Introduction to Methods of Applied Mathematicsor

Advanced Mathematical Methods for Scientists and Engineers

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

January 24, 2004

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

Contents

Anti-Copyright xxiv

Preface xxv0.1 Advice to Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv0.2 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv0.3 Warnings and Disclaimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi0.4 Suggested Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii0.5 About the Title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii

I Algebra 1

1 Sets and Functions 21.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Single Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Inverses and Multi-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Transforming Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 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 47

3 Differential Calculus 483.1 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.6 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.6.1 Application: Using Taylors Theorem to Approximate Functions. . . . . . . . . . . . . . . . . . 683.6.2 Application: Finite Difference Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.7 LHospitals Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.8.1 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.8.2 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.8.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.8.4 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.8.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.8.6 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

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3.8.7 LHospitals Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.10 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.11 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.12 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4 Integral Calculus 1164.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.3 The Fundamental Theorem of Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.4 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.4.1 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.6.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.6.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.6.3 The Fundamental Theorem of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.6.4 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.6.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.9 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1504.10 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5 Vector Calculus 1545.1 Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545.2 Gradient, Divergence and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

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5.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1685.6 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1775.7 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

III Functions of a Complex Variable 179

6 Complex Numbers 1806.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.2 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1846.3 Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1886.4 Arithmetic and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.5 Integer Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.6 Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1976.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2016.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2086.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

7 Functions of a Complex Variable 2397.1 Curves and Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2397.2 The Point at Infinity and the Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . . . . 2427.3 A Gentle Introduction to Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2467.4 Cartesian and Modulus-Argument Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2467.5 Graphing Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2497.6 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2527.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.8 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2687.9 Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2707.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

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7.11 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2977.12 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

8 Analytic Functions 3608.1 Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3608.2 Cauchy-Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3678.3 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3728.4 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

8.4.1 Categorization of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3778.4.2 Isolated and Non-Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

8.5 Application: Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3838.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3888.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3968.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

9 Analytic Continuation 4379.1 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4379.2 Analytic Continuation of Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4409.3 Analytic Functions Defined in Terms of Real Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 442

9.3.1 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4469.3.2 Analytic Functions Defined in Terms of Their Real or Imaginary Parts . . . . . . . . . . . . . . 450

9.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4549.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4569.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

10 Contour Integration and the Cauchy-Goursat Theorem 46210.1 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46210.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

10.2.1 Maximum Modulus Integral Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46610.3 The Cauchy-Goursat Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

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10.4 Contour Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46910.5 Moreras Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47110.6 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47310.7 Fundamental Theorem of Calculus via Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

10.7.1 Line Integrals and Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47410.7.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

10.8 Fundamental Theorem of Calculus via Complex Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 47510.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47810.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48210.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

11 Cauchys Integral Formula 49311.1 Cauchys Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49411.2 The Argument Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50111.3 Rouches Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50211.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50511.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50911.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

12 Series and Convergence 52512.1 Series of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

12.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52512.1.2 Special Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52712.1.3 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

12.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53612.2.1 Tests for Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53712.2.2 Uniform Convergence and Continuous Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 539

12.3 Uniformly Convergent Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53912.4 Integration and Differentiation of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54712.5 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

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12.5.1 Newtons Binomial Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55312.6 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55512.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560

12.7.1 Series of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56012.7.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56612.7.3 Uniformly Convergent Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56612.7.4 Integration and Differentiation of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . 56812.7.5 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56912.7.6 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

12.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57412.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

13 The Residue Theorem 62613.1 The Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62613.2 Cauchy Principal Value for Real Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634

13.2.1 The Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63413.3 Cauchy Principal Value for Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63913.4 Integrals on the Real Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64313.5 Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64713.6 Fourier Cosine and Sine Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64913.7 Contour Integration and Branch Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65213.8 Exploiting Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655

13.8.1 Wedge Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65513.8.2 Box Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658

13.9 Definite Integrals Involving Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65913.10Infinite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66213.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66613.12Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68013.13Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686

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IV Ordinary Differential Equations 772

14 First Order Differential Equations 77314.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77314.2 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775

14.2.1 Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77514.3 One Parameter Families of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77714.4 Integrable Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779

14.4.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78014.4.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78214.4.3 Homogeneous Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786

14.5 The First Order, Linear Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79114.5.1 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79114.5.2 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79214.5.3 Variation of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

