Introduction to Methods of Applied Mathematics

or

Advanced Mathematical Methods for Scientists and Engineers

Sean Mauchhttp://www.its.caltech.edu/sean

January 24, 2004

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

2

Contents

Anti-Copyright xv

Preface xvii0.1 Advice to Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii0.2 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii0.3 Warnings and Disclaimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii0.4 Suggested Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii0.5 About the Title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

I Algebra 1

1 Sets and Functions 31.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Single Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Inverses and Multi-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Transforming Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Vectors 172.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 The Kronecker Delta and Einstein Summation Convention . . . . . . . . . . . 192.1.3 The Dot and Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Sets of Vectors in n Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

II Calculus 31

3 Differential Calculus 333.1 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.6 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6.1 Application: Using Taylors Theorem to Approximate Functions. . . . . . . . 453.6.2 Application: Finite Difference Schemes . . . . . . . . . . . . . . . . . . . . . . 47

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3.7 LHospitals Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.8.1 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.8.2 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.8.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.8.4 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.8.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.8.6 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.8.7 LHospitals Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.10 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.11 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.12 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4 Integral Calculus 754.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3 The Fundamental Theorem of Integral Calculus . . . . . . . . . . . . . . . . . . . . . 804.4 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4.1 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.6.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.6.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.6.3 The Fundamental Theorem of Integration . . . . . . . . . . . . . . . . . . . . 864.6.4 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.6.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.9 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.10 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5 Vector Calculus 995.1 Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2 Gradient, Divergence and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.6 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.7 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

III Functions of a Complex Variable 117

6 Complex Numbers 1196.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.2 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.3 Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.4 Arithmetic and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.5 Integer Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.6 Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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6.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7 Functions of a Complex Variable 1537.1 Curves and Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537.2 The Point at Infinity and the Stereographic Projection . . . . . . . . . . . . . . . . . 1557.3 A Gentle Introduction to Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . 1577.4 Cartesian and Modulus-Argument Form . . . . . . . . . . . . . . . . . . . . . . . . . 1577.5 Graphing Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . . . . . 1597.6 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.8 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1697.9 Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1707.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1807.11 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1877.12 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

8 Analytic Functions 2238.1 Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2238.2 Cauchy-Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2278.3 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2308.4 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

8.4.1 Categorization of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . 2338.4.2 Isolated and Non-Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . 235

8.5 Application: Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2368.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2398.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2448.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

9 Analytic Continuation 2699.1 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2699.2 Analytic Continuation of Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2719.3 Analytic Functions Defined in Terms of Real Variables . . . . . . . . . . . . . . . . . 271

9.3.1 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2749.3.2 Analytic Functions Defined in Terms of Their Real or Imaginary Parts . . . . 276

9.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2799.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2809.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

10 Contour Integration and the Cauchy-Goursat Theorem 28510.1 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28510.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

10.2.1 Maximum Modulus Integral Bound . . . . . . . . . . . . . . . . . . . . . . . . 28710.3 The Cauchy-Goursat Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28810.4 Contour Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28910.5 Moreras Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29010.6 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29110.7 Fundamental Theorem of Calculus via Primitives . . . . . . . . . . . . . . . . . . . . 292

10.7.1 Line Integrals and Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . 29210.7.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

10.8 Fundamental Theorem of Calculus via Complex Calculus . . . . . . . . . . . . . . . 29210.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29510.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29710.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

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11 Cauchys Integral Formula 30511.1 Cauchys Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30511.2 The Argument Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30911.3 Rouches Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31111.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31211.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31511.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

12 Series and Convergence 32512.1 Series of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

12.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32512.1.2 Special Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32612.1.3 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

12.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33112.2.1 Tests for Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 33212.2.2 Uniform Convergence and Continuous Functions. . . . . . . . . . . . . . . . . 333

12.3 Uniformly Convergent Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33312.4 Integration and Differentiation of Power Series . . . . . . . . . . . . . . . . . . . . . 33712.5 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

12.5.1 Newtons Binomial Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34112.6 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34212.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

12.7.1 Series of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34412.7.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34712.7.3 Uniformly Convergent Power Series . . . . . . . . . . . . . . . . . . . . . . . . 34712.7.4 Integration and Differentiation of Power Series . . . . . . . . . . . . . . . . . 34912.7.5 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34912.7.6 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

12.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35312.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

13 The Residue Theorem 38313.1 The Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38313.2 Cauchy Principal Value for Real Integrals . . . . . . . . . . . . . . . . . . . . . . . . 387

13.2.1 The Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38713.3 Cauchy Principal Value for Contour Integrals . . . . . . . . . . . . . . . . . . . . . . 39013.4 Integrals on the Real Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39313.5 Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39513.6 Fourier Cosine and Sine Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39713.7 Contour Integration and Branch Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . 39813.8 Exploiting Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

13.8.1 Wedge Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40013.8.2 Box Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

13.9 Definite Integrals Involving Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . 40313.10Infinite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40413.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40713.12Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41613.13Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

