Mathai A, Haubold H - Special Functions for Applied Scientists.pdf

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A.M. Mathai
•
Hans J. Haubold
Special Functions
for Applied Scientists
123
A.M. Mathai
McGill University
Montreal, QC
Canada
and
Centre for Mathematical Sciences
Pala Campus, Kerla
India
mathai@math.mcgill.ca
Hans J. Haubold
Office for Outer Space Affairs
UN, Vienna
Austria
hans.haubold@unvienna.org
ISBN: 978-0-387-75893-0
e-ISBN: 978-0-387-75894-7
DOI: 10.1007/978-0-387-75894-7
Library of Congress Control Number:2007942158
c 2008 Springer Science+Business Media, LLC
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Preface
The first S.E.R.C. (Science and Engineering Research Council of the Department of
Science and Technology, Government of India, New Delhi) School on Special Func-
tions was sponsored by DST (Department of Science and Technology), New Delhi,
and conducted by CMS (Centre for Mathematical Sciences) for six weeks in 1995.
The second S.E.R.C. School on Special Functions and Functions of Matrix Argu-
ment was sponsored by DST, Delhi, and conducted by CMS for five weeks during
the period 29th May to 30th June 2000. In the second School, the main lectures were
given by Dr. H.L. Manocha of Delhi, India, Dr. S. Bhargava of Mysore, India, Dr.
K. Srinivasa Rao and Dr. R. Jagannathan of Chennai, India and Dr. A.M.Mathai of
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. Moothathu and Dr. Yageen Thomas of the University of
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. The first two Schools were conducted in Trivandrum (Thiruvananthapuram)
area. Dr. A.M. Mathai was the Director and Dr. K.S.S. Nambooripad was the Co-
Director of these two Schools.
The third S.E.R.C. School on Special Functions and Functions of Matrix Argu-
ment: Recent Developments and Recent Applications in Statistics and Astrophysics,
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
lecture notes to be collected from the main lecturers in advance, compiled and dis-
tributed prior to the start of the School. The main lectures for the 3rd School were
given by Dr. Hans J. Haubold of the Office of Outer Space Affairs of the United
Nations, Dr. Serge B. Provost, Professor of Actuarial Sciences and Statistics of the
University of Western Ontario, Canada, Dr. R.K. Saxena of Jodhpur, India, Dr. S.
Bhargava of Mysore, India and Dr. A.M. Mathai of Montreal, Canada. Supplemen-
tary lectures were given by Dr. A. Sukumaran Nair (Chairman, CMS), Dr. K.K.
Jose (Director-in-Charge, CMS Pala Campus), Dr. R.N. Pillai, Dr. Yageen Thomas,
Dr. V. Seetha Lekshmi, Dr. Alice Thomas, Dr. E. Sandhya, Dr. S. Satheesh and
v
vi
Preface
Dr. K. Jayakumar. Problem sessions were supervised by Dr. Sebastian George and
other lecturers. Extra training in the use of statistical packages, LATEX and Math-
ematica/Maple were given by Joy Jacob, Seemon Thomas and K.M. Kurian of the
Department of Statistics of St. Thomas College, Pala. A special lecture sequence on
Matlab was conducted by Alexander Haubold of Columbia University, U.S.A.
The 4th S.E.R.C. School on Special Functions and Functions of Matrix Argu-
ment: Recent Advances and Applications to Stochastic Processes, Statistics and As-
trophysics, sponsored by DST, Delhi, was conducted by CMS for five weeks from
6th March to 7th April 2006 at its Pala Campus. The main lectures were scheduled
to be given by Dr. Hans Joachim Haubold of the Office of Outer Space Affairs of the
United Nations, Dr. P.N. Rathie of the University of Brasilia, Brazil, Dr. S. Bhargava
of Mysore, India, Dr. R.K. Saxena of Jodhpur, India, Dr. K.K. Jose of Pala, India and
Dr. A.M. Mathai of Montreal, Canada. Supplementary lectures were given by Dr. A.
Sukumaran Nair (Chairman, CMS), Dr. K.S.S. Nambooripad (Director-in-Charge,
CMS Trivandrum Campus), Dr. R. Y Denis of Gorakhpur, India, Dr. N. Unnikrish-
nan Nair (former Vice-Chancellor of Cochin University of Science and Technology,
Kerala, India), Dr. Yageen Thomas and Dr. K. Jayakumar. One week was devoted
to Stochastic Processes and recent advances in this area in the 4th School. Problem
sessions were supervised by Dr. Joy Jacob, Dr. Sebastian George and other lecturers.
