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THE CALCULUS OF FINITE DIFFERENCES
MACM1LLAN AND CO., LIMITED
LONDON • BOMBAY • CALCUTTA * MADRAS MELBOURNE
THE MACM1LLAN COMPANY
NEW YORK • BOSTON • CHICAGO DALLAS ' ATLANTA • SAN FRANCISCO
THH MACMILLAN COMPANY OF CANADA, L1MITKD
TORONTO
THE CALCULUS
OF
FINITE DIFFERENCES
»Y L. M. MJLNETHQMSON
MA, K.R.S.E.
ASSISTANT PROfKSSOR OF MATHEMATICS IN THE ROYAL NAVAL COLLKGE, GREENWICH
MACMILLAN AND CO., LIMITED
ST. MARTIN'S STREET, LONDON
1933
F1UNTEI) IN GUJfiAT BRITAIN
PREFACE
THE object of this book is to provide a simple and connected account of the subject of Finite Differences and to present the theory in a form which can be readily applied.
Two distinct reasons impelled me to undertake this work. First, in my lectures at Greenwich to junior members of the Royal Corps of Naval Constructors I have occasion to treat certain aspects of difference equations ; secondly, the calculation of tables of elliptic functions and integrals, on which I have been recently engaged, gave rise to several interesting practical difficulties which had to be overcome. For both these causes my attention has been directed towards the subject and the lack of a suitable textbook upon which to draw was brought to my notice. The only com prehensive English treatise, namely Boole's Finite Differences, is long since out of print, and in most respects out of date. My first idea was to revise Boole's book, but on looking into the matter it appeared that such a course would be unsatisfactory, if not impracticable. I therefore decided to write a completely new work in which not only the useful material of Boole should find a place, but in which room should also be found for the more modern developments of the finite calculus.
My aim throughout has been to keep in mind the needs of the beginner, so that the book may be regarded as suitable for a first course as well as for more advanced reading. I do not, however, believe that the needs of the beginner in a mathematical subject are best served by eschewing all but the most elementary mathe matical apparatus. Rather, his interest in the subject may well form an adequate opportunity for enlarging his outlook on the science of mathematics, so that he may the better be enabled to distinguish and appreciate the connection of the whole system and the relative dependency of its several parts. Consequently I have not hesitated to use the mathematical process or terminology which has appeared to me most appropriate to the immediate object in view. On the other hand whenever this course seems to
vi PREFACE
lead beyond the elementary matters, with which all who embark on the»reading of a mathematical book must be presumed to be acquainted, I have included the necessary definitions or proofs as part of the text or have given accessible references to treatises which can ordinarily be found in any mathematical library. In this way it has been possible to treat the subject in a simple yet rigorous manner.
The subjectmatter falls naturally into two main divisions which may be subsumed under the headings Interpolation and Difference Equations. The pioneer in interpolation was undoubtedly Briggs, whose work was largely of an arithmetical character. Newton was the originator of the systematic theory and his divided difference formula is really the fundamental basis of all the usual methods of polynomial interpolation. Gregory was probably an independent discoverer in the same field.*
The present work therefore starts with divided differences in Chapter I ; and in a general sense Chapters III, IV and VII may be regarded as elaborations of Newton's work. Chapter V on reciprocal differences, describes a method of interpolation, due to Thiele, by means of rational functions, which is more general than polynomial interpolation, and which will possibly be new to many English readers. Chapter VI introduces the generalisations, due to Norlund, of Bernoulli's polynomials, but here they are treated by a symbolic method, which seemed to me to be as effi cacious and in many ways more suitable than Norlund's method, which is founded upon a different principle. By means of these generalisations the subject of numerical differentiation and integra tion assumes a unified aspect which hardly seems to be attainable without them. Chapters I to VII therefore form a suitable intro ductory course and will make very little demand on the reader's previous mathematical knowledge. I have tried to meet the requirements of those who wish to make numerical applications by giving the formulae in a manner suited to direct use with a table of data. The numerical illustrations scattered through these chapters are mainly of a simple kind which can be easily worked, for it is not my purpose to obscure principles by unnecessary arith metic. The subjectmatter of some of these examples is perhaps
* See H. W. Turnbull, p. 101, footnote.