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IV Ordinary Differential Equations 471

14 First Order Differential Equations 47314.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47314.2 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

14.2.1 Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47414.3 One Parameter Families of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 47514.4 Integrable Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

14.4.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47714.4.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47814.4.3 Homogeneous Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . 480

14.5 The First Order, Linear Differential Equation . . . . . . . . . . . . . . . . . . . . . . 48314.5.1 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48314.5.2 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48414.5.3 Variation of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

14.6 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48614.6.1 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . 486

14.7 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48914.8 Equations in the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

14.8.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49014.8.2 Regular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49214.8.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49514.8.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

14.9 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49814.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50014.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50214.12Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51314.13Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

15 First Order Linear Systems of Differential Equations 51515.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51515.2 Using Eigenvalues and Eigenvectors to find Homogeneous Solutions . . . . . . . . . . 51515.3 Matrices and Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 51815.4 Using the Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52215.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52615.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52915.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

16 Theory of Linear Ordinary Differential Equations 54716.1 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54716.2 Nature of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54816.3 Transformation to a First Order System . . . . . . . . . . . . . . . . . . . . . . . . . 55016.4 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

16.4.1 Derivative of a Determinant. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55016.4.2 The Wronskian of a Set of Functions. . . . . . . . . . . . . . . . . . . . . . . 55116.4.3 The Wronskian of the Solutions to a Differential Equation . . . . . . . . . . . 552

16.5 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55416.6 The Fundamental Set of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55516.7 Adjoint Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55616.8 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55916.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56016.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56116.11Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56516.12Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566

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17 Techniques for Linear Differential Equations 56717.1 Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

17.1.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56717.1.2 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57017.1.3 Higher Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

17.2 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57317.2.1 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574

17.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57617.4 Equations Without Explicit Dependence on y . . . . . . . . . . . . . . . . . . . . . . 57717.5 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57717.6 *Reduction of Order and the Adjoint Equation . . . . . . . . . . . . . . . . . . . . . 57817.7 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58017.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58417.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

18 Techniques for Nonlinear Differential Equations 60118.1 Bernoulli Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60118.2 Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60218.3 Exchanging the Dependent and Independent Variables . . . . . . . . . . . . . . . . . 60418.4 Autonomous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60518.5 *Equidimensional-in-x Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60718.6 *Equidimensional-in-y Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60818.7 *Scale-Invariant Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61018.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61118.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61318.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614

19 Transformations and Canonical Forms 62119.1 The Constant Coefficient Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62119.2 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

19.2.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62319.2.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 624

19.3 Transformations of the Independent Variable . . . . . . . . . . . . . . . . . . . . . . 62419.3.1 Transformation to the form u + a(x) u = 0 . . . . . . . . . . . . . . . . . . 62419.3.2 Transformation to a Constant Coefficient Equation . . . . . . . . . . . . . . . 625

19.4 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62619.4.1 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62619.4.2 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628

19.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63019.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63219.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

20 The Dirac Delta Function 63720.1 Derivative of the Heaviside Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 63720.2 The Delta Function as a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63820.3 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63920.4 Non-Rectangular Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 63920.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64120.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64320.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644

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21 Inhomogeneous Differential Equations 64921.1 Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64921.2 Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 65021.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652

21.3.1 Second Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 65221.3.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 654

21.4 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . . . . . 65621.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 658

21.5.1 Eliminating Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . 65821.5.2 Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions65921.5.3 Existence of Solutions of Problems with Inhomogeneous Boundary Conditions 659

21.6 Green Functions for First Order Equations . . . . . . . . . . . . . . . . . . . . . . . 66121.7 Green Functions for Second Order Equations . . . . . . . . . . . . . . . . . . . . . . 662

21.7.1 Green Functions for Sturm-Liouville Problems . . . . . . . . . . . . . . . . . 66821.7.2 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67021.7.3 Problems with Unmixed Boundary Conditions . . . . . . . . . . . . . . . . . 67121.7.4 Problems with Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . 672

21.8 Green Functions for Higher Order Problems . . . . . . . . . . . . . . . . . . . . . . . 67421.9 Fredholm Alternative Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67721.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68221.11Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68621.12Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68821.13Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71021.14Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

22 Difference Equations 71322.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71322.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71422.3 Homogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71522.4 Inhomogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71622.5 Homogeneous Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . 71722.6 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71922.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72122.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72222.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723

23 Series Solutions of Differential Equations 72523.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725

23.1.1 Taylor Series Expansion for a Second Order Differential Equation . . . . . . . 72823.2 Regular Singular Points of Second Order Equations . . . . . . . . . . . . . . . . . . . 733

23.2.1 Indicial Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73523.2.2 The Case: Double Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73623.2.3 The Case: Roots Differ by an Integer . . . . . . . . . . . . . . . . . . . . . . 738

23.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74323.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74323.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74523.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74823.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74923.8 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76323.9 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764

