Special Functions for Applied Scientists

479 Pages·2008·5.939 MB·English

by  A. M. Mathai, Hans J. Haubold (auth.)

Special Functions for Applied Scientists
A.M. Mathai • Hans J. Haubold
Special Functions
for Applied Scientists
123
A.M.Mathai HansJ.Haubold
McGillUniversity OfficeforOuterSpaceAffairs
Montreal,QC UN,Vienna
Canada Austria
and [e-mail]
CentreforMathematicalSciences
PalaCampus,Kerla
India
[e-mail]
ISBN:978-0-387-75893-0 e-ISBN:978-0-387-75894-7
DOI:10.1007/978-0-387-75894-7
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Preface
ThefirstS.E.R.C.(ScienceandEngineeringResearchCounciloftheDepartmentof
ScienceandTechnology,GovernmentofIndia,NewDelhi)SchoolonSpecialFunc-
tionswassponsoredbyDST(DepartmentofScienceandTechnology),NewDelhi,
andconductedbyCMS(CentreforMathematicalSciences)forsixweeksin1995.
The second S.E.R.C. School on Special Functions and Functions of Matrix Argu-
mentwassponsoredbyDST,Delhi,andconductedbyCMSforfiveweeksduring
theperiod29thMayto30thJune2000.InthesecondSchool,themainlectureswere
givenbyDr.H.L.ManochaofDelhi,India,Dr.S.BhargavaofMysore,India,Dr.
K.SrinivasaRaoandDr.R.JagannathanofChennai,IndiaandDr.A.M.Mathaiof
Montreal, Canada. Supplementary lectures were given and problem sessions were
supervised by Dr. K.S.S. Nambooripad and Dr. S.R. Chettiyar of CMS, Dr. R.N.
Pillai(retired),Dr.T.S.K.MoothathuandDr.YageenThomasoftheUniversityof
Kerala and Dr. E. Krishnan of the University College Trivandrum. Lecture Notes
were brought out as Publication No.31 of CMS soon after the School was com-
pleted.ThefirsttwoSchoolswereconductedinTrivandrum(Thiruvananthapuram)
area. Dr. A.M. Mathai was the Director and Dr. K.S.S. Nambooripad was the Co-
DirectorofthesetwoSchools.
ThethirdS.E.R.C.SchoolonSpecialFunctionsandFunctionsofMatrixArgu-
ment:RecentDevelopmentsandRecentApplicationsinStatisticsandAstrophysics,
sponsored by DST, Delhi, was conducted for five weeks from 14th March to 15th
April 2005 by CMS at its Pala Campus, Kerala, India. This time DST, wanted the
lecturenotestobecollectedfromthemainlecturersinadvance,compiledanddis-
tributedpriortothestartoftheSchool.Themainlecturesforthe3rdSchoolwere
given by Dr. Hans J. Haubold of the Office of Outer Space Affairs of the United
Nations,Dr.SergeB.Provost,ProfessorofActuarialSciencesandStatisticsofthe
University of Western Ontario, Canada, Dr. R.K. Saxena of Jodhpur, India, Dr. S.
BhargavaofMysore,IndiaandDr.A.M.MathaiofMontreal,Canada.Supplemen-
tary lectures were given by Dr. A. Sukumaran Nair (Chairman, CMS), Dr. K.K.
Jose(Director-in-Charge,CMSPalaCampus),Dr.R.N.Pillai,Dr.YageenThomas,
Dr. V. Seetha Lekshmi, Dr. Alice Thomas, Dr. E. Sandhya, Dr. S. Satheesh and
v
vi Preface
Dr.K.Jayakumar.ProblemsessionsweresupervisedbyDr.SebastianGeorgeand
otherlecturers.Extratrainingintheuseofstatisticalpackages,LATEXandMath-
ematica/MapleweregivenbyJoyJacob,SeemonThomasandK.M.Kurianofthe
DepartmentofStatisticsofSt.ThomasCollege,Pala.Aspeciallecturesequenceon
MatlabwasconductedbyAlexanderHauboldofColumbiaUniversity,U.S.A.
