Author | Dyke, Philip P. G. author |
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Title | An Introduction to Laplace Transforms and Fourier Series [electronic resource] / by Philip P. G. Dyke |
Imprint | London : Springer London : Imprint: Springer, 2001 |
Connect to | http://dx.doi.org/10.1007/978-1-4471-0505-3 |
Descript | XII, 250 p. 2 illus. online resource |
1. The Laplace Transform -- 1.1 Introduction -- 1.2 The Laplace Transform -- 1.3 Elementary Properties -- 1.4 Exercises -- 2. Further Properties of the Laplace Transform -- 2.1 Real Functions -- 2.2 Derivative Property of the Laplace Transform -- 2.3 Heavisideโs Unit Step Function -- 2.4 Inverse Laplace Transform -- 2.5 Limiting Theorems -- 2.6 The Impulse Function -- 2.7 Periodic Functions -- 2.8 Exercises -- 3. Convolution and the Solution of Ordinary Differential Equations -- 3.1 Introduction -- 3.2 Convolution -- 3.3 Ordinary Differential Equations -- 3.3.1 Second Order Differential Equations -- 3.3.2 Simultaneous Differential Equations -- 3.4 Using Step and Impulse Functions -- 3.5 Integral Equations -- 3.6 Exercises -- 4. Fourier Series -- 4.1 Introduction -- 4.2 Definition of a Fourier Series -- 4.3 Odd and Even Functions -- 4.4 Complex Fourier Series -- 4.5 Half Range Series -- 4.6 Properties of Fourier Series -- 4.7 Exercises -- 5. Partial Differential Equations -- 5.1 Introduction -- 5.2 Classification of Partial Differential Equations -- 5.3 Separation of Variables -- 5.4 Using Laplace Transforms to Solve PDEs -- 5.5 Boundary Conditions and Asymptotics -- 5.6 Exercises -- 6. Fourier Transforms -- 6.1 Introduction -- 6.2 Deriving the Fourier Transform -- 6.3 Basic Properties of the Fourier Transform -- 6.4 Fourier Transforms and PDEs -- 6.5 Signal Processing -- 6.6 Exercises -- 7. Complex Variables and Laplace Transforms -- 7.1 Introduction -- 7.2 Rudiments of Complex Analysis -- 7.3 Complex Integration -- 7.4 Branch Points -- 7.5 The Inverse Laplace Transform -- 7.6 Using the Inversion Formula in Asymptotics -- 7.7 Exercises -- A. Solutions to Exercises -- B. Table of Laplace Transforms -- C. Linear Spaces -- C.1 Linear Algebra -- C.2 Gramm-Schmidt Orthonormalisation Process