Page 8 - Essentials of applied mathematics for scientists and engineers

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

viii ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
6.2.7 Heaviside Step .................................................. 99
6.2.8 The Dirac Delta Function ....................................... 100
6.2.9 Transforms of Derivatives ....................................... 100
6.2.10 Laplace Transforms of Integrals .................................. 101
6.2.11 Derivatives of Transforms . ...................................... 101
6.3 Linear Ordinary Differential Equations with Constant Coefficients . ........ 102
6.4 Some Important Theorems ............................................. 103
6.4.1 Initial Value Theorem...........................................103
6.4.2 Final Value Theorem ........................................... 103
6.4.3 Convolution ................................................... 103
6.5 Partial Fractions ....................................................... 104
6.5.1 Nonrepeating Roots ............................................ 104
6.5.2 Repeated Roots. ................................................107
6.5.3 Quadratic Factors: Complex Roots . .............................. 108
Problems . . . ................................................ 109
Further Reading ....................................................... 110

7. Complex Variables and the Laplace Inversion Integral ......................... 111
7.1 Basic Properties ........................................................ 111
7.1.1 Limits and Differentiation of Complex Variables:
Analytic Functions. .............................................115
Integrals .................................................... 117
7.1.2 The Cauchy Integral Formula....................................118
Problems . . . ................................................ 120

8. Solutions with Laplace Transforms. ..........................................121
8.1 Mechanical Vibrations. .................................................121
Problems . . . ................................................ 125
8.2 Diffusion or Conduction Problems . . . ................................... 125
Problems . . . ................................................ 134
8.3 Duhamel’s Theorem....................................................135
Problems . . . ................................................ 138
Further Reading ....................................................... 139

9. Sturm–Liouville Transforms. . ...............................................141
9.1 A Preliminary Example: Fourier Sine Transform ..........................141
9.2 Generalization: The Sturm–Liouville Transform: Theory .................. 143
9.3 The Inverse Transform ................................................. 146