An Introduction to Laplace Transforms and Fourier Series

The Laplace transform is a useful tool for solving practical mathematical problems. It has a wealth of applications (for example, engineers use it to solve engineering problems) and is often known to applied mathematicians as operational calculus. This book takes introduces the basics and demands no more of the reader than a knowledge of elementary caclulus.

Written by a very experienced author in undergraduate mathematics Based on the course the author is teaching at the present time Also accessible to engineers and physicists with a relevant mathematical background Includes supplementary material: sn.pub/extras

Auteur
The Laplace transform is a useful tool for solving practical mathematical
problems. It has a wealth of applications (for
example, engineers use it to solve engineering problems) and is often
known to applied mathematicians as operational
calculus. This book takes introduces the basics and demands no more of the
reader than a knowledge of elementary
caclulus.

Contenu

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 Heaviside's 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.