14.6 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79614.6.1 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . 797

14.7 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80114.8 Equations in the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

14.8.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80314.8.2 Regular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80614.8.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81214.8.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814

14.9 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81614.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81914.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82214.12Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84314.13Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844

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15 First Order Linear Systems of Differential Equations 84615.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84615.2 Using Eigenvalues and Eigenvectors to find Homogeneous Solutions . . . . . . . . . . . . . . . . . . . 84715.3 Matrices and Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85215.4 Using the Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86015.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86515.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87015.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872

16 Theory of Linear Ordinary Differential Equations 90016.1 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90016.2 Nature of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90116.3 Transformation to a First Order System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90516.4 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905

16.4.1 Derivative of a Determinant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90516.4.2 The Wronskian of a Set of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90616.4.3 The Wronskian of the Solutions to a Differential Equation . . . . . . . . . . . . . . . . . . . . 908

16.5 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91116.6 The Fundamental Set of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91316.7 Adjoint Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91516.8 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91916.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92016.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92216.11Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92816.12Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929

17 Techniques for Linear Differential Equations 93017.1 Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 930

17.1.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93117.1.2 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935

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17.1.3 Higher Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93717.2 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 940

17.2.1 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94217.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94517.4 Equations Without Explicit Dependence on y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94617.5 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94717.6 *Reduction of Order and the Adjoint Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94817.7 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95117.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95717.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 960

18 Techniques for Nonlinear Differential Equations 98418.1 Bernoulli Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98418.2 Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98618.3 Exchanging the Dependent and Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 99018.4 Autonomous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99218.5 *Equidimensional-in-x Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99518.6 *Equidimensional-in-y Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99718.7 *Scale-Invariant Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100018.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100118.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100418.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006

19 Transformations and Canonical Forms 101819.1 The Constant Coefficient Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101819.2 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021

19.2.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102119.2.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022

19.3 Transformations of the Independent Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102419.3.1 Transformation to the form u + a(x) u = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024

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19.3.2 Transformation to a Constant Coefficient Equation . . . . . . . . . . . . . . . . . . . . . . . . 102519.4 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027

19.4.1 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102719.4.2 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029

19.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103219.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103419.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035

20 The Dirac Delta Function 104120.1 Derivative of the Heaviside Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104120.2 The Delta Function as a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104320.3 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104520.4 Non-Rectangular Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104620.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104820.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105020.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052

21 Inhomogeneous Differential Equations 105921.1 Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105921.2 Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106121.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065

21.3.1 Second Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106521.3.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068

21.4 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . 107121.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074

21.5.1 Eliminating Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 107421.5.2 Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions . . . . . . . . . 107621.5.3 Existence of Solutions of Problems with Inhomogeneous Boundary Conditions . . . . . . . . . . 1077

21.6 Green Functions for First Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107921.7 Green Functions for Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082

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21.7.1 Green Functions for Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 109221.7.2 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109521.7.3 Problems with Unmixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 109821.7.4 Problems with Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1100

21.8 Green Functions for Higher Order Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110421.9 Fredholm Alternative Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110921.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111721.11Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112321.12Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112621.13Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116421.14Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165

22 Difference Equations 116622.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116622.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116822.3 Homogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116922.4 Inhomogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117122.5 Homogeneous Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117422.6 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117722.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117922.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118022.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1181

23 Series Solutions of Differential Equations 118423.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184

23.1.1 Taylor Series Expansion for a Second Order Differential Equation . . . . . . . . . . . . . . . . 118823.2 Regular Singular Points of Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198

23.2.1 Indicial Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120123.2.2 The Case: Double Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120323.2.3 The Case: Roots Differ by an Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206

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23.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121623.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121623.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121923.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122423.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122523.8 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124823.9 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249

24 Asymptotic Expansions 125124.1 Asymptotic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125124.2 Leading Order Behavior of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125524.3 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126324.4 Asymptotic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127024.5 Asymptotic Expansions of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272

24.5.1 The Parabolic Cylinder Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272

25 Hilbert Spaces 127825.1 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127825.2 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128025.3 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128125.4 Linear Independence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128325.5 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128325.6 Gramm-Schmidt Orthogonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128425.7 Orthonormal Function Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128725.8 Sets Of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128825.9 Least Squares Fit to a Function and Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129425.10Closure Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129725.11Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130225.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130325.13Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304