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24 Asymptotic Expansions 76524.1 Asymptotic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76524.2 Leading Order Behavior of Differential Equations . . . . . . . . . . . . . . . . . . . . 76724.3 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77224.4 Asymptotic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77724.5 Asymptotic Expansions of Differential Equations . . . . . . . . . . . . . . . . . . . . 777

24.5.1 The Parabolic Cylinder Equation. . . . . . . . . . . . . . . . . . . . . . . . . 777

25 Hilbert Spaces 78125.1 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78125.2 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78225.3 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78325.4 Linear Independence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78425.5 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78425.6 Gramm-Schmidt Orthogonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78425.7 Orthonormal Function Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78625.8 Sets Of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78725.9 Least Squares Fit to a Function and Completeness . . . . . . . . . . . . . . . . . . . 79025.10Closure Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79225.11Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79525.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79625.13Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79725.14Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798

26 Self Adjoint Linear Operators 79926.1 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79926.2 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79926.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80126.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80226.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

27 Self-Adjoint Boundary Value Problems 80527.1 Summary of Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80527.2 Formally Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80627.3 Self-Adjoint Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80727.4 Self-Adjoint Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80827.5 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81127.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81327.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81427.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815

28 Fourier Series 81728.1 An Eigenvalue Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81728.2 Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81928.3 Least Squares Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82128.4 Fourier Series for Functions Defined on Arbitrary Ranges . . . . . . . . . . . . . . . 82428.5 Fourier Cosine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82628.6 Fourier Sine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82728.7 Complex Fourier Series and Parsevals Theorem . . . . . . . . . . . . . . . . . . . . . 82828.8 Behavior of Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82928.9 Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83528.10Integrating and Differentiating Fourier Series . . . . . . . . . . . . . . . . . . . . . . 83528.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83828.12Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84328.13Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845

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29 Regular Sturm-Liouville Problems 87329.1 Derivation of the Sturm-Liouville Form . . . . . . . . . . . . . . . . . . . . . . . . . 87329.2 Properties of Regular Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . 87429.3 Solving Differential Equations With Eigenfunction Expansions . . . . . . . . . . . . 88129.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88529.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88829.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889

30 Integrals and Convergence 90530.1 Uniform Convergence of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90530.2 The Riemann-Lebesgue Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90630.3 Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906

30.3.1 Integrals on an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 90630.3.2 Singular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907

31 The Laplace Transform 90931.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90931.2 The Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910

31.2.1 f(s) with Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91231.2.2 f(s) with Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91431.2.3 Asymptotic Behavior of f(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . 916

31.3 Properties of the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 91731.4 Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 91931.5 Systems of Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . 92031.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92231.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92631.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928

32 The Fourier Transform 94732.1 Derivation from a Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94732.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948

32.2.1 A Word of Caution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94932.3 Evaluating Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 950

32.3.1 Integrals that Converge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95032.3.2 Cauchy Principal Value and Integrals that are Not Absolutely Convergent. . 95232.3.3 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953

32.4 Properties of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 95432.4.1 Closure Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95432.4.2 Fourier Transform of a Derivative. . . . . . . . . . . . . . . . . . . . . . . . . 95532.4.3 Fourier Convolution Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 95532.4.4 Parsevals Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95732.4.5 Shift Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95832.4.6 Fourier Transform of x f(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959

32.5 Solving Differential Equations with the Fourier Transform . . . . . . . . . . . . . . . 95932.6 The Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . 960

32.6.1 The Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 96032.6.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961

32.7 Properties of the Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . 96232.7.1 Transforms of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96232.7.2 Convolution Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96232.7.3 Cosine and Sine Transform in Terms of the Fourier Transform . . . . . . . . 964

32.8 Solving Differential Equations with the Fourier Cosine and Sine Transforms . . . . . 96532.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96632.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 970

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32.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972

33 The Gamma Function 98733.1 Eulers Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98733.2 Hankels Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98833.3 Gauss Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98933.4 Weierstrass Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99033.5 Stirlings Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99133.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99533.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99633.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997

34 Bessel Functions 99934.1 Bessels Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99934.2 Frobeneius Series Solution about z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 999

34.2.1 Behavior at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100134.3 Bessel Functions of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003

34.3.1 The Bessel Function Satisfies Bessels Equation . . . . . . . . . . . . . . . . . 100334.3.2 Series Expansion of the Bessel Function . . . . . . . . . . . . . . . . . . . . . 100434.3.3 Bessel Functions of Non-Integer Order . . . . . . . . . . . . . . . . . . . . . . 100534.3.4 Recursion Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100734.3.5 Bessel Functions of Half-Integer Order . . . . . . . . . . . . . . . . . . . . . . 1009

34.4 Neumann Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101034.5 Bessel Functions of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . 101234.6 Hankel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101334.7 The Modified Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101334.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101634.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101934.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1020

V Partial Differential Equations 1033

35 Transforming Equations 103535.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103635.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103735.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1038

36 Classification of Partial Differential Equations 103936.1 Classification of Second Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . 1039

