.4 We can evaluate the derivative of xx by using the identity ab = eb ln a.

d

dxxx =

d

dxex lnx

= ex lnxd

dx(x lnx)

= xx(1 lnx+ x1x

)

= xx(1 + ln x)

58

Inverse Functions. If we have a function y(x), we can consider x as a function of y, x(y). For example, ify(x) = 8x3 then x(y) = 2 3

y; if y(x) = x+2

x+1then x(y) = 2y

y1 . The derivative of an inverse function is

d

dyx(y) =

1dydx

.

Example 3.3.5 The inverse function of y(x) = ex is x(y) = ln y. We can obtain the derivative of the logarithm fromthe derivative of the exponential. The derivative of the exponential is

dy

dx= ex .

Thus the derivative of the logarithm is

d

dyln y =

d

dyx(y) =

1dydx

=1

ex=

1

y.

3.4 Implicit Differentiation

An explicitly defined function has the form y = f(x). A implicitly defined function has the form f(x, y) = 0. A fewexamples of implicit functions are x2 + y2 1 = 0 and x+ y+ sin(xy) = 0. Often it is not possible to write an implicitequation in explicit form. This is true of the latter example above. One can calculate the derivative of y(x) in termsof x and y even when y(x) is defined by an implicit equation.

Example 3.4.1 Consider the implicit equation

x2 xy y2 = 1.

This implicit equation can be solved for the dependent variable.

y(x) =1

2

(x

5x2 4

).