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25.14Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305

26 Self Adjoint Linear Operators 130726.1 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130726.2 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130826.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131126.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131226.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313

27 Self-Adjoint Boundary Value Problems 131427.1 Summary of Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131427.2 Formally Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131527.3 Self-Adjoint Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131827.4 Self-Adjoint Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131827.5 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132327.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132627.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132727.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328

28 Fourier Series 133028.1 An Eigenvalue Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133028.2 Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133328.3 Least Squares Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133728.4 Fourier Series for Functions Defined on Arbitrary Ranges . . . . . . . . . . . . . . . . . . . . . . . . . 134128.5 Fourier Cosine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134428.6 Fourier Sine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134528.7 Complex Fourier Series and Parsevals Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134628.8 Behavior of Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134928.9 Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135828.10Integrating and Differentiating Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358

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28.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136328.12Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137128.13Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373

29 Regular Sturm-Liouville Problems 142029.1 Derivation of the Sturm-Liouville Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142029.2 Properties of Regular Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142229.3 Solving Differential Equations With Eigenfunction Expansions . . . . . . . . . . . . . . . . . . . . . . 143329.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143929.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144329.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445

30 Integrals and Convergence 147030.1 Uniform Convergence of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147030.2 The Riemann-Lebesgue Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147130.3 Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1472

30.3.1 Integrals on an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147230.3.2 Singular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473

31 The Laplace Transform 147531.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147531.2 The Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477

31.2.1 f(s) with Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148031.2.2 f(s) with Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148431.2.3 Asymptotic Behavior of f(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488

31.3 Properties of the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148931.4 Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149231.5 Systems of Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 149531.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149731.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1504

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31.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507

32 The Fourier Transform 153932.1 Derivation from a Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153932.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1541

32.2.1 A Word of Caution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154432.3 Evaluating Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545

32.3.1 Integrals that Converge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154532.3.2 Cauchy Principal Value and Integrals that are Not Absolutely Convergent. . . . . . . . . . . . . 154832.3.3 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1550

32.4 Properties of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155232.4.1 Closure Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155232.4.2 Fourier Transform of a Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155332.4.3 Fourier Convolution Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155432.4.4 Parsevals Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155732.4.5 Shift Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155932.4.6 Fourier Transform of x f(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1559

32.5 Solving Differential Equations with the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 156032.6 The Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1562

32.6.1 The Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156232.6.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563

32.7 Properties of the Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 156432.7.1 Transforms of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156432.7.2 Convolution Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156632.7.3 Cosine and Sine Transform in Terms of the Fourier Transform . . . . . . . . . . . . . . . . . . 1568

32.8 Solving Differential Equations with the Fourier Cosine and Sine Transforms . . . . . . . . . . . . . . . 156932.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157132.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157832.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1581

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33 The Gamma Function 160533.1 Eulers Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160533.2 Hankels Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160733.3 Gauss Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160933.4 Weierstrass Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161133.5 Stirlings Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161333.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161833.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161933.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1620

34 Bessel Functions 162234.1 Bessels Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162234.2 Frobeneius Series Solution about z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1623

34.2.1 Behavior at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162634.3 Bessel Functions of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1628

34.3.1 The Bessel Function Satisfies Bessels Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 162934.3.2 Series Expansion of the Bessel Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163034.3.3 Bessel Functions of Non-Integer Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163334.3.4 Recursion Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163634.3.5 Bessel Functions of Half-Integer Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1639

34.4 Neumann Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164034.5 Bessel Functions of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164434.6 Hankel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164634.7 The Modified Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164634.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165034.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165534.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1657

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V Partial Differential Equations 1680

35 Transforming Equations 168135.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168235.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168335.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1684

36 Classification of Partial Differential Equations 168536.1 Classification of Second Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1685

36.1.1 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168636.1.2 Parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169136.1.3 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1692

36.2 Equilibrium Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169436.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169636.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169736.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698

37 Separation of Variables 170437.1 Eigensolutions of Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170437.2 Homogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . 170437.3 Time-Independent Sources and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 170637.4 Inhomogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 170937.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171037.6 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171337.7 General Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171637.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171837.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173437.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1739

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38 Finite Transforms 182138.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182538.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182638.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827

