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The chapter derives conditions for convergence by restricting s to real values and extends the treatment to complex-valued functions using real and imaginary parts. 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When it exists, it is written as {F(t)} and determines f(s).","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Why does convergence depend on the value of s?",{"text":117,"@type":113},"If the Laplace integral converges for a particular real value s = σ, then it converges for all s ≥ σ. The chapter uses this to analyze when transforms exist and when they do not.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the chapter use piecewise continuity in the existence theorem?",{"text":121,"@type":113},"It considers functions F(t) with finite discontinuities, described as piecewise continuous. It then states that such functions are integrable on a finite interval and uses this to establish sufficient conditions for the existence of the Laplace transform.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},567473,1792568586,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145,"preview_page_count":4},2336478945635,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Laplace Transforms  \nLaplace Transforms  \nJ.WILLIAMS  \nJ. WILLIAMS  \nGEORGE ALLEN  \nUNWIN  \n8  \nProblem Solvers  \nEdited by L.MarderSenior Lecturer in Mathematics,University of Southampton  \nNo.10  \nLaplace Transforms  \nProblem Solvers  \n1.ORDINARY DIFFERENTIAL EQUATIONS—J.Heading  \n2.CALCULUS OF SEVERAL VARIABLES—L.Marder  \n3.VECTOR ALGEBRA—L.Marder  \n4.ANALYTICAL MECHANICS—D.F.Lawden  \n5.CALCULUS OF ONE VARIABLE—K.Hirst  \n6.COMPLEX NUMBERS—J.Williams  \n7.VECTOR FIELDS—L.Marder  \n8.MATRICES AND VECTOR SPACES—F.Brickell  \n9.CALCULUS OF VARIATIONS—J.W.Craggs  \n10.LAPLACE TRANSFORMS-—J.Williams  \n11.STATISTICS I—A.K.Shahani &P.K.Nandi  \n12.FOURIER SERIES AND BOUNDARY VALUE PROBLEMS—W.E.Williams  \n13.ELECTROMAGNETISM—D.F.Lawden  \nLaplace Transforms  \nJ.WILLIAMS  \nSenior Lecturer in Applied MathematicsUniversity of Exeter  \nLONDON·GEORGE ALLEN &UNWIN LTD  \nFirst published in 1973  \nThis book is copyright under the Berne Convention.All rightsare reserved.Apart from any fair dealing for the purpose ofprivate study,research,criticism or review,as permitted underthe Copyright Act,1956,no part of this publication may bereproduced,stored in a retrieval system,or transmitted,inany form or by any means,electronic,electrical,chemical,mechanical,optical,photocopying,recording or otherwise,without the prior permission of the copyright owner.Enquiries should be addressed to the publishers.  \nCGeorge Allen &Unwin Ltd,1973  \nISBN 004512020x hardback0045120218 paper  \nTo my wife,Anita  \nContents  \npage 1  \n1 THE LAPLA CE TRANSFORM  \n1.1 Definition of Laplace transform  \n1  \n3  \n1.2 Existence theorem  \n1.3 Operational rules of Laplace transforms7  \n1.4 Laplace transform of M(t)F(t)where M(t)is a polynomial 14  \n1.5 Laplace transform of F(t)/(t+a),a≥015  \n1.6 Laplace transforms of derivatives17  \n1.7 Laplace transforms of integrals20  \n1.8 Laplace transforms via differential equations22  \n1.9 Miscellaneous functions252 INVERSION OF THE LAPLA CE TRANSFORM29  \n2.1 The inverse29  \n2.2 Null functions and uniqueness29  \n2.3 Operational rules of inverse Laplace transforms29  \n2.4 Inverse Laplace transforms of s-derivatives37  \n2.5 Division by(s+a)\"38  \n2.6 Method of partial fractions40  \n2.7 Convolution rule42  \n2.8 Heaviside formula443 ELEMENTARY APPLICATIONS48  \n3.1 Evaluation of integrals48  \n3.2 Solution of ordinary differential equations with constantcoefficients51  \n3.3 Solution of ordinary differential equations withpolynomial coefficients57  \n3.4 Simultaneous ordinary differential equations58  \n3.5 Integral equations of the convolution type61  \n3.6 Linear difference equations634 INVERSION THEOREM67  \n4.1 The inversion theorem67  \n4.2 The inversion theorem when f(s)has a branch point705 BOUNDARY VALUE PROBLEMS76  \n5.1 Fundamentals76  \n5.2 Conduction of heat in one dimension78  \n5.3 Longitudinal vibrations of a uniform bar80TABLE1 Index of Operational Rules for Laplace transforms85TABLE 2 List of Useful Laplace Transforms88A NSWERS TO EXERCISES90  \nINDEX93  \n# Chapter 1\n\nThe Laplace Transform  \n1.1 Definition of Laplace Transform Given a function F(t)defined forall real t≥0,the integral  \n(1.1)  \nif it exists for some values of s,is said to be the Laplace transform of F(t).It is written {F(t)}and determines a function f(s),say,of the parameter sat those values of s for which the integral exists,i.e.  \n(1.2)  \nFor the moment we restrict s to real values,in which case it can be shownthat if the integral in (1.1)converges for s=σ it will converge for alls≥σ.It is convenient to use a capital letter to denote the originalfunction F(t)and the corresponding lower case letter to indicate itsLaplace transform f(s).  \nWhen F(t)is a complex function of the real variable t≥0,we writeF(t)=X(t)+iY(t),where X(t)and Y(t)are real functions.Then,providedx(s)and y(s)exist,  \nΦ{F(t)}=f(s)=2{X(t)}+i 2{Y(t)}=x(s)+i y(s)  \nwhere x(s)=  {X(t)},y(s)={Y(t)}  \nProblem 1.1 Find {cosh pt},Φ{sinh pt}where p is a real positiveconstant.  \nSolution.Since coshwe have,  \nprovided s","cbCaiokvs4pAUQzB","https://ap.wps.com/l/cbCaiokvs4pAUQzB","pdf",3402279,101,"English","# Chapter 1\n## The Laplace Transform\n## Definition of Laplace Transform\n## Existence Theorem","[{\"question\":\"What does it mean for F(t) to have a Laplace transform, and how is it defined?\",\"answer\":\"Given F(t) for t ≥ 0, the Laplace transform is defined as an integral that exists for some values of s. When it exists, it is written as {F(t)} and determines f(s).\"},{\"question\":\"Why does convergence depend on the value of s?\",\"answer\":\"If the Laplace integral converges for a particular real value s = σ, then it converges for all s ≥ σ. The chapter uses this to analyze when transforms exist and when they do not.\"},{\"question\":\"How does the chapter use piecewise continuity in the existence theorem?\",\"answer\":\"It considers functions F(t) with finite discontinuities, described as piecewise continuous. It then states that such functions are integrable on a finite interval and uses this to establish sufficient conditions for the existence of the Laplace transform.\"}]","Laplace Transforms - Chapter 1 - The Laplace Transform | PDF",1791866537,255]