PREFACE vii
of an unusual nature, but this is intentional in order to lend variety to the applications. In the chapters on Interpolation I have followed Steffensen's excellent example in laying much stress on the remainder term, which measures the error committed in using an interpo lation formula. Indeed, no formula has been given which is unaccompanied by a means of estimating the remainder.
The part of the book which deals with difference equations begins with Chapter VIII, which expounds Norlund's method of treating the summation problem. Chapter IX applies these methods to elaborating the theory of the Gamma function. In Chapter X, I have attempted to give a consecutive account of the salient properties of factorial series, which, I hope, will prove interesting in itself. The object of this chapter is to develop the properties of the series in which the solutions of difference equations find their natural expression. Chapter XI discusses the difference equation of the first order ; the linear case is completely elucidated, and certain amenable nonlinear forms are treated. This chapter includes an investigation of the exact difference equation of the first order. The methods of this, and of succeeding chapters, are illustrated by simple worked examples in the text. Chapter XII considers the properties of the general linear equation, including the application of generalised continued fractions treated by matrix methods. Chapter XIII deals with the important case of constant coefficients. Here the theory is complete, in the sense that the solution can be explicitly obtained. I have dealt with this equation at some length both by Boole's method and by a method of my own, which seems well adapted to applications of a geometrical or physical nature, and which is analogous to Heaviside's method for differential equations. The linear equation with constant coefficients has recently come into prominence in connection with various physical and mechanical problems ; for example in the theory of Structures. Chapters XIV and XVI develop the solution of linear difference equations with variable coefficients by means of Boole's operators, which I have generalised in order to render the treatment more complete. Chapter XV gives an alternative treatment founded on Norlund's use of Laplace's transformation. Chapter XVII gives two fundamental theorems of PoincarS and Perron on the asymptotic properties of the solutions of a certain type of linear difference equation. The proof of Perron's theorem
viii PREFACE
is made to depend upon the properties of a certain class of simul taneous linear equations in infinitely many unknowns. The theory is so interesting and so closely connected with finite differences that it has seemed worth while to give Perron's treatment in extenso.
Operational and symbolic methods have been freely used through out the book, and it is hoped that the manner of presentation here given will be found free from the objections often associated with their use. Indeed it has always seemed to me that symbolic methods constitute the essence of the finite calculus. My choice of notations has therefore been made with a view to facilitating the statement and application of operational methods, and to stressing the analogies with the infinitesimal calculus.
In stating theorems I have as far as possible associated the name of the discoverer as sufficient indication of the origin, but it must not be assumed that the method of presentation is in every case that in which the theorem was originally given. Indeed in the case of the work of the older analysts it would be easy, but un profitable, to point out defects and lack of rigour in many of their proofs.
My labour in correcting the proof sheets has been greatly lightened by Professor H. W. Turnbull, F.R.S., who has read the first proof and made many valuable suggestions both mathematical and historical ; and by Dr. A. C. Aitken, F.R.S.E., who has performed the same kindly office, has supplied many original examples, and has verified the numerical work. To both these friends I wish to express my lively thanks for assistance which has helped me to remove many imperfections both of expression and demonstration. For any blemishes which may remain I am solely responsible, but I am led to express the hope that the work will be found to be free from important errors. I take this opportunity of expressing my thanks to the officials of the Glasgow University Press for the ready way in which they have met my somewhat exacting require ments.
L. M. MILNETHOMSON.
MATHEMATICS DEPARTMENT, ROYAL NAVAL COLLEGE, GREENWICH, July 1933.