39 The Diffusion Equation 183139.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183239.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183439.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1835

40 Laplaces Equation 184140.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184140.2 Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1841

40.2.1 Two Dimensional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184240.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184340.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184640.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1847

41 Waves 185941.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186041.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186641.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868

42 Similarity Methods 188842.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189242.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189342.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1894

43 Method of Characteristics 189743.1 First Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189743.2 First Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1898

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43.3 The Method of Characteristics and the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 190043.4 The Wave Equation for an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190143.5 The Wave Equation for a Semi-Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190243.6 The Wave Equation for a Finite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190443.7 Envelopes of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190543.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190843.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191043.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1911

44 Transform Methods 191844.1 Fourier Transform for Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191844.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192044.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192044.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192244.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192644.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1928

45 Green Functions 195045.1 Inhomogeneous Equations and Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 195045.2 Homogeneous Equations and Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 195145.3 Eigenfunction Expansions for Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195345.4 The Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195845.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196045.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197145.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1974

46 Conformal Mapping 203446.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203546.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203846.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2039

xx

47 Non-Cartesian Coordinates 205147.1 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205147.2 Laplaces Equation in a Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205247.3 Laplaces Equation in an Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2055

VI Calculus of Variations 2059

48 Calculus of Variations 206048.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206148.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207548.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2079

VII Nonlinear Differential Equations 2166

49 Nonlinear Ordinary Differential Equations 216749.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216849.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217349.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2174

50 Nonlinear Partial Differential Equations 219650.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219750.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220050.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2201

VIII Appendices 2220

A Greek Letters 2221

xxi

B Notation 2223

C Formulas from Complex Variables 2225

D Table of Derivatives 2228

E Table of Integrals 2232

F Definite Integrals 2236

G Table of Sums 2238

H Table of Taylor Series 2241

I Continuous Transforms 2244I.1 Properties of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2244I.2 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2247I.3 Table of Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2250I.4 Table of Fourier Transforms in n Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2253I.5 Table of Fourier Cosine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2254I.6 Table of Fourier Sine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2255

J Table of Wronskians 2257

K Sturm-Liouville Eigenvalue Problems 2259

L Green Functions for Ordinary Differential Equations 2261

M Trigonometric Identities 2264M.1 Circular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2264M.2 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2266

xxii

N Bessel Functions 2269N.1 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2269

O Formulas from Linear Algebra 2270

P Vector Analysis 2271

Q Partial Fractions 2273

R Finite Math 2276

S Physics 2277

T Probability 2278T.1 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2278T.2 Playing the Odds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2279

U Economics 2280

V Glossary 2281

W whoami 2283

xxiii

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.

xxiv

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.

xxv

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@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.

xxvi

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.

xxvii

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} is the set of the integer 1, the pure imaginary number =

1 and the

transcendental numbers e = 2.7182818 . . . and = 3.1415926 . . .. For elements of a set, we do not count multiplicities.We regard the set {1, 2, 2, 3, 3, 3} as identical to the set {1, 2, 3}. Order is not significant in sets. The set {1, 2, 3} isequivalent to {3, 2, 1}.

In enumerating the elements of a set, we use ellipses to indicate patterns. We denote the set of positive integers as{1, 2, 3, . . .}. We also denote sets with the notation {x|conditions on x} for sets that are more easily described thanenumerated. This is read as the set of elements x such that . . . . x S is the notation for x is an element of theset 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 = {. . . ,3,2,1, 0, 1, 2, 3 . . .} is the set of integers. (Z is for Zahlen, the German word for number.)

2

Q = {p/q|p, q Z, q 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 decimal expansions. 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, . . .}.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. 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.

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

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 Snis 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 a

4

particular element we can use any of the notations: f(x) = y, f : x 7 y or simply x 7 y. f is the identity map 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}.

Examples.

Finite polynomials, f(x) =n

k=0 akxk, ak R, and the exponential function, f(x) = ex, are examples 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 the greatest integer lessthan or equal to x. Likewise, the least integer function, f(x) = dxe, is the least integer greater than or equal tox.

The -jectives. A function is injective if for each x1 6= x2, f(x1) 6= f(x2). In other words, distinct elements aremapped to distinct elements. f is surjective if for each y in the codomain, there is an x such that y = f(x). If afunction is both injective and surjective, then it is bijective. A bijective function is also called a one-to-one mapping.