The 4th S.E.R.C. School on Special Functions and Functions of Matrix Argu-
ment:RecentAdvancesandApplicationstoStochasticProcesses,StatisticsandAs-
trophysics,sponsoredbyDST,Delhi,wasconductedbyCMSforfiveweeksfrom
6thMarchto7thApril2006atitsPalaCampus.Themainlectureswerescheduled
tobegivenbyDr.HansJoachimHauboldoftheOfficeofOuterSpaceAffairsofthe
UnitedNations,Dr.P.N.RathieoftheUniversityofBrasilia,Brazil,Dr.S.Bhargava
ofMysore,India,Dr.R.K.SaxenaofJodhpur,India,Dr.K.K.JoseofPala,Indiaand
Dr.A.M.MathaiofMontreal,Canada.SupplementarylecturesweregivenbyDr.A.
Sukumaran Nair (Chairman, CMS), Dr. K.S.S. Nambooripad (Director-in-Charge,
CMSTrivandrumCampus),Dr.R.YDenisofGorakhpur,India,Dr.N.Unnikrish-
nanNair(formerVice-ChancellorofCochinUniversityofScienceandTechnology,
Kerala, India), Dr. Yageen Thomas and Dr. K. Jayakumar. One week was devoted
toStochasticProcessesandrecentadvancesinthisareainthe4thSchool.Problem
sessionsweresupervisedbyDr.JoyJacob,Dr.SebastianGeorgeandotherlecturers.
The5thS.E.R.C.Schoolinthissequence,titled“SpecialFunctionsandFunctions
of Matrix Argument: Recent Advances and Applications in Stochastic Processes,
Statistics, Wavelet Analysis and Astrophysics” was conducted by CMS at its Pala
Campus from 23rd April to 25th May 2007. The lecture notes for the 5th School
were assembled by November 2006 and were printed and distributed as Publica-
tionNo.34inthePublicationsSeriesofCMS.Themainlectureswerescheduledto
begivenbyDr.HansJ.Haubold(OfficeofOuterSpaceAffairsoftheUnitedNa-
tions,Austria),Dr.H.M.Srivastava(UniversityofVictoria,Canada),Dr.R.Y.Denis
(Gorakhpur University, India), Dr.R.K. Saxena (Jodhpur University, India), Dr. S.
Bhargava(MysoreUniversity,India),Dr.D.V.Pai(IITBombay,India),Dr.Yageen
Thomas(UniversityofKerala,India),Dr.K.K.Jose(MahatmaGandhiUniversity,
India), Dr. J.J. Xu (China/Canada), Dr. K. Jayakumar (Calicut University, India)
andDr.A.M.Mathai(Canada/India).ButDr.H.M.Srivastava,Dr.S.Bhargavaand
Dr.Xucouldnotreachthevenueontimeduetounexpectedemergencies.Supple-
mentarylecturesweregivenbyDr.A.SukumaranNair(Chairman,CMS),Dr.R.N.
Pillai (former Head, Department of Statistics, University of Kerala, India), Dr. N.
Mukunda(IISc,Bangalore,India)andtheproblemsessionsweresupervisedbyDr.
Sebastian George, Dr. Joy Jacob, Dr. Seemon Thomas and the lecturers. For the
2005,2006and2007Schools,Dr.A.M.MathaiwastheDirectorandDr.K.K.Jose
wastheCo-DirectoroftheSchools.
TheparticipantsfortheS.E.R.C.Schoolsareselectedonall-Indiabasis.Allthe
expenses of the selected candidates, total number of seats is 30, including travel,
accommodation,food,lecturematerials,localtravelsetcaremetbytheDST,Delhi,
GovernmentofIndia.Foreignparticipationisallowedundertheconditionsthatthe
Preface vii
participantsmustcomewiththeirownreturnairticketsandmustattendallthelec-
tures,andproblemsessionsandtakealltheexaminationsandtestsfromthebegin-
ningtilltheend.Alltheirlocalexpensesaremetandlecturematerialsareprovided
bytheSchools.ThereisnofeeforattendingtheSchools.TheSchoolsaremainly
research orientation courses aimed at young faculty members in colleges and uni-
versitiesacrossIndia,below35yearsofage,andfreshgraduateswithM.Sc,M.Phil,
Ph.Ddegreesbelow30yearsofage,inMathematics/Statistics/TheoreticalPhysics.
In most of the Indian universities rigid compartmentalization is the order and as a
resultaM.Scgraduateinmathematicsmaynothaveeventakenaverybasiccourse
inprobabilityandstatistics.Eventhoughbasicdifferentialandintegralcalculusand
matrix theory are required subjects for statistics and physics students, these stu-
dentsmayhaveforgottenthesesubjectsbecausetheyteachthecompartmentalized
topics in their own areas and do not usually do research even in their narrow ar-
easoftheirownfields.Formaintainingtheirjobsandgettingregularincrementsin
their salaries and all other monetary and other benefits, research work and further
reading and learning process are not required of them. As a result, the quality of
teachingandtheinformationpassedontothestudentsgodownfromyeartoyear.
