P.P.G. Dyke

An Introduction to Laplace Transforms and Fourier Series

• Broschiertes Buch

Produktbeschreibung

This introduction to Laplace transforms and Fourier series is aimed at second year students in applied mathematics. It is unusual in treating Laplace transforms at a relatively simple level with many examples. Mathematics students do not usually meet this material until later in their degree course but applied mathematicians and engineers need an early introduction. Suitable as a course text, it will also be of interest to physicists and engineers as supplementary material.

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This book has been primarily written for the student of mathematics who is in the second year or the early part of the third year of an undergraduate course. It will also be very useful for students of engineering and the physical sciences for whom Laplace Transforms continue to be an extremely useful tool. The book demands no more than an elementary knowledge of calculus and linear algebra of the type found in many first year mathematics modules for applied subjects. For mathematics majors and specialists, it is not the mathematics that will be challenging but the applications to the real world. The author is in the privileged position of having spent ten or so years outside mathematics in an engineering environment where the Laplace Transform is used in anger to solve real problems, as well as spending rather more years within mathematics where accuracy and logic are of primary importance. This book is written unashamedly from the point of view of the applied mathematician. TheLaplace Transform has a rather strange place in mathematics. There is no doubt that it is a topic worthy of study by applied mathematicians who have one eye on the wealth of applications; indeed it is often called Operational Calculus.

Produktdetails

• Springer Undergraduate Mathematics Series
• Verlag: Springer / Springer London / Springer, Berlin
• Artikelnr. des Verlages: 978-1-85233-015-6
• 2001
• Seitenzahl: 264
• Erscheinungstermin: 15. Oktober 1999
• Englisch
• Abmessung: 235mm x 178mm x 15mm
• Gewicht: 440g
• ISBN-13: 9781852330156
• ISBN-10: 1852330155
• Artikelnr.: 09218317

Autorenporträt

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.

Inhaltsangabe

- The Laplace Transform

- Further Properties

- Convolution and the Solutions

- Fourier Series

- Partial Differential Equations

- Fourier Transforms

- Complex Variables and Laplace Transforms

- Appendices:
... A: Answers to Exercises
... B: Table of Laplace Transforms
... C: Linear Spaces

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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.