A Student's Guide to Laplace Transforms

Description

The Laplace transform is a useful mathematical tool encountered by students of physics, engineering, and applied mathematics, within a wide variety of important applications in mechanics, electronics, thermodynamics and more. However, students often struggle with the rationale behind these transforms, and the physical meaning of the transform results. Using the same approach that has proven highly popular in his other Student's Guides, Professor Fleisch addresses the topics that his students have found most troublesome; providing a detailed and accessible description of Laplace transforms and how they relate to Fourier and Z-transforms. Written in plain language and including numerous, fully worked examples. The book is accompanied by a website containing a rich set of freely available supporting materials, including interactive solutions for every problem in the text, and a series of podcasts in which the author explains the important concepts, equations, and graphs of every section of the book. Cover Half-title Series information Title page Copyright information About this book Contents Preface Acknowledgments 1 The Fourier and Laplace Transforms 1.1 Definition of the Laplace Transform 1.2 Phasors and Frequency Spectra 1.3 How These Transforms Work 1.4 Transforms as Inner Products 1.5 Relating Laplace F(s) to Fourier F(ω) 1.6 Inverse Transforms 1.7 Problems 2 Laplace-Transform Examples 2.1 Constant Functions 2.2 Exponential Functions 2.3 Sinusoidal Functions 2.4 t[sup(n)] Functions 2.5 Hyperbolic Functions 2.6 Problems 3 Properties of the Laplace Transform 3.1 Linearity 3.2 Time and Frequency Shifting 3.3 Scaling 3.4 Time-Domain Differentiation 3.5 Time-Domain Integration 3.6 Multiplication and Division of f(t) by t 3.7 Transform of Periodic Functions 3.8 Convolution 3.9 Initial- and Final-Value Theorems 3.10 Problems 4 Applications of the Laplace Transform 4.1 Differential Equations 4.2 Mechanical Oscillations 4.3 Electric-Circuit Oscillations 4.4 Heat Flow 4.5 Waves 4.6 Transmission Lines 4.7 Problems 5 The Z-Transform 5.1 Introduction to the Z-Transform 5.2 Examples of the Z-transform 5.3 Characteristics of the Z-transform 5.4 Problems Further Reading Index