Cohen A. - Numerical Methods for Laplace Transform Inversion (Springer, 2007).pdf

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NUMERICAL METHODS FOR
LAPLACE TRANSFORM INVERSION
Numerical Methods and Algorithms
VOLUME 5
Series Editor:
Claude Brezinski
Université des Sciences et Technologies de Lille, France
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NUMERICAL METHODS FOR
LAPLACE TRANSFORM INVERSION
By
ALAN M. COHEN
Cardiff University
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Library of Congress Control Number: 2006940349
ISBN-13: 978-0-387-28261-9
e-ISBN-13: 978-0-387-68855-8
Printed on acid-free paper.
AMS Subject Classifications: 44A10, 44-04, 65D30, 65D32, 65Bxx
2007 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY
10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in
connection with any form of information storage and retrieval, electronic adaptation, computer software,
or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are
not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject
to proprietary rights.
Printed in the United States of America.
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Contents
Preface ...................................viii
Acknowledgements.............................xii
Notation...................................xiii
1BasicResults 1
1.1Introduction.............................. 1
1.2TransformsofElementaryFunctions................ 2
1.2.1ElementaryPropertiesofTransforms............ 3
1.3TransformsofDerivativesandIntegrals .............. 5
1.4InverseTransforms.......................... 8
1.5Convolution.............................. 9
1.6TheLaplaceTransformsofsomeSpecialFunctions....... 11
1.7Di®erenceEquationsandDelayDi®erentialEquations...... 14
1.7.1z-Transforms......................... 16
1.8MultidimensionalLaplaceTransforms ............... 18
2InversionFormulaeandPracticalResults 23
2.1TheUniquenessProperty...................... 23
2.2TheBromwichInversionTheorem................. 26
2.3ThePost-WidderInversionFormula................ 37
2.4InitialandFinalValueTheorems.................. 39
2.5SeriesandAsymptoticExpansions................. 42
2.6Parseval'sFormulae ......................... 43
3TheMethodofSeriesExpansion 45
3.1ExpansionasaPowerSeries..................... 45
3.1.1Analternativetreatmentofseriesexpansions....... 49
3.2ExpansionintermsofOrthogonalPolynomials.......... 49
3.2.1LegendrePolynomials.................... 50
3.2.2ChebyshevPolynomials................... 52
3.2.3LaguerrePolynomials.................... 55
3.2.4ThemethodofWeeks.................... 58
3.3Multi-dimensionalLaplacetransforminversion.......... 66

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