دانلود کتاب Laplace Transform Theory

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Preface CHAPTER 1 THE D-OPERATOR 1 1.1 Operational Techniques 1 1.2 Simple D-algebra 2 1.3 The Inverse Operator 4 1.4 Expansion of the Inverse Operator 6 Worked Examples 7 Exercises 10 Answers 11 CHAPTER 2 THE LAPLACE INTEGRAL 12 2.1 Conditions for Existence 12 2.2 Abscissae of Convergence 13 2.3 Elementary Transforms 14 2.4 Simple Applications 17 Worked Examples 18 Exercises 20 Answers 21 CHAPTER 3 SIMPLE PROPERTIES OF THE LAPLACE TRANSFORM 22 3.1 Linear Property 22 3.2 Change of Scale 22 3.3 Shift of the Origin 23 3.4 Periodic Functions 23 3.5 Multiplication by a Simple Power 24 3.6 Expressions involving Inverse Powers of t 25 3.7 Transform of Integrals 25 3.8 Transform of an Integral Function 26 3.9 Examples illustrating 3.8 27 3.10 Behaviour for Large Isl 28 Exercises 28 Answers 29 CHAPTER 4 THE INVERSE INTEGRAL 30 4.1 The Inversion Problem 30 4.2 The Fourier Transform 30 4.3 The Inverse as a Contour Integral 32 4.4 Examples of Inversion 35 Exercises 40 CHAPTER 5 THE CONVOLUTION OR FALTUNG INTEGRAL 43 5.1 The Repeated Integral 43 5.2 Simple Application to Differential Equations 45 5.3 A Generalization of 5.2 46 5.4 Transform of a Product 48 5.5 A Simple Illustration 49 Exercises 49 Answers 50 CHAPTER 6 APPLICATION TO ORDINARY LINEAR DIFFERENTIAL EQUATIONS 51 6.1 Limitations of the Technique 51 6.2 The Bessel Equation 52 6.3 Another Related Equation 53 6.4 The Laguerre Equation 53 6.5 The Hermite Equation 57 6.6 The Laplace Linear Equation 58 6.7 A Further Illustrative Example 60 Exercises 61 Answers 62 CHAPTER 7 PARTIAL DIFFERENTIAL EQUATIONS 64 7.1 Limitations of the Technique 64 7.2 A Simple Equation of the First Order 65 7.3 The Diffusion Equation 66 7.4 The Wave Equation 68 7.4.1 The infinite string 68 7.4.2 The finite string 69 7.4.3 The axially-symmetric equation 70 7.4..4 The equation in three dimensions 71 Exercises 73 Answers 74 CHAPTER 8 LINEAR INTEGRAL EQUATIONS 8.1 Types of Integral Equations 8.2 The Difference Kernel 8.2.1 The equation of the first kind 8.2.2 The equation of the second kind 8.3 The Wiener-Hopf Technique 8.4 The Milne Equation Exercises Answers CHAPTER 9 LINEAR DIFFERENCE EQUATIONS 9.1 The Difference Equations 9.2 Transform of a Step Function 9.3 Simple Applications Worked Examples 9.4 Inhomogeneous Equations Worked Examples 9 .5 Variable Coefficients Worked Examples Exercises Answers CHAPTER 10 ASYMPTOTIC FORMULAE 10.1 Elementary Asymptotic Results 10.2 Asymptotic Form of a Transform for Large (s( 10.3 Asymptotic Series of a Transform for Large Isf 10.4 Application to the Exponential Integral 10.5 Asymptotic Behaviour off(t) 10.5.1 Single-valued function 10.5.2 Function with a branch point Worked Examples 10.6 Terms involving Logarithms A Simple Example Exercises Answers APPENDIX 1 THE FOURIER INTEGRAL THEOREM 107 APPENDIX 2 THE EXPONENTIAL INTEGRAL 112 APPENDIX 3 OPERATIONAL FORMULAE 115 APPENDIX 4 TABLE OF THE MORE COMMON LAPLACE TRANSFORMS 118 References 120 Index 121