Inordertoremedythissituationalittlebit,S.E.R.C.Schoolsinvariousareaswere
startedbyDST,Delhi.Dr.A.M.MathaiwasaskedtorunS.E.R.C.Schoolsinmath-
ematics. For taking up a challenging research problem in any applied area a good
backgroundinbasicmathematics,probabilityandstatisticsisrequired.Inorderto
givethebasicideasinprobabilityandstatisticsandtobringanumberoftopicsin
the area of Special Functions and Functions of Matrix Argument and their appli-
cations to the current research level, these S.E.R.C. Schools on Special Functions
wereestablished.ThecurrentsequenceoffiveSchoolsreallyachievedtheaimand
almost all the participants in the first four Schools have become research oriented
towardsacareerinresearchandteaching,andtheparticipantsofthe5thSchoolare
alsoexpectedtofollowthesamefootstepsoftheirpredecessors.
Chapter 1 introduces elementary classical special functions. Gamma, beta, psi,
zetafunctions,hypergeometricfunctionsandtheassociatedspecialfunctions,gen-
eralizationstoMeijer’sGandFox’sH-functionsarealsoexaminedhere.Discussion
isconfinedtobasicpropertiesandsomeapplications.Introductiontostatisticaldis-
tribution theory is given here. Some recent extensions of Dirichlet integrals and
Dirichlet densities are also given. A glimpse into multivariable special functions
suchasAppell’sfunctionsandLauricellafunctionsisalsogiven.Specialfunctions
assolutionsofdifferentialequationsarealsoexaminedhere.
Chapter 2 is devoted to fractional calculus. Fractional integrals and fractional
derivativesofvariouskindsarediscussedhere.Thentheirapplicationstoreaction-
diffusion problems in physics, input-output analysis and Mittag-Leffler stochas-
tic processes are examined here. Chapter 3 deals with q-hypergeometric or basic
hypergeometric functions and Chapter 4 goes into basic hypergeometric functions
andRamanujan’sworkonellipticandthetafunctions.Chapter5examinesthetopic
ofSpecialFunctionsandLieGroups.
viii Preface
Chapters 6 to 9 are devoted to applications of Special Functions to various ar-
eas. Applications to stochastic processes, geometric infinite divisibility of random
variables,Mittag-Lefflerprocesses,α-Laplaceprocesses,densityestimation,order
statisticsandvariousastrophysicsproblems,aredealtwithinthesechapters.
Chapter10isdevotedtoWaveletAnalysis.Anintroductiontowaveletanalysisis
givenhere.Chapter11dealswiththeJacobiansofmatrixtransformations.Various
typesofmatrixtransformationsandtheassociatedJacobiansaregivenhere.Chapter
12isdevotedtothediscussionoffunctionsofmatrixargument.Onlytherealcase
isconsideredhere.Functionsofmatrixargumentandthepathwaymodels,recently
introducedbyMathai(2005),arealsodiscussedhere,alongwiththeirapplications
tovariousareas.
InalltheS.E.R.C.SchoolsconductedundertheDirectorshipofDr.A.M.Mathai
aseriouseffortismadesothattheparticipantsabsorbthematerialscoveredinthe
Schools. The classes started at 8.30 am and went until 6.00 pm. The first lecture
of 08.30 to 10.30 was followed by a problem session from 10.30 to 13.00 hrs on
the materials covered in the first lecture. The second lecture of the day was from
14.00 to 16.00 hrs followed by problem session from 16.00 to 18.00 hrs. At the
end of every week a written examination was conducted, followed by a personal
interview of each participant by the lecturer of that week in the form of an oral
examination.Cumulativegradesofsuchweeklyexaminationsappearedinthefinal
certificates distributed to them. The main aim was to inculcate in them a habit of
long and sustained hard work, which would help them in their careers whatever
they may be. During the first week the participants, especially the teachers from
collegesanduniversities,founditdifficulttoadjusttotheroutineoflonghoursof
hard work but starting from the second week, in all the Schools, the participants
started enjoying, especially the problem sessions, because for the first time, they
started understanding and appreciating the meanings and significance of theorems
thattheylearntormemorizedwhentheywerestudents.
