Mittag-Leffler Functions, Related Topics and Applications: Theory and Applications

454 Pages·2014·4.271 MB·English

by  Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin (auth.)

Springer Monographs in Mathematics
Rudolf Gorenflo
Anatoly A. Kilbas
Francesco Mainardi
Sergei V. Rogosin
Mittag-Leffler
Functions,
Related Topics
and Applications
Springer Monographs in Mathematics
Moreinformationaboutthisseriesat
http://www.springer.com/series/3733
Rudolf Gorenflo • Anatoly A. Kilbas •
Francesco Mainardi • Sergei V. Rogosin
Mittag-Leffler Functions,
Related Topics
and Applications
123
RudolfGorenflo AnatolyA.Kilbas(July20,1948-June28,
FreeUniversityBerlinMathematical 2010)
Institute BelarusianStateUniversityDepartment
Berlin ofMathematicsandMechanics
Germany Minsk
Belarus
FrancescoMainardi SergeiV.Rogosin
UniversityofBolognaDepartment BelarusianStateUniversityDepartment
ofPhysics ofEconomics
Bologna Minsk
Italy Belarus
ISSN1439-7382 ISSN2196-9922(electronic)
SpringerMonographsinMathematics
ISBN978-3-662-43929-6 ISBN978-3-662-43930-2(eBook)
DOI10.1007/978-3-662-43930-2
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014949196
MathematicsSubjectClassification:33E12,26A33,34A08,45K05,44Axx,60G22
©Springer-VerlagBerlinHeidelberg2014
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To thememoryof ourcolleagueandfriend
AnatolyKilbas

Preface
ThestudyoftheMittag-Lefflerfunctionanditsvariousgeneralizationshasbecome
averypopulartopicinmathematicsanditsapplications.However,duringthetwen-
tieth century, this function was practically unknown to the majority of scientists,
sinceitwasignoredinmostcommonbooksonspecialfunctions.Asanoteworthy
exception the handbook Higher Transcendental Functions, vol. 3, by A. Erdelyi
etal.deservestobementioned.
NowtheMittag-LefflerfunctionisleavingitsisolatedroleasCinderella(using
thetermcoinedbyF.G.Tricomifortheincompletegammafunction).
The recent growing interest in this function is mainly due to its close relation
totheFractionalCalculusandespeciallytofractionalproblemswhichcomefrom
applications.
Ourdecisiontowritethisbookwasmotivatedbythe needtofillthe gapinthe
literature concerning this function, to explain its role in modern pure and applied
mathematics,andtogivethereaderanideaofhowonecanusesuchafunctionin
theinvestigationofmodernproblemsfromdifferentscientificdisciplines.
ThisbookisafruitofcollaborationbetweenresearchersinBerlin,Bolognaand
Minsk. It has highly profited from visits of SR to the Department of Physics at
the University of Bologna and from several visits of RG to Bologna and FM to
the Department of Mathematics and Computer Science at Berlin Free University
undertheEuropeanERASMUSexchange.RGandSRappreciatethedeepscientific
atmosphereattheUniversityofBolognaandtheperfectconditionstheymetthere
forintensiveresearch.
We are saddened that our esteemed and always enthusiastic co-author Anatoly
A. Kilbas is nolongerwith us, havinglosthis life in a tragic accidenton 28June
2010in the South of Russia. We will keep him, and ourinspiringjointwork with
him,inlivingmemory.
Berlin,Germany RudolfGorenflo
Bologna,Italy FrancescoMainardi
Minsk,Belarus SergeiRogosin
March2014
vii

Contents
1 Introduction .................................................................. 1
2 HistoricalOverviewoftheMittag-LefflerFunctions .................... 7
2.1 AFewBiographicalNotesonGöstaMagnusMittag-Leffler ....... 7
2.2 TheContentsofthe Five PapersbyMittag-Leffler
onNewFunctions ..................................................... 9
2.3 FurtherHistoryofMittag-LefflerFunctions ......................... 12
3 TheClassicalMittag-LefflerFunction..................................... 17
3.1 DefinitionandBasicProperties....................................... 17
3.2 RelationstoElementaryandSpecialFunctions...................... 19
3.3 RecurrenceandDifferentialRelations................................ 21
3.4 IntegralRepresentationsandAsymptotics ........................... 23
3.5 DistributionofZeros .................................................. 30
3.6 FurtherAnalyticProperties ........................................... 35
3.7 TheMittag-LefflerFunctionofaRealVariable...................... 39
3.7.1 IntegralTransforms.......................................... 39
3.7.2 TheCompleteMonotonicityProperty ...................... 46
3.7.3 RelationtoFractionalCalculus.............................. 48
3.8 HistoricalandBibliographicalNotes................................. 51
3.9 Exercises............................................................... 53
4 TheTwo-ParametricMittag-LefflerFunction............................ 55
4.1 SeriesRepresentationandPropertiesofCoefficients................ 56
4.2 ExplicitFormulas:RelationstoElementaryandSpecial
Functions............................................................... 57
4.3 DifferentialandRecurrenceRelations................................ 58
4.4 IntegralRelationsandAsymptotics................................... 60
4.5 The Two-Parametric Mittag-Leffler Function
asanEntireFunction.................................................. 65
4.6 DistributionofZeros .................................................. 67
ix