Riemann-Liouville Fractional Derivatives and the

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1、Riemann-LiouvilleFractionalDerivativesandtheTaylor-RiemannSeriesJoakimMunkhammarU.U.D.M.ProjectReport2004:7Examensarbeteimatematik,10poängHandledareochexaminator:AndreasStrömbergssonJuni2004DepartmentofMathematicsUppsalaUniversityRIEMANN-LIOUVILLEFRACTIONALDERIVATIVESANDTHETAYLOR-RIEMANNSERIESJ.

2、D.MUNKHAMMARABSTRACT.Inthispaperwegivesomebackgroundtheoryonthecon-ceptoffractionalcalculus,inparticulartheRiemann-Liouvilleoperators.WetheninvestigatetheTaylor-RiemannseriesusingOsler'stheoremandobtaincertaindoubleinniteseriesexpansionsofsomeelementaryfunc-tions.Intheprocessofthiswegiveaproofo

3、ftheconvergenceofanalternativeformofHeaviside'sseries.ASemi-TaylorseriesisintroducedasthespecialcaseoftheTaylor-Riemannserieswhen
,andsomeofitsrelationstospecialfunctionsareobtainedviacertaingeneratingfunctionsarisingincomplexfractionalcalculus.1.INTRODUCTION1.1.Riemann-Liouvilleoperator

4、.Theconceptofnon-integralorderofin-tegrationcanbetracedbackthetothegenesisofdifferentialcalculusitself:thephilosopherandcreatorofmoderncalculusG.W.Leibnizmadesomere-marksonthethemeaningandpossibilityoffractionalderivativeoforderinthelate17:thcentury.Howeverarigorousinvestigationwasrstcar-riedou

5、tbyLiouvilleinaseriesofpapersfrom1832-1837,wherehedenedtherstoutcastofanoperatoroffractionalintegration.Laterinvestiga-tionsandfurtherdevelopmentsbyamongothersRiemannledtothecon-structionoftheintegral-basedRiemann-Liouvillefractionalintegralopera-tor,whichhasbeenavaluablecornerstoneinfractiona

6、lcalculuseversince.PriortoLiouvilleandRiemann,Eulertooktherststepinthestudyoffrac-tionalintegrationwhenhestudiedthesimplecaseoffractionalintegralsofmonomialsofarbitraryrealorderintheheuristicfashionofthetime;ithasbeensaidtohaveleadhimtoconstructtheGammafunctionforfractionalpowersofthefactorial[

7、2,p.243].AnearlyattemptbyLiouvillewaslaterpuriedbytheSwedishmathematicianHolmgren[10],whoin1865madeimportantcontributionstothegrowingstudyoffractionalcalculus.ButitwasRiemann[4]whoreconstructedittotAbel'sintegralequation,a