36.1.1 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104036.1.2 Parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104336.1.3 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043

36.2 Equilibrium Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104436.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104636.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104736.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048

37 Separation of Variables 105137.1 Eigensolutions of Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . 105137.2 Homogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . 105137.3 Time-Independent Sources and Boundary Conditions . . . . . . . . . . . . . . . . . . 105237.4 Inhomogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . 105437.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 105537.6 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056

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37.7 General Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105837.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105937.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106937.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072

38 Finite Transforms 111938.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112138.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112238.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123

39 The Diffusion Equation 112739.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112839.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112939.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1130

40 Laplaces Equation 113540.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113540.2 Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135

40.2.1 Two Dimensional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113540.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113640.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113840.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1139

41 Waves 114741.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114841.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115241.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154

42 Similarity Methods 116742.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117042.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117142.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172

43 Method of Characteristics 117543.1 First Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117543.2 First Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 117643.3 The Method of Characteristics and the Wave Equation . . . . . . . . . . . . . . . . . 117643.4 The Wave Equation for an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . 117743.5 The Wave Equation for a Semi-Infinite Domain . . . . . . . . . . . . . . . . . . . . . 117843.6 The Wave Equation for a Finite Domain . . . . . . . . . . . . . . . . . . . . . . . . . 117943.7 Envelopes of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118043.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118243.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118343.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184

44 Transform Methods 118944.1 Fourier Transform for Partial Differential Equations . . . . . . . . . . . . . . . . . . 118944.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119044.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119044.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119244.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119544.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197

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45 Green Functions 121145.1 Inhomogeneous Equations and Homogeneous Boundary Conditions . . . . . . . . . . 121145.2 Homogeneous Equations and Inhomogeneous Boundary Conditions . . . . . . . . . . 121145.3 Eigenfunction Expansions for Elliptic Equations . . . . . . . . . . . . . . . . . . . . . 121345.4 The Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121545.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121745.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122445.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226

46 Conformal Mapping 126146.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126246.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126446.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265

47 Non-Cartesian Coordinates 127347.1 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127347.2 Laplaces Equation in a Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127347.3 Laplaces Equation in an Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275

VI Calculus of Variations 1279

48 Calculus of Variations 128148.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128248.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129148.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294

VII Nonlinear Differential Equations 1345

49 Nonlinear Ordinary Differential Equations 134749.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134849.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135149.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1352

50 Nonlinear Partial Differential Equations 136550.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136650.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136850.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369

VIII Appendices 1381

A Greek Letters 1383

B Notation 1385

C Formulas from Complex Variables 1387

D Table of Derivatives 1389

E Table of Integrals 1391

F Definite Integrals 1393

G Table of Sums 1395

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H Table of Taylor Series 1397

I Continuous Transforms 1399I.1 Properties of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399I.2 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1401I.3 Table of Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403I.4 Table of Fourier Transforms in n Dimensions . . . . . . . . . . . . . . . . . . . . . . 1405I.5 Table of Fourier Cosine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406I.6 Table of Fourier Sine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407

J Table of Wronskians 1409

K Sturm-Liouville Eigenvalue Problems 1411

L Green Functions for Ordinary Differential Equations 1413

M Trigonometric Identities 1415M.1 Circular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415M.2 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416

N Bessel Functions 1419N.1 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1419

O Formulas from Linear Algebra 1421

P Vector Analysis 1423

Q Partial Fractions 1425

R Finite Math 1427

S Physics 1429

T Probability 1431T.1 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1431T.2 Playing the Odds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1431

U Economics 1433

V Glossary 1435

W whoami 1437

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xiv

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xv

xvi

Preface

During the summer before my final undergraduate year at Caltech I set out to write a math textunlike any other, namely, one written by me. In that respect I have succeeded beautifully. Unfor-tunately, the text is neither complete nor polished. I have a Warnings and Disclaimers sectionbelow that is a little amusing, and an appendix on probability that I feel concisesly captures theessence of the subject. However, all the material in between is in some stage of development. I amcurrently 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 Saffman for advising me on this project and the Caltech SURFprogram for providing the funding for me to write the first edition of this book.

0.3 Warnings and Disclaimers

This book is a work in progress. It contains quite a few mistakes and typos. I would greatlyappreciate your constructive criticism. You can reach me at sean@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.

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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 purelycoincidental.

This is by far the most amusing section of this book.

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

Finding the typos and mistakes in this book is left as an exercise for the reader. (Eye ewesa 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 friendswithin ten days of obtaining this text you will suffer great misfortune and other nastiness.

The surgeon general has determined that excessive studying is detrimental to your social life.

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 texthas a boquet that is light and fruity, with some earthy undertones. It is ideal with dinner or as anapertif. 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 orElementary in the title. If it is an Intermediate text it will be incomprehensible. If it isAdvanced then not only will it be incomprehensible, it will have low production qualities, i.e. acrappy typewriter font, no graphics and no examples. There is an exception to this rule: When thetitle also contains the word Scientists or Engineers the advanced book may be quite suitable foractually learning the material.