The 5th S.E.R.C. School in this sequence, titled “Special Functions and Functions
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-
tion No.34 in the Publications Series of CMS. The main lectures were scheduled to
be given by Dr. Hans J. Haubold (Office of Outer Space Affairs of the United Na-
tions, Austria), Dr. H.M. Srivastava (University of Victoria, Canada), Dr. R.Y. Denis
(Gorakhpur University, India), Dr.R.K. Saxena (Jodhpur University, India), Dr. S.
Bhargava (Mysore University, India), Dr. D.V. Pai (IIT Bombay, India), Dr. Yageen
Thomas (University of Kerala, India), Dr. K.K. Jose (Mahatma Gandhi University,
India), Dr. J.J. Xu (China/Canada), Dr. K. Jayakumar (Calicut University, India)
and Dr. A.M. Mathai (Canada/India). But Dr. H.M. Srivastava, Dr. S. Bhargava and
Dr. Xu could not reach the venue on time due to unexpected emergencies. Supple-
mentary lectures were given by Dr. A. Sukumaran Nair (Chairman,CMS), Dr. R.N.
Pillai (former Head, Department of Statistics, University of Kerala, India), Dr. N.
Mukunda (IISc, Bangalore, India) and the problem sessions were supervised by Dr.
Sebastian George, Dr. Joy Jacob, Dr. Seemon Thomas and the lecturers. For the
2005, 2006 and 2007 Schools, Dr. A.M. Mathai was the Director and Dr. K.K. Jose
was the Co-Director of the Schools.
The participants for the S.E.R.C. Schools are selected on all-India basis. All the
expenses of the selected candidates, total number of seats is 30, including travel,
accommodation, food, lecture materials, local travels etc are met by the DST, Delhi,
Government of India. Foreign participation is allowed under the conditions that the
Preface
vii
participants must come with their own return air tickets and must attend all the lec-
tures, and problem sessions and take all the examinations and tests from the begin-
ning till the end. All their local expenses are met and lecture materials are provided
by the Schools. There is no fee for attending the Schools. The Schools are mainly
research orientation courses aimed at young faculty members in colleges and uni-
versities across India, below 35 years of age, and fresh graduates with M.Sc, M.Phil,
Ph.D degrees below 30 years of age, in Mathematics/Statistics/Theoretical Physics.
In most of the Indian universities rigid compartmentalization is the order and as a
result a M.Sc graduate in mathematics may not have even taken a very basic course
in probability and statistics. Even though basic differential and integral calculus and
matrix theory are required subjects for statistics and physics students, these stu-
dents may have forgotten these subjects because they teach the compartmentalized
topics in their own areas and do not usually do research even in their narrow ar-
eas of their own fields. For maintaining their jobs and getting regular increments in
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
teaching and the information passed on to the students go down from year to year.
In order to remedy this situation a little bit, S.E.R.C. Schools in various areas were
started by DST, Delhi. Dr. A.M. Mathai was asked to run S.E.R.C. Schools in math-
ematics. For taking up a challenging research problem in any applied area a good
background in basic mathematics, probability and statistics is required. In order to
give the basic ideas in probability and statistics and to bring a number of topics in
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
were established. The current sequence of five Schools really achieved the aim and
almost all the participants in the first four Schools have become research oriented
towards a career in research and teaching, and the participants of the 5th School are
also expected to follow the same footsteps of their predecessors.
Chapter 1 introduces elementary classical special functions. Gamma, beta, psi,
zeta functions, hypergeometric functions and the associated special functions, gen-
eralizations to Meijer’s G and Fox’s H-functions are also examined here. Discussion
is confined to basic properties and some applications. Introduction to statistical dis-
tribution theory is given here. Some recent extensions of Dirichlet integrals and
Dirichlet densities are also given. A glimpse into multivariable special functions
such as Appell’s functions and Lauricella functions is also given. Special functions
as solutions of differential equations are also examined here.
Chapter 2 is devoted to fractional calculus. Fractional integrals and fractional
derivatives of various kinds are discussed here. Then their applications to reaction-
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
and Ramanujan’s work on elliptic and theta functions. Chapter 5 examines the topic
of Special Functions and Lie Groups.

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