Page 7 - Essentials of applied mathematics for scientists and engineers

MOBK070-FM MOBKXXX-Sample.cls March 22, 2007 13:6

CONTENTS vii
4.1.1 Definitions . ..................................................... 45
4.1.2 Ordinary Points and Series Solutions .............................. 46
4.1.3 Lessons: Finding Series Solutions for Differential Equations
with Ordinary Points.............................................48
Problems . . .................................................. 48
4.1.4 Regular Singular Points and the Method of Frobenius............... 49
4.1.5 Lessons: Finding Series Solution for Differential Equations with
Regular Singular Points .......................................... 54
4.1.6 Logarithms and Second Solutions . ................................ 55
Problems . . .................................................. 57
4.2 Bessel Functions ........................................................ 58
4.2.1 Solutions of Bessel’s Equation. ....................................58
Here are the Rules ............................................ 61
4.2.2 Fourier–Bessel Series. ............................................64
Problems . . .................................................. 68
4.3 Legendre Functions . .................................................... 69
4.4 Associated Legendre Functions . .......................................... 72
Problems . . .................................................. 73
Further Reading ........................................................ 74

5. Solutions Using Fourier Series and Integrals ................................... 75
5.1 Conduction (or Diffusion) Problems . ..................................... 75
5.1.1 Time-Dependent Boundary Conditions. ...........................80
5.2 Vibrations Problems ..................................................... 83
Problems . . .................................................. 88
5.3 Fourier Integrals ........................................................ 89
Problem . . . .................................................. 93
Further Reading ........................................................ 93

6. Integral Transforms: The Laplace Transform .................................. 95
6.1 The Laplace Transform .................................................. 95
6.2 Some Important Transforms ............................................. 96
6.2.1 Exponentials . ................................................... 96
6.2.2 Shifting in the s -domain . ........................................ 96
6.2.3 Shifting in the Time Domain ..................................... 96
6.2.4 Sine and Cosine ................................................. 97
6.2.5 Hyperbolic Functions . ........................................... 97
m
6.2.6 Powers of t: t .................................................. 97