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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 bylisting the elements between braces. For example: {e, , , 1} is the set of the integer 1, the pureimaginary 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 identicalto the set {1, 2, 3}. Order is not significant 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 ofpositive integers as {1, 2, 3, . . .}. We also denote sets with the notation {x|conditions on x} for setsthat 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 havex 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 = {. . . ,3,2,1, 0, 1, 2, 3 . . .} is the set of integers. (Z is for Zahlen, the German wordfor number.)

Q = {p/q|p, q Z, q 6= 0} is the set of rational numbers. (Q is for quotient.) 1

R = {x|x = a1a2 an.b1b2 } is the set of real numbers, i.e. the set of numbers with decimalexpansions. 2

C = {a + b|a, b R, 2 = 1} is the set of complex numbers. is the square root of 1. (Ifyou 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, . . .}. We use a superscript to denote the sets of negative numbers.

Z0+, Q0+ and R0+ are the sets of non-negative integers, rationals and reals, respectively. Forexample, 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}

1 Note that with this description, we enumerate each rational number an infinite 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.

2Guess what R is for.

3

The cardinality or order of a set S is denoted |S|. For finite sets, the cardinality is the numberof 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. Thisis denoted, S = T . Inequality is S 6= T , of course. S is a subset of T , S T , if every element of Sis an element of T . S is a proper subset of T , S T , if S T and S 6= T . For example: The emptyset is a subset of every set, S. The rational numbers are a proper subset of the real numbers,Q R.

Operations. The union of two sets, S T , is the set whose elements are in either of the two sets.The union of n sets,

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 setwhose elements are in both of the two sets. In other words, the intersection of two sets in the set ofelements that the two sets have in common. The intersection of n sets,

nj=1Sj S1 S2 Sn

is the set whose elements are in all of the sets Sj . If two sets have no elements in common, ST = ,then the sets are disjoint. If T S, then the difference between S and T , S \T , is the set of elementsin 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 theelements x X into elements y Y . This is expressed as f : X Y or X f Y . If such a functionis well-defined, then for each x X there exists a unique element of y such that f(x) = y. The setX is the domain of the function, Y is the codomain, (not to be confused with the range, which weintroduce shortly). To denote the value of a function on a particular element we can use any ofthe notations: f(x) = y, f : x 7 y or simply x 7 y. f is the identity map on X if f(x) = x for allx 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, f(x) =nk=0 akx

k, ak R, and the exponential function, f(x) = ex, areexamples of single valued functions which map real numbers to real numbers.

The greatest integer function, f(x) = bxc, is a mapping from R to Z. bxc is defined as thegreatest integer less than or equal to x. Likewise, the least integer function, f(x) = dxe, is theleast 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, distinctelements are mapped to distinct elements. f is surjective if for each y in the codomain, there is anx such that y = f(x). If a function is both injective and surjective, then it is bijective. A bijectivefunction 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, (aone-to-one mapping).

f(x) = x2 is a bijection from R+ to R+. f is not injective from R to R+. For each positive yin the range, there are two values of x such that y = x2.

f(x) = sinx is not injective from R to [1..1]. For each y [1..1] there exists an infinitenumber of values of x such that y = sinx.

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 = f1(y) where f1 is the inverse of f . If y = f(x) is a one-to-onefunction, then f1(y) is also a one-to-one function. In this case, x = f1(f(x)) = f(f1(x)) forvalues of x where both f(x) and f1(x) are defined. For example lnx, which maps R+ to R is theinverse of ex. x = eln x = ln(ex) for all x R+. (Note the x R+ ensures that lnx is defined.)

If y = f(x) is a many-to-one function, then x = f1(y) is a one-to-many function. f1(y) is amulti-valued function. We have x = f(f1(x)) for values of x where f1(x) is defined, howeverx 6= f1(f(x)). There are diagrams showing one-to-one, many-to-one and one-to-many functions inFigure 1.2.

Example 1.3.1 y = x2, a many-to-one function has the inverse x = y1/2. For each positive y, thereare two values of x such that x = y1/2. y = x2 and y = x1/2 are graphed in Figure 1.3.

5

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

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 denotethe positive branch as y =

x; the negative branch is y =

x. We call

x the principal branch of

x1/2. Note thatx 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 = sinx. The inverse is x = arcsin y. For each y [1..1] there are an infinite number of values x such that x = arcsin y. In Figure 1.5 is a graph ofy = sinx and a graph of a few branches of y = arcsinx.

Figure 1.5: y = sinx and y = arcsinx

Example 1.3.2 arcsinx has an infinite number of branches. We will denote the principal branch

6

by Arcsinx which maps [1..1] to[2 ..

