The Laplace Transformation

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 8
Contents......Page 10
1. Absolutely Integrable Piecewise Continuous Functions......Page 12
2. The Fourier Transform of an Absolutely Integrable Piecewise Continuous Function......Page 16
3. The Fourier Integral Theorem......Page 20
4. Completion of The Proof of The Fourier Integral Theorem. The Laplace Version of The Fourier Integral Theorem......Page 26
5. The Laplace Transform of a Right-sided Function......Page 33
6. The Laplace Transform of exp (- t2)......Page 39
7. The Laplace Transform of the Product of a Right-sided Function by t and of the Integral of a Right-sided Function over the Interval [0,1]......Page 46
8. Functions of Exponential Type......Page 51
9. The Characterization of Functions of Exponential Type......Page 56
10. The Polynomials of Laguerre......Page 62
11. Bessel's Differential Equation......Page 68
12. The Recurrence and other Relations Connecting Bessel Functions.......Page 76
13. The Problem of the Vibrating String......Page 86
14. The Solution of the Problem of the Vibrating String......Page 92
15. The Generalized Vibrating String Problem......Page 98
16. The Solution of the Generalized Vibrating String Problem......Page 105
17. The Asymptotic Series for j; exp (-0) dz......Page 110
18. The Asymptotic Series for (27p)i exp (- p)I (p), arg p 1 < it/2, The Hankel Functions......Page 117
19. The Asymptotic Series for and......Page 125
Bibliography......Page 135
Index......Page 136