CONTENTS
PAGK
INTRODUCTION xxi
NOTATIONS xxii
CHAPTER I DIVIDED DIFFERENCES
10. Definitions 1
11. Newton's interpolation formula with divided differences   2
116. Rolle's Theorem 4
12. Remainder term in Newton's formula ..... 5
13. Divided differences are symmetric functions of the arguments 7
131. Divided differences of xn  7
14. Lagrange's interpolation formula 8
15. Expression of divided differences by means of determinants  9
16. Divided differences expressed by definite integrals 10
17. Divided differences expressed by contour integrals    11
18. Divided differences with repeated arguments 12
19. Interpolation polynomials 14
EXAMPLES I  17
CHAPTER II DIFFERENCE OPERATORS
20. Difference notation ........ 20
201. Central difference notation  22
21. Difference quotients 23
2105. Partial difference quotients 24
211. Difference quotients of factorial expressions 25
212. Expansion of a polynomial in factorials 27
213. Successive difference quotients of a polynomial 28
214. Difference quotients of ax 29
22. Properties of the operator A        30
<•>
23. The operator y 31
b ix
x CONTENTS
PAGE
24. The operator Ew 31
241. Herschel's Theorem 32
242. ^(E")ajr = a*^(O 32
243. <£(£")«*« (*) = a" <Mow £*>(*) 32
25. Relations between A, E", # 33
4l>
251. The analogue of Leibniz' Theorem 34
252. Difference quotients of axvx 35
253. Difference quotients of zero 36
254. Difference quotients in terms of derivates 37
26. The summation operator p"1 37
261. A theorem on the value of p^1 ut 39
262. Relation between sums and functional values 39
263. Moments 40
264. Partial summation 41
27. Summation of finite series 42
271. Summation of factorial expressions of the form x(m)   42
272. Polynomials 43
273. Factorial expressions of the form x(~m) 44
274. A certain type of rational function  45
275. The form ax </>(#), <t>(x) a polynomial 46
276. The form vx<j>(x), <£(#) a polynomial 46
277. Unclassified forms 48
EXAMPLES II 49
CHAPTER III INTERPOLATION
30. Divided differences for equidistant arguments .... 56
31. Newton's interpolation formula (forward differences)   57
311. Newton's interpolation formula (backward differences)   59
312. The remainder term 61
32. Interpolation formulae of Gauss  63
33. Stirling's interpolation formula 67
34. Bessel's interpolation formula 68
341. Modified Bessel's formula 71
35. Everett's interpolation formula 72
36. Steffensen's interpolation formula 74
37. Interpolation without differences 75
381. Aitken's linear process of interpolation by iteration   76
382. Aitken's quadratic process 78
3'83. Neville's process of iteration 81
EXAMPLES III 84
CONTENTS xi
CHAPTER IV NUMERICAL APPLICATIONS OF DIFFERENCES
PAGE
40. Differences when the interval is subdivided 87
41. Differences of a numerical table ...... 88
42. Subtabulation 91
43. Inverse interpolation ........ 95
44. Inverse interpolation by divided differences .... 96
4'5. Inverse interpolation by iterated linear interpolation   97
46. Inverse interpolation by successive approximation 99
47. Inverse interpolation by reversal of series     100
EXAMPLES IV 101
CHAPTER V RECIPROCAL DIFFERENCES
51. Definition of reciprocal differences  104
52. Thielo's interpolation formula       106
53. Matrix notation for continued fractions     108
54. Reciprocal differences expressed by determinants   110
55. Reciprocal differences of a quotient      112
56. Some properties of reciprocal differences     114
57. The remainder in Thielo's formula       116
5*8. Reciprocal derivates ; the confluent case     117
59. Thielo's Theorem 119
EXAMPLES V 122
CHAPTER VI THE POLYNOMIALS OF BERNOULLI AND EULER
60. The </> polynomials 124
601. The p polynomials 126
61. Definition of Bernoulli's polynomials 126
611. Fundamental properties of Bernoulli's polynomials    127
62. Complementary argument theorem 128
63. Relation between polynomials of successive orders    129
64. Relation of Bernoulli's polynomials to factorials    129
6401. The integral of the factorial 131