2

]. Note that x = sin(arcsinx), but x 6= arcsin(sinx).

y = Arcsinx in Figure 1.6.

Figure 1.6: y = Arcsinx

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

Example 1.3.4 Consider arccos(1/2). cosx and a portion of arccosx are graphed in Figure 1.8.The equation cosx = 1/2 has the two solutions x = /3 in the range x (..]. We use theperiodicity of the cosine, cos(x+ 2) = cosx, to find the remaining solutions.

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

Figure 1.8: y = cosx and y = arccosx

1.4 Transforming Equations

Consider the equation g(x) = h(x) and the single-valued function f(x). A particular value of xis a solution of the equation if substituting that value into the equation results in an identity. Indetermining the solutions of an equation, we often apply functions to each side of the equation inorder to simplify its form. We apply the function to obtain a second equation, f(g(x)) = f(h(x)). Ifx = is a solution of the former equation, (let = g() = h()), then it is necessarily a solution oflatter. 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.

7

Suppose that x = is a solution of the latter, f(g()) = f(h()). That f(x) is a one-to-one mappingimplies 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 definedfor that domain). For example, we can apply f(x) = x3 or f(x) = ex, considered as mappings onR, to the equation x = 1. The equations x3 = 1 and ex = e each have the unique solution x = 1 forx R.

In general, we must take care in applying functions to equations. If we apply a many-to-onefunction, we may introduce spurious solutions. Applying f(x) = x2 to the equation x = 2 results inx2 =

2

4 , which has the two solutions, x = {2 }. Applying f(x) = sinx results in x

2 = 2

4 , whichhas an infinite number of solutions, x = {2 + 2n |n Z}.

We do not generally apply a one-to-many, (multi-valued), function to both sides of an equationas this rarely is useful. Rather, we typically use the definition of the inverse function. Consider theequation

sin2 x = 1.

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= sinx, we cannot simplify the left side of the equation. Instead we could use thedefinition of f(x) = x1/2 as the inverse of the x2 function to obtain

sinx = 11/2 = 1.

Now note that we should not just apply arcsin to both sides of the equation as arcsin(sinx) 6= x.Instead we use the definition of arcsin as the inverse of sin.

x = arcsin(1)

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

x ={2+ n | n Z

}

8

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

Exercise 1.2Consider the equation

x+ 1y 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 3 is linearly related to the temperature measured indegrees Fahrenheit 4. Water freezes at 0 C = 32 F and boils at 100 C = 212 F . Write thetemperature in degrees Celsius as a function of degrees Fahrenheit.Hint, Solution

Exercise 1.5Consider the function graphed in Figure 1.9. Sketch graphs of f(x), f(x + 3), f(3 x) + 2, andf1(x). You may use the blank grids in Figure 1.10.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 bacteria are 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 rationalquadratic polynomials. Give possible formulas for p(x) and q(x).Hint, Solution

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

3 Originally, it was called degrees Centigrade. centi because there are 100 degrees between the two calibrationpoints. 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 pointwas 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.

9

Figure 1.9: Graph of the function.

Figure 1.10: Blank grids.

1.6 Hints

Hint 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.

10

1 2

1

2

2 4 6 8 10

1

2

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

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) = x0rt, 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 the leading coefficient of q(x) to be unity.

f(x) =p(x)q(x)

=ax2 + bx+ cx2 + x+

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

Hint 1.8Write the polynomial in factored form.

11

1.7 Solutions

Solution 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+ 1y 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 divisionby zero. There are no solutions for these two values of y and we assume from this point thaty 6= 2. We multiply by (y 2)(y + 2).

(x+ 1)(y + 2) = (x+ 1)(x 1)

For x = 1, the equation becomes the identity 0 = 0. Now we consider x 6= 1. We divide byx+ 1 to obtain the 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.

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Figure 1.12: The solutions of x+1y2 =x21y24 .

12

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

0 0 there exists a > 0 such that|y(x) | < for all 0 < |x | < . That is, there is an interval surrounding the point x = forwhich the function is within of . See Figure 3.1. Note that the interval surrounding x = is adeleted neighborhood, that is it does not contain the point x = . Thus the value of the function atx = need not be equal to for the limit to exist. Indeed the function need not even be defined atx = .

x

y

+

+

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

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

Example 3.1.1 Show that

limx1

x2 = 1.

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

33

|x 1| < . First we obtain 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.

Example 3.1.2 Recall that the value of the function y() need not be equal to limx y(x) for thelimit to exist. We show an example of this. 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). Thisis the limit as x approaches from above. Mathematically: limx+ exists if for any > 0 thereexists a > 0 such that |y(x) | < for all 0 < x < . The left limit limx y(x) is definedanalogously.

Example 3.1.3 Consider the function, sin x|x| , defined for x 6= 0. (See Figure 3.2.) The left and rightlimits exist as x approaches zero.

limx0+

sinx|x|

= 1, limx0

sinx|x|

= 1

However the limit,

limx0

sinx|x|

,

does not exist.