Thelecturenotesarewrittenupinastyleforself-study.Eachtopicisdeveloped
fromfirstprincipleswithlotsofworkedexamplesandexercises.Hencethematerial
inthisbookcanbeusedforself-studyandwillhelpanyonetounderstandthebasic
ideasintheareaofSpecialFunctionsandFunctionsofMatrixArgumentandthey
willbeabletomakeuseoftheseresultsintheirownproblemsinappliedareas,espe-
ciallyinStatistics,PhysicsandEngineeringproblems.Applicationsinvariousareas
areillustratedinthisbook.Insightsintorecentdevelopmentsintheapplicationsof
fractionalcalculus,inthedevelopmentsinvariousothertopicsarealsogiveninthe
booksothatthereaderswhoareinterestedinanyofthetopicsdiscussedinthebook
candirectlygointoaresearchprobleminthetopics.
SeveralpeoplehavecontributedenormouslyforthesuccessoftheS.E.R.C.Schools
and in making the publications of four Lecture Notes and this final publication of
summarized lecture notes possible. Dr. B.D. Acharya, Advisor to Government of
IndiaandDr.AshokK.SinghoftheMathematicalSciencesDivisionofDST,New
Delhi,arethedrivingforcebehindthere-energizedmathematicalactivitiesinIndia
Preface ix
now. They were kind enough to pursue the matter and get the funds released for
runningtheSchoolsaswellasforthepreparationofvariouspublications,including
this one. Since the basic materials for this publication are supplied by various
lecturersintheformoftheirlecturenotes,therewillbesomeoverlaps.Veryobvious
inconsistenciesareremovedbutsomeoverlappingmaterialsarelefttheretomakethe
discussionsself-contained.Dr.R.K.Saxena,Dr.S.Bhargava,Dr.H.J.Haubold,Dr.
P.N.Rathie,Dr.K.K.Jose,Dr.K.Jayakumar,Dr.H.L.Manocha,Dr.R.Jagannathan,
Dr. K. Srinivasa Rao, Dr. K.S.S. Nambooripad, Dr. Serge B. Provost, Dr. Yageen
Thomas, Dr. D.V. Pai, and Dr. A.M. Mathai are thanked for making their notes
available in advance for the S.E.R.C. Schools. Most of the material was typeset
at CMS office by Miss K.H. Soby, Dr. Joy Jacob, Seemon Thomas, Dr. Sebastian
George, Dr. K.K. Jose and Dr. A.M. Mathai. Part of the material was typeset by
Barbara Haubold of the United Nations, Vienna Office, fully free of charge as a
voluntary service to the Schools. Notes and programs for a series of lectures on
MatlabweresuppliedtoCMSbyAlexanderHauboldofColumbiaUniversity,USA.
Thosenotesarenotincludedinthisbooktokeepthematerialswithinthefocusof
thebook.Dr.SebastianGeorge,Dr.JoyJacob,Dr.SeemonThomas,K.M.Kurian,
Jaisymol Thomas and Ashly P. Jose, who spent a lot of time in running problem
sessions,inrunningseparatesessionsontheuseofstatisticalpackagesandMaple
program,useofLATEXetcandforcheckingthetypedmaterials,arethanked.CMS
would like to thank each and every one who helped to make S.E.R.C. Schools on
SpecialFunctionsagrandsuccessandwhohelpedtomakethispublicationpossible.
The authors would like to express their sincere thanks to the Department of
ScienceandTechnology,GovernmentofIndia,NewDelhi,India,forthefinancial
assistanceundertheprojectNo.SR/S4/MS:287/05titled“BuildingupaCoreGroup
of Researchers/Faculty and Facilities at CMS, Trivandrum and Pala Campuses”,
whichenabledthecollaborationonthisbookprojectpossible.Weexpressoursin-
ceregratitudetotheManagementandespeciallytoDr.MathewJohnK.(Principal),
St. Thomas College Pala, Kerala, India, for providing all facilities in the College
duringthepreparationofthisbook.