641. Expansion of x(n) in powers of x 133
642. Expansion of xv in factorials       133
643. Generating functions of Bernoulli's numbers    134
xii
PAG]
65. Bernoulli's polynomials of the first ordefN^^^ 136
6501. Sum of the vth powers of the first n integers*1  137
651. Bernoulli's numbers of the first order  137
6*511 . EulerMaclaurin Theorem for polynomials  139
652. Multiplication Theorem 141
653. Bernoulli's polynomials in the interval (0, 1 )  141
66. The 73 polynomials  142
67. Definition of Euler's polynomials  143
6'71. Fundamental properties of Euler's polynomials    144
672. Complementary argument theorem ....  145
673. Euler's polynomials of successive orders  145
68. Euler's polynomials of the first order 146
681. Euler's numbers of the first order 147
6*82. Boole's Theorem for polynomials 149
EXAMPLES VI 150
CHAPTER VII NUMERICAL DIFFERENTIATION AND INTEGRATION
70. The first order derivate 154
701. Derivates of higher order   155
702. Markoff's formula 157
703. Derivates from Stirling's formula  159
704. Derivates from Bessel's formula  161
705. Differences in terms of derivates       162
71. Numerical integration         162
7101. Mean Value Theorem 163
711. Integration by Lagrange's interpolation formula    164
712. Equidistant arguments 165
713. Remainder term, n odd         166
714. Remainder term, n even        167
72. Cotes' formulae 168
721. Trapezoidal rule 170
722. Simpson's rule 171
723. Formulae of G. F. Hardy and Weddle 171
73. Quadrature formulae of the open type 172
731. Method of Gauss 173
733. Method of Tschebyscheff 177
74. Quadrature formulae involving differences     180
741. Laplace's formula 181
742. Laplace's formula applied to differential equations    183
743. Central difference formulae 184
CONTENTS xiii
PAGE
75. EulerMaclaurin formula  187
751. Application to finite summation    191
76. Gregory's formula 191
77. Summation formula of Lubbock      193
EXAMPLES VII 196
CHAPTER VIII THE SUMMATION PROBLEM
80. Definition of the principal solution or sum  201
81. Properties of the sum  204
811. Sum of a polynomial  208
812. Repeated summation 208
815. Proof of the existence of the principal solution (real variable)  209
816. Bernoulli's polynomials Bv (x \ co) 213
82. Differentiation of the sum       213
821. Asymptotic behaviour of the sum for large values of x   214
822. Asymptotic behaviour of the sum for small values of co   216
83. Fourier series for the sum ....... 218
84. Complex variable. Notation. Residue Theorem    220
841. Application of Cauchy's residue theorem .... 222
85. Extension of Jhe theory 226
8«53. The sum of the exponential function     231
86. Functions with only one singular point  232
87. An expression for F (x \ co) 238
EXAMPLES VI11 238
CHAPTER IX THE PSI FUNCTION AND THE GAMMA FUNCTION
90. The function^' (x \ co) 241
901. Differentiation of the Psi function  241
903. Partial and repeated summation 243
91. Asymptotic behaviour for large values of # . . . 244
911. Partial fraction development of ^ (x \ co )  ... 245
92. Multiplication theorem for the Psi function  ... 246
922. Fourier series for ¥ (x) 247
93. Gauss' integral for ¥ (x) 247
932. Poisson's integral 248
94. Complementary argument theorem for the Psi function   249
95. The Gamma function 249
xiv CONTENTS
PAGE
952. Schlomilch's infinite product for F(x + l)  250
953. Infinite products expressed by means of T (x) ... 251
954. Complementary argument theorem for T (x)     251
955. The residues of T (a;) 252
956. Determination of the constant c 252
96. Stirling's series for log T (x +h) 253
961. An important limit 254
966. Generalised Gamma function T (x \ co) 255
967. Some definite integrals 256
968. Multiplication theorem of the Gamma function    257
97. Euler's integral f or T (x) 257
972. Complementary Gamma function Tl (x) 258
9*8. Hypergeometric series and function F (a, b ; c ; x)    260
982. Hypergeometric function when x = 1 261
984. The Beta function B(#, y) 262
986. Definite integral for the hypergeometric function    264
