Jordan Charles - Calculus of Finite Differences, 1950.pdf

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CALCULUS
OF
FINITE DIFFERENCES
BY
INTRODUCTION
BY
SECOND
EDITION
CHELSEA PUBLISHING COMPANY
NEW YORK, N.Y.
1950
i
/
4:
PHINTED
IN THE IlNlTED STATES OF AMEHItX
INTRODUCTION
There is more than mere coincidence in the fact that the
recent rapid growth in the theory and application of mathema-
tical statistics has been accompanied by a revival in interest in
the Calculus of Finite Differences. The reason for this pheno-
mena is clear: the student of mathematical statistics must now
regard the finite calculus as just as important a tool and pre-
requisite as the infinitesimal calculus.
To my mind, the progress that has been made to date in
the development of the finite calculus has been marked and
stimulated by the appearence of four outstanding texts.
The first of these was the treatise by George Boole that
appeared in 1860. I do not me& by this to underestimate the
valuable contributions of earlier writers on this subject or to
overlook the elaborate work of Lacroix.’
I merely wish to state
that Boole was the first to present’ this subject in a form best
suited to the needs of student and teacher.
The second milestone was the remarkable work of Narlund
that appeared in 1924. This book presented the first rigorous
treatment of the subject, and was written from the point of
view of the mathematician rather .than the statistician. It was
most oportune. ’
Steffensen’s
Interpolation,
the third of the four texts to
which ‘I have referred, presents an excellent treatment of one
section of the Calculus of Finite Differences, namely interpola-
tion and summation formulae, and merits the commendation of
both mathematicians and statisticians.
I do not hesitate to predict that the fourth of the texts that
1
Volume
3
of
Trait6 du Calcul Diffdrentiel et du Calcul Int&pl,
entitled
Trait6 des diffbences et des shies.
S. F, Lacroix, 1819.
vi
I have in mind, Professor Jordan’s Calculus of Finite Differen-
ces, is destined to remain the classic treatment of this subject
- especially for statisticians - for many years to come.
Although an inspection of the table of contents reveals a
coverage so extensive that the work of more than 600 pages
might lead one at first to regard this book as an encyclopedia
on the subject, yet a reading of any chapter of the text will
impress the reader as a friendly lecture revealing an ununsual
appreciation of both rigor and the computing technique so im-
portant to the statistician.
The author has made a most thorough study of the literature
that has appeared during the last two centuries on the calculus
of finite differences and has not hesitated in resurrecting for-
gotten journal contributions and giving them the emphasis that
his long experience indicates they deserve in this day of mathe-
matical statistics.
In a word, Professor Jordan’s work is a most readable and
detailed record of lectures on the Calculus of Finite Differences
which will certainly appeal tremendously to the statistician and
which could have been written only by one possessing a deep
appreciation of mathematical statistics.
Harry C. Carver.
’
THE AUTHOR’S PREFACE
This book, a result of nineteen years’ lectures on the Cal-
culus of Finite Differences, Probability, and Mathematical Sta-
tistics in the Budapest University of Technical and Economical
Sciences, and based on the venerable works of
Stirling, Euler
and
Boole,
has been written especially for practical use, with
the object of shortening and facilitating the labours of the Com-
puter. With this aim in view, some of the old and neglected,
though useful, methods have been utilized and further developed:
as for instance
Stirling’s
methods of summation,
Boole’s
symbo-
lical methods, and
Laplace’s
method of Generating Functions,
which last is especially helpful for the resolution of equations of
partial differences,
The great practical value of
Newton’s
formula is shown;
this is in general little appreciated by the Computer and the
Statistician, who as a rule develop their functions in power
series, although they are primarily concerned with the differences
and sums of their functions, which in this case are hard to
compute, but easy with the- use of
Newton’s
formula. Even for
-
-
interpolation -itis more advisable to employ
Newton’s
expansion
than to expand the function into a power series.
The importance of
Stirling’s
numbers in Mathematical Cal-
culus has not yet been fully recognised, and they are seldom
used. This is especially due to the fact that different authors
have reintroduced them under different definitions and notations,
often not knowing, or not mentioning, that they deal with the
same numbers. Since
Stirling’s
numbers are as important as
Bernoulli’s,
or even more so, they should occupy a central posi-
tion in the Calculus of Finite Differences, The demonstration of

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