Examples.

The exponential function f(x) = ex, considered as a mapping from R to R+, is bijective, (a one-to-one mapping).

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

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

5

Injective Surjective Bijective

Figure 1.1: Depictions of Injective, Surjective and Bijective Functions

1.3 Inverses and Multi-Valued Functions

If y = f(x), then we can write x = 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 lnx, which maps R+ to R is the inverse of ex. x = elnx = 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 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.

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.

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.

6

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

Figure 1.4: y =x

7

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 = sinx and a graph of a few branchesof y = arcsin x.

Figure 1.5: y = sinx and y = arcsin x

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 = 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.

Example 1.3.4 Consider arccos(1/2). cosx and a portion of arccos x are graphed in Figure 1.8. The equationcosx = 1/2 has the two solutions x = /3 in the range x (..]. We use the periodicity of the cosine,

8

Figure 1.7: y = x3 and y = x1/3

cos(x+ 2) = cosx, to find the remaining solutions.

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

Figure 1.8: y = cos x 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 x is a solution of theequation if substituting that value into the equation results in an identity. In determining the solutions of an equation,we often apply functions to each side of the equation in order to simplify its form. We apply the function to obtaina second equation, f(g(x)) = f(h(x)). If x = is a solution of the former equation, (let = g() = h()), then it

9

is necessarily a solution of latter. This is because f(g()) = f(h()) reduces to the identity f() = f(). If f(x) isbijective, then the converse is true: any solution of the latter equation is a solution of the former equation. Supposethat x = is a solution of the latter, f(g()) = f(h()). That f(x) is a one-to-one mapping implies that g() = h().Thus x = is a solution of the former equation.

It is always safe to apply a one-to-one, (bijective), function to an equation, (provided it is defined for that domain).For example, we can apply f(x) = x3 or f(x) = ex, considered as mappings on R, to the equation x = 1. Theequations x3 = 1 and ex = e each have the unique solution x = 1 for x R.

In general, we must take care in applying functions to equations. If we apply a many-to-one function, we mayintroduce spurious solutions. Applying f(x) = x2 to the equation x =

2results in x2 =

2

4, which has the two solutions,

x = {2}. Applying f(x) = sin x results in x2 = 2

4, which has 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 equation as this rarely is useful.Rather, we typically use the definition of the inverse function. Consider the equation

sin2 x = 1.

Applying the function f(x) = x1/2 to the equation would not get us anywhere.(sin2 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.Now note that we should not just apply arcsin to both sides of the equation as arcsin(sinx) 6= x. Instead we use thedefinition 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 enumeratethe solutions.

x ={

2+ n | n Z

}

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 3 is linearly related to the temperature measured in degrees Fahrenheit 4.Water freezes at 0 C = 32 F and boils at 100 C = 212 F . Write the temperature in degrees Celsius as a functionof degrees Fahrenheit.

3 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.

4 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

13

1 2

1

2

2 4 6 8 10

1

2

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

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.

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.

14

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 all 0 < |x | < . That is, there is an interval surrounding the point x = for which the function is within of. See Figure 3.1. Note that the interval surrounding x = is a deleted neighborhood, that is it does not contain thepoint x = . Thus the value of the function at x = need not be equal to for the limit to exist. Indeed the functionneed not even 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 .

48

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.

49

Example 3.1.2 Recall that the value of the function y() need not be equal to limx y(x) for the limit to exist. Weshow 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). 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

sin x

|x|,

does not exist.

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).

50

Figure 3.2: Plot of sin(x)/|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 the desired result:|f(x)g(x)| < . Trying to ensure that |f(x)||g(x)| < /2 is hard because of the |f(x)| factor. We will replacethat factor with a constant. We want to write |f(x)||| < /2 as |f(x)| < /(2||), but this is problematic forthe case = 0. We fix these two problems and then proceed. We choose 1 such that |f(x) | < 1 for |x | < 1.

51

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)

)= .

52

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 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 the function isequal to its limit, that is, limx y(x) = y(). Note that this one condition is actually the three conditions: y() is

53

defined, 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.

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.

54

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

B