988. Single loop integral for the Beta function .... 265
989. Double loop integral for the Beta function .... 266
EXAMPLES IX 267
CHAPTER X FACTORIAL SERIES
100. Associated factorial series 272
1002. Convergence of factorial series 273
1004. Region of convergence 275
1006. Region of absolute convergence 276
1007. Abel's identities 276
1008. The upper limit of a sequence 277
1009. Abscissa of convergence. Landau's Theorem    279
10091. Ma jorant inverse factorial series 283
101. Series of inverse factorials 284
1011. Uniform convergence of inverse factorial series    284
1013. The poles of 12 (x) 287
1015. Theorem of unique development 288
102. Application of Laplace's integral ; generating function   288
1022. Order of singularity and the convergence abscissa    292
103. The transformation (x, x +m) 293
1032. The transformation (x, a:/co) 294
104. Addition and multiplication of inverse factorial series   295
1042. Differentiation of inverse factorial series .... 297
1043. An asymptotic formula 298
CONTENTS xv
PAGE
1044. Integration of inverse factorial series 299
105. Finite difference and sum of factorial series .... 300
106. Newton's factorial series 302
10*61. Uniform convergence of Newton's series 302
1063. Null series 304
1064. Unique development 305
1065. Expansion in Newton's series ; reduced series  306
1067. Abscissa of convergence of Newton's series .... 309
107. Majorant properties 310
108. Eulcr's transformation of series 311
1082. Generating function 312
1083. Laplace's integral and Newton's series 314
1085. Expansion of the Psi function in Newton's series    315
109. Application to the hypergeometric function    316
EXAMPLES X 317
CHAPTER XI THE DIFFERENCE EQUATION OF THE FIRST ORDER
110. Genesis of difference equations 322
1101. The linear difference equation of the first order    324
111. The homogeneous linear equation 324
112. Solution by means of the Gamma function. Rational cocllicients 327
113. Complete linear equation of the first order .... 328
1131. The case of constant coefficients 329
1132. Application of ascending continued fractions ...  330
1133. Incomplete Gamma functions 331
1134. Application of Prym's functions 332
1 1 4. The exact difference equation of the first order    334
1141. Multipliers 339
1142. Multipliers independent of x 339
1143. Multipliers independent of u 340
115. Independent variable absent. Haldane's method    341
1151. Boole's iterative method 343
11*6. Solution by differencing. Clairaut's form .... 344
117. Equations homogeneous in u 346
118. Riccati's form 346
119. Miscellaneous forms 347
EXAMPLES XI 348
xvi CONTENTS
CHAPTER XII
GENERAL PROPERTIES OF THE LINEAR DIFFERENCE EQUATION
PAGE
120. The homogeneous linear difference equation    351
1201. Existence of solutions 352
121. Fundamental system of solutions 353
1211. Casorati's Theorem 354
1212. Heymann's Theorem 357
1214. Relations between two fundamental systems .... 359
1216. A criterion for linear independence 360
122. Symbolic highest common factor 361
1222. Symbolic lowest common multiple 363
1224. Reducible equations 366
123. Reduction of order when a solution is known .... 367
124. Functional derivates 369
125. Multiple solutions of a difference equation .... 370
126. Multipliers. Adjoint equation 372
127. The complete linear equation. Variation of parameters   374
1272. Polynomial coefficients 377
128. Solution by means of continued fractions .... 378
EXAMPLES XII 381
CHAPTER XIII
THE LINEAR DIFFERENCE EQUATION WITH CONSTANT COEFFICIENTS
130. Homogeneous equations 384
1302. Boole's symbolic method 387
131. Complete equation 388
132. Boole's operational method   392
1321. Case I, <j>(x) = xm 393
1322. Case II, <f>(x) = ax 396
1323. Caselll, </> (x) = axR (x) 398
1324. The general case 398
1325. Broggi's method for the particular solution .... 401
1326. Solution by undetermined coefficients 403
133. Particular solution by contour integrals 404
1332. Laplace's integral 407