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

34

Properties of Limits. Let limx

f(x) and limx

g(x) exist.

limx

(af(x) + bg(x)) = a limx

f(x) + b limx

g(x).

limx

(f(x)g(x)) =(limx

f(x))(

limx

g(x)).

limx

(f(x)g(x)

)=

limx f(x)limx g(x)

if limx

g(x) 6= 0.

Example 3.1.4 We prove that if limx f(x) = and limx g(x) = exist then

limx

(f(x)g(x)) =(limx

f(x))(

limx

g(x)).

Since the limit exists for f(x), we know that for all > 0 there exists > 0 such that |f(x) | < for |x | < . Likewise for g(x). We seek to show that for all > 0 there exists > 0 such that|f(x)g(x) | < for |x | < . We proceed by writing |f(x)g(x) |, in terms of |f(x) |and |g(x) |, which we know how to bound.

|f(x)g(x) | = |f(x)(g(x) ) + (f(x) )| |f(x)||g(x) |+ |f(x) |||

If we choose a such that |f(x)||g(x) | < /2 and |f(x) ||| < /2 then we will have thedesired result: |f(x)g(x) | < . Trying to ensure that |f(x)||g(x) | < /2 is hard because ofthe |f(x)| factor. We will replace that factor with a constant. We want to write |f(x) ||| < /2as |f(x) | < /(2||), but this is problematic for the case = 0. We fix these two problems andthen proceed. We choose 1 such that |f(x)| < 1 for |x | < 1. This gives us the desired form.

|f(x)g(x) | (||+ 1)|g(x) |+ |f(x) |(||+ 1), for |x | < 1

Next we choose 2 such that |g(x) | < /(2(|| + 1)) for |x | < 2 and choose 3 such that|f(x) | < /(2(||+ 1)) for |x | < 3. Let be the minimum of 1, 2 and 3.

|f(x)g(x) | (||+ 1)|g(x) |+ |f(x) |(||+ 1) < 2+

2, for |x | <

|f(x)g(x) | < , for |x | <

We conclude that the limit of a product is the product of the limits.

limx

(f(x)g(x)) =(limx

f(x))(

limx

g(x))= .

35

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 > 0there exists a > 0 such that |y(x) | < for all x in the neighborhood0 < |x | < . The left and right limits,

limx

y(x) = and limx+

y(x) =

denote the limiting value as x approaches respectively from below and above.The neighborhoods are respectively < x < 0 and 0 < x < .Properties of Limits. Let lim

xu(x) and lim

xv(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 lim

xv(x) 6= 0.

3.2 Continuous Functions

Definition of Continuity. A function y(x) is said to be continuous at x = if the value of thefunction is equal to its limit, that is, limx y(x) = y(). Note that this one condition is actuallythe three conditions: y() is defined, limx y(x) exists and limx y(x) = y(). A function iscontinuous if it is continuous at each point in its domain. A function is continuous on the closedinterval [a, b] if the function is continuous for each point x (a, b) and limxa+ y(x) = y(a) andlimxb y(x) = y(b).

Discontinuous Functions. If a function is not continuous at a point it is called discontinuousat that point. If limx y(x) exists but is not equal to y(), then the function has a removablediscontinuity. It is thus named because we 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 arenot equal, then the function has a jump discontinuity at that point. If either the left or right limitof a function does not exist, then the function is said to have an infinite discontinuity at that point.

Example 3.2.1 sin xx 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.

36

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

Properties of Continuous Functions.

Arithmetic. If u(x) and v(x) are continuous at x = then u(x)v(x) and u(x)v(x) are continuousat 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)) is continuous at x = . The composition of continuous functions is a continuousfunction.

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 ofthe point such that 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] such that u() = . This is known as the intermediate value theorem. A corollary ofthis is that if u(a) and u(b) are of 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, there is at least one point [a, b] such that u() u(x) for all x [a, b] andthere 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 thefunction is bounded on the interval and the interval can be divided into a finite number of intervalson each of which the function is continuous. For example, the greatest integer function, bxc, ispiecewise continuous. (bxc is defined to the the greatest integer less than or equal to x.) SeeFigure 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 meansthat 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 soit is a function of alone and independent of then the function is said to be uniformly continuouson the interval. A sufficient condition for uniform continuity is that the function is continuous on aclosed interval.

37

3.3 The Derivative

Consider a function y(x) on the interval (x . . . x +x) for some x > 0. We define the incrementy = y(x + x) y(x). The average rate of change, (average velocity), of the function on theinterval is yx . The average rate of change is the slope of the secant line that passes through thepoints (x, y(x)) and (x+x, y(x+x)). See Figure 3.5.

y

x

y

x

Figure 3.5: The increments x and y.