Pala,Kerala,India A.M.Mathai
1st February,2007 H.J.Haubold
Glossary of Symbols
(b) PochhammerSymbol Notation1.0.1 1
r
n(cid:1)!(cid:2) factorialnornfactorial Notation1.0.2 2
n numberofcombinationsofn Notation1.0.3 2
r
takenratatime
R(·) realpartof(·) Notation1.1.1 3
Γ(z) gammafunction Notation1.1.1 3
γ Euler’sconstant Notation1.1.2 3
π pi,mathematicalconstant Definition1.1.6 4
∧ wedgeorskewsymmetricproduct Notation1.1.3 6
J Jacobian Definition1.1.7 6
(a)
B (x) generalizedBernoullipolynomial Noation1.1.4 9
k
B (x) Bernoulipolynomialoforderk Notation1.1.4 9
k
B Bernoullinumberoforderk Notation1.1.4 9
k
ψ(·) psifunction Section1.2 12
ζ(ρ) Riemannzetafunction Section1.2.1 12
ζ(ρ,a) generalizedzetafunction Section1.2.1 12
M (s) Mellintransformof f Section1.3 14
f
L (t) Laplacetransformof f Section1.3.2 17
f
F (t) Fouriertransformof f Section1.3.2 18
f
P(A) probabilityoftheeventA Section1.4 19
E(·) expectedvalue Notation1.5.1 28
Var(·) varianceof(·) Definition1.5.2 29
Cov(·) covariance Exercises1.5.7 34
B(α,β) betafunction Section1.6 34
D(α,...,α;
1 k
α ) Dirichletfunction Notation1.6.2 37
k+1
F (·) hypergeometricseries Section1.7 42
p q
γ(α;b) incompletegammafunction Section1.7 43
Γ(α;b) incompletegammafunction Section1.7 43
b(α,β;t) incompletebetafunction Section1.7 44
B(α,β;t) incompletebetafunction Section1.7 44
xi
xii GlossaryofSymbols
Gm,n[·] G-function Section1.8 49
p,q
Jν(·) Besselfunction Exercise1.8 54
Iν(·) Besselfunction Exercises1.8 54
Hm,n(·) H-function Section1.9 54
p,q
Eα,β(·) Mittag-Lefflerfunction Section1.9 57
f (·),f (·), Lauricellafunction Section1.10 58
A B
f (·),f (·) Lauricellafunction Section1.10 58,59
C D
(1) (2)
Hν (x),Hν (x) Besselfunctionofthethirdkind Section1.11.6 73
(α)
L (x) Laguerrepolynomial Section1.11.7 74
n
P (x) Legendrepolynomial Section1.11.8 74
m
Eα(x),Eα,β(x) Mittag-Lefflerfunctions Section2.1 79
h (z,n) hyperbolicfunctionofordern Definition2.1.3 80
r
K (z,n) trigonometricfunctionofordern Definition2.1.4 80
r
erf,erfc errorfunctions Definition2.1.5 80
E(ν,α) Mellin-Rossfunction Definition2.1.6 81
t
Rα(β,t) Robotov’sfunction Definition2.1.7 81
F(s)=L{f(t);s} Laplacetransform Notation2.2.1 85
L−1{f(s);t} inverseLaplacetransform Notation2.2.2 85
δ
E (x) generalizedMittag-Leffler Notation2.3.1 91
β,γ
function
In, D−n Riemann-Liouvillefractional
a x a x
integrals Notation2.4.1 98
−α α
D , I Riemann-Liouvilleleft-sided
a x a+
fractionalintegrals Definition2.4.2 99
α −α
I , D Riemann-Liouvilleright-sided
x b x b
fractionalintegrals Definition2.4.3 99
xW∞α, xI∞α: Weylintegraloforderα Notation2.4.4 103
α α
xD∞, D−: Weylfractionalderivatives Notation2.4.5 103
α α
−∞Wx ,I+ Weylintegrals Notation2.4.6 104
α
C D Caputoderivative Section2.4.9 108
a x
0
m{f(x);s},f∗(s) Mellintransform Section2.5 109
m−1{f∗(s);x} inverseMellintransform Section2.5 109
I[f(x)], Koberoperatorofthefirstkind Notation2.6.1 112
I[α,η: f(x)] Koberoperatorofthefirstkind Notation2.6.1 112
I(α,η)f(x)Eα,η f Koberoperatorofthefirstkind Notation2.6.1 112
0,x
In,α f Koberoperatorofthefirstkind Notation2.6.1 112
x
R[f(x)] Koberoperator,secondkind Notation2.6.2 112
R[α,ζ: f(x)] Koberoperator,secondkind Notation2.6.2 112
R(α,ζ)f(x) Koberoperator,secondkind Notation2.6.2 112
α,ζ ζ,α
Kx,∞ f,Kx f Koberoperator,secondkind Notation2.6.2 112
I[·],K[·] generalizedKoberoperators Section2.7 114
Iα,β,η(·) fractionalintegral/derivative Definition2.7.7 115
0,x
ofthefirstkind