134. Equations reducible to equations with constant coefficients  408
CONTENTS xvii
PAGE
135. MilneThomson's operational method 410
1351. Operations on unity 411
1352. Operations on a given function X 412
1353. Application to linear equations witli constant coefficients  413
1354. Simultaneous equations 415
1355. Applications of the method   415
136. Simultaneous equations 420
137. Sylvester's nonlinear equations 420
138. Partial difference equations with constant coefficients   423
1381. Alternative method 425
1382. Equations resolvable into first order equations    426
1383. Laplace's method 427
EXAMPLES XIII 429
CHAPTER XIV
THE LINEAR DIFFERENCE EQUATION WITH RATIONAL COEFFICIENTS. OPERATIONAL METHODS
140. The operator p 434
1401. The operator n 436
1402. Inverse operations with TT 437
1403. The operators ni9 pA 439
141. Theorem I 439
1411. Theorem II 440
1412. Theorem III 440
1413. Theorem IV 442
1414. Theorem V 443
142. Formal solution in series 445
1421. Solution in Newton's series 448
1422. Exceptional cases 451
143. Asymptotic forms of the solutions 457
1431. Solutions convergent in a halfplane on the left    459
144. The complete equation 460
145. Monomial difference equations 461
146. Binomial equations 465
147. Transformation of equations. Theorems VI, VIT, VIII   467
1471. Equation with linear coefficients 469
1473. Theec{ua,tioTL(ax*+bx+c)u(x)+(ejc+f)u(xl)+gu(x2) = 0 472
2
1475. The equation (ax2 + bx +c) Aw +(ex +f) Aw +gu  0    474
148. Bronwin's method 475
14*9. Linear partial difference equations 475
xviii CONTENTS
PAGE
1 491 . Laplace's method for partial equations 476
EXAMPLES XIV 477
CHAPTER XV
THE LINEAR DIFFERENCE EQUATION WITH RATIONAL COEFFICIENTS. LAPLACE'S TRANSFORMATION
150. Laplace's transformation 478
151. Canonical systems of solutions 482
152. Factorial series for the canonical solutions .... 485
153. Asymptotic properties ,  487
15*31. Casorati's determinant 488
154. Partial fraction series 490
155. Laplace's difference equation 491
1551. Reducible cases 493
1552. Hypergeometric solutions  494
1553. Partial fraction series 495
1554. Relations between the canonical systems 496
1555. The case a^ =a2 498
156. Equations not of normal form 500
EXAMPLES XV 501
CHAPTER XVI
EQUATIONS WHOSE COEFFICIENTS ARE EXPRESSIBLE BY FACTORIAL SERIES
160. Theorem IX 504
1601. Theorem X 505
161. First normal form 508
162. Operational solution of an equation of the first normal form  509
163. Convergence of the formal solution 511
164. Example of solution        516
165. Second normal form 518
166. Note on the normal forms 519
EXAMPLES XVI 521
CHAPTER XVII THE THEOREMS OF POINCARfi AND PERRON
170. The linear equation with constant coefficients  523
171. Poincari's Theorem 526
CONTENTS xix
PAGE
17 2. Continued fraction solution of the second order equation   532
173. Sum equations 534
174. Homogeneous sum equations with, constant coefficients.
Theorem I 537
175. A second transformation of sum equations .... 539
176. General solution of sum equations. Theorem II 542
177. Difference equations of Poincare's type. Perron's Theorem  548
EXAMPLES XVII 550
INDEX 553
INTRODUCTION
LET f(x) be a given function of the variable x. The Differential Calculus is concerned with the properties of
which is still a function of the single variable x. On the other hand the Calculus of Differences is concerned with the properties of
CO
which is a function of the two variables x and o>.
More generally, in contrast with the Infinitesimal Calculus, the Finite Calculus is concerned with the values of a function at a set of isolated points and with such properties as may be derivable there from.
Suppose then that we are given the numbers f(xl),f(x2),f(x^)9 ... , and an argument x different from XL, x2) x3, ... . Among the subjects of enquiry which naturally present themselves are the following.
(i) The determination of f(x) from the given functional values. This is the Interpolation problem.
(ii) The determination of
/'(*), \bfWdx.