If the slope of the secant line has a limit as x approaches zero then we call this slope thederivative or instantaneous rate of change of the function at the point x. We denote the derivativeby dydx , which is a nice notation as the derivative is the limit of

yx as x 0.

dydx

limx0

y(x+x) y(x)x

.

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

(x),y or Dy.

A function is said to be differentiable at a point if the derivative exists there. Note that differ-entiability implies continuity, 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 aboveand below.

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

ddx[x2]= lim

x0

(x+x)2 x2

x= lim

x0(2x+x)

= 2x

38

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.

Properties. Let u(x) and v(x) be differentiable. Let a and b be constants. Some fundamentalproperties of derivatives are:

ddx

(au+ bv) = adudx

+ bdvdx

Linearity

ddx

(uv) =dudxv + u

dvdx

Product Rule

ddx

(uv

)=v dudx u

dvdx

v2Quotient Rule

ddx

(ua) = aua1dudx

Power Rule

ddx

(u(v(x))) =dudv

dvdx

= 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.

ddx

(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)x

v2(x)

=v dudx u

dvdx

v2

39

Trigonometric Functions. Some derivatives of trigonometric functions are:

ddx

sinx = cosxddx

arcsinx =1

(1 x2)1/2ddx

cosx = sinx ddx

arccosx =1

(1 x2)1/2ddx

tanx =1

cos2 xddx

arctanx =1

1 + x2ddx

ex = exddx

lnx =1x

ddx

sinhx = coshxddx

arcsinhx =1

(x2 + 1)1/2

ddx

coshx = sinhxddx

arccoshx =1

(x2 1)1/2ddx

tanhx =1

cosh2 xddx

arctanhx =1

1 x2

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

ddxxx =

ddx

ex ln x

= ex ln xddx

(x lnx)

= xx(1 lnx+ x 1x)

= xx(1 + lnx)

Inverse Functions. If we have a function y(x), we can consider x as a function of y, x(y). Forexample, if y(x) = 8x3 then x(y) = 3

y/2; if y(x) = x+2x+1 then x(y) =

2yy1 . The derivative of an

inverse function isddyx(y) =

1dydx

.

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

dydx

= ex .

Thus the derivative of the logarithm is

ddy

ln y =ddyx(y) =

1dydx

=1ex

=1y.

3.4 Implicit Differentiation

An explicitly defined function has the form y = f(x). A implicitly defined function has the formf(x, y) = 0. A few examples of implicit functions are x2+ y2 1 = 0 and x+ y+sin(xy) = 0. Oftenit is not possible to write an implicit equation in explicit form. This is true of the latter exampleabove. One can calculate the derivative of y(x) in terms of x and y even when y(x) is defined by animplicit equation.

Example 3.4.1 Consider the implicit equation

x2 xy y2 = 1.

40

This implicit equation can be solved for the dependent variable.

y(x) =12

(x

5x2 4

).

We can differentiate this expression to obtain

y =12

(1 5x

5x2 4

).

One can obtain the same result without first solving for y. If we differentiate the implicit equation,we obtain

2x y xdydx

2y dydx

= 0.

We can solve this equation for dydx .dydx

=2x yx+ 2y

We can differentiate this expression to obtain the second derivative of y.

d2ydx2

=(x+ 2y)(2 y) (2x y)(1 + 2y)

(x+ 2y)2

=5(y xy)(x+ 2y)2

Substitute in the expression for y.

= 10(x2 xy y2)(x+ 2y)2

Use the original implicit equation.

= 10(x+ 2y)2

3.5 Maxima and Minima

A differentiable function is increasing where f (x) > 0, decreasing where f (x) < 0 and stationarywhere f (x) = 0.

A function f(x) has a relative maxima at a point x = if there exists a neighborhood around such that f(x) f() for x (x , x + ), > 0. The relative minima is defined analogously.Note that this definition does not require that the function be differentiable, or even continuous.We refer to relative maxima and minima collectively are relative extrema.

Relative Extrema and Stationary Points. If f(x) is differentiable and f() is a relative ex-trema then x = is a stationary point, f () = 0. We can prove this using left and right limits.Assume that f() is a relative maxima. Then there is a neighborhood (x , x+ ), > 0 for whichf(x) f(). Since f(x) is differentiable the derivative at x = ,

f () = limx0

f( +x) f()x

,

exists. This in turn means that the left and right limits exist and are equal. Since f(x) f() for < x < the left limit is non-positive,

f () = limx0

f( +x) f()x

0.

41

Since f(x) f() for < x < + the right limit is nonnegative,

f () = limx0+

f( +x) f()x

0.

Thus we have 0 f () 0 which implies that f () = 0.

It is not true that all stationary points are relative extrema. That is, f () = 0 does not implythat x = is an extrema. Consider the function f(x) = x3. x = 0 is a stationary point sincef (x) = x2, f (0) = 0. However, x = 0 is neither a relative maxima nor a relative minima.

It is also not true that all relative extrema