J a
These are the problems of Numerical Differentiation and Integration. Extending our enquiries in another direction we are led to con sider the properties of the functions f(x) denned by the equation
where g(x) is a given function. This constitutes the Summation problem, which is analogous to the problem of integration in the Integral Calculus.
xxii INTRODUCTION
On the theory of summation we are able to found in a satisfactory manner the theory of the Gamma function which plays such an important part in the Calculus of Differences.
Consideration of more general relations between /(#),/(# fco), ... , /(xfnco) brings us to the study of difference equations, which are analogous to the differential equations of the Infinitesimal Calculus.
NOTATIONS
THE following list, which is intended only for reference, contains the symbols, operators, and functions which occur most frequently in this book. The numbers refer to the sections where explanations are given.
SYMBOLS
(m\ __ m(m 1) ._. . (mn+ 1) \n)~ nT '
lim f(x), the limit of f(x) when
x~*a x tends to a.
lim sup xn, Upper limit, 1008.
n»oo
f(x) ~ g(x), 822.
==, symbolic equivalence, 241.
R(x), the real part of #, 84.
Rn(%)> Rn, Remainder Terms, 11, 711.
 a? , argz, 84.
or, Arbitrary Periodic Function, 111.
... xn], Divided Difference, 10.
= x(x  co) (x  2co) . . . (# wco + co), 211.
co)1, 211.
n
A0m, Difference Quotient of w Zero, 253.
Bernoulli's Numbers of order n, 61. ), Euler's Numbers of order n,
67,
Bv, Bernoulli's Numbers, 65. EV) Euler's Numbers, 68. 0(zn), 98.
D,
Ax
"
V,
Difference Operator, 20. {JL§, Central Difference Oper
ators, 201.
Difference Quotient Oper
ator, 21.
Differentiation Operator, 21. > Partial Difference Quotient,
2105. 23.
NOTATIONS xxiii
OPERATORS
E", 24.
p1, Summation Operator, 26. p, pn, Reciprocal Difference, 51. r, rn, Reciprocal Derivate, 58.
, Sum Operator, 80.
TC, p, 1401, 140. TC,, plf 1403.
FUNCTIONS ex, exp x, Exponential Function.
Bin\x), Bernoulli's Polynomial of order n, 61.
Bv(x\ co), Generalised Bernoulli's Polynomial, 816.
E^\x)9 Euler's Polynomial of order n, 67.
Bv(x\ Bernoulli's Polynomial, 65.
Ev(x), Euler's Polynomial, 68.
Pm(x), Periodic Bernoulli Func tion, 75.
B(z, y), Beta Function, 984 r(x), Gamma Function, 95. rl(x), Complementary Gamma
Function, 972. r(xd)), Generalised Gamma
Function, 966. ^(x  co), Psi Function, 90. ¥(x), Psi Function, 90. F(x\ co), Sum Function, 80. F(a, b ; c ; x), Hypergeometric
Function, 98. Q(ic), Factorial Series Sum
Function, 101.
M] DIVIDED DIFFERENCES
or
f(x) = f(xj + (x  a^) [x^2] + (x xj (x 
+ (xx1)(xx2)...(xxn)[xx1x2...xn], or
(1) /(z)=/(ZiH2(z*i)(z^
*l
(2) where Rn (x) = (x  x^ (x  x2) . . . (x  xn) [xx^2 . . . xn].
This is Newton's general interpolation formula with the remainder term Rn(x). The formula is of course a pure identity and is therefore true without any restriction on the form of f(x).
By means of this formula the evaluation of a f unction /(x) whose value is known for the values xl9 x2, ... , xn of the variable is reduced to the problem of evaluating the remainder term Rn (x). Should this term be known or negligible, the required value/(#) can be calculated from Newton's formula.
It should be observed that no particular rule is laid down for the sequence of the arguments x, xv x2, ... " xn, which need not be in ascending or descending order of magnitude.
Iff(x) be a polynomial of degree n — 1 in x, then
xxl
is of degree n  2 in x. Hence the operation of taking the divided difference of a polynomial lowers the degree by unity. Conse quently the divided difference of the (n  1 )th order of a polynomial of degree n  1 is constant, and therefore the divided difference of the nth order is zero, that is, [xx± ... xn] — 0.
In this case En(x) == 0, so that the value of f(x) given by the formula
nl
(3) f(x) = f(xj + ^l(xx1)...(x xa) [X& . . . xs+l]
*=!
is exact.
Iff(x) be not a polynomial, we see that
Rn(x) = 0, for x = xl9 x2, ... , xn,
4 DIVIDED DIFFERENCES [11
so that the righthand member of (3) yields the polynomial of degree n~ 1 whose value coincides with the value of f(x) for
X = a?j_, XZi • •• , Xn.
Example. Find approximately the real root of the equation «
y32y5 = 0. Let x = y3  2y  5.
This relation defines a function y~f(x). We want the value of /(O). Attributing suitable values to y we obtain the folio wfng table of divided differences.
x f(x)
1941 19
+ •10627 1000 20 0060
+ • 09425 + 0005
+ 0061 21 0044
+ 08425 +0003
+ 1248 22 0034
+ 07582 + 2567 23
Thus approximately, using (3) above, we have y = 20+ 1 x 09425 + 1 x 061 x 0044 + 1 x 061 x 1248 x 0003 = 209454.
The corresponding value of a? is  000013 and the above value of y is in error by about one unit in the last digit.
1 15. Rolle's Theorem. In order to discuss the form of the remainder term in Newton's formula we need the following theorem known as Rolle's theorem.
If the function f(x) be continuous and differentiate in the interval a^x^ib, where a, b are two roots of the equation f(x) = 0, then the equation f (x) = 0 has at least one root interior to the interval (a, b).
Proof. If f(x) have the constant value zero, the theorein is evident. If f(x) be not constantly zero, it will take positive values pr negative values. Suppose that f(x) takes positive values. r£hen
115] DIVIDED DIFFERENCES , 5
f(x) being continuous will attain a maximum value M for some point £ such that a < £ < b. Thus, if h be positive,
h
is negative or zero and hence the limit when A>0, namely /'(!•), cannot be a positive number, that is,
Similarly, by considering the ratio
we prove that
/'(5)>o.
Thus we have /'(£) = 0 and the theorem is proved. In the same way we prove the theorem for the case when f(x) takes negative values.
1 *2. The Remainder Term. From 11 (2), we have
nl
where Pn _x (a;) = f(xj f X (x  xi)  •  (x ~ x*) lxixz • • • xs+i\>
*i
so that Pni(x) is a polynomial of degree w1, and its (wl)th derivate is
(1) P<tf>(x) = (nl)\[x1xt...xn].
Hitherto f(x) has been unrestricted. We now suppose that in the interval .(a, 6) bounded by the greatest and least of x, xv x2, ... , xn the function f(t) of the real variable ty and its first nl derivates are finite and continuous and that/<n)(£) exists.
Then since Rn(t) vanishes when t = xvx2,...9 xn, by Rolle's theorem it follows that Rn (t) vanishes at n — 1 points of (a, b) and therefore by a second application of the theorem that Rn (t) vanishes at w2 points of (a, 6). Proceeding in this way we see that fifi1"" r*(>j) = 0 where TJ is some point of (a, b).
Thus /*1>)Pfcri1)) = 0,
6 DIVIDED DIFFERENCES [12
so that
(2) [Xix2 ••• xn\ ~ /n~~ j\j 9
which is a formula expressing the divided difference of order n — 1 in terms of the (nl)th derivate of f(x) at some point of (a, b). Hence we have
[xx^... xn\ — — j , where £ is some point of (a, 6). We have therefore
W\ 7? (r\ — (r Y \ (T Y \ (v T \ ^ ^'
\°) l\,n\JU) — {JU — JU^^Ji — d,%) ... \Jb — xn) ™t '
where ^ is some point of (a, 6).
This important result enables us to find an upper limit to the error committed in omitting the remainder term, provided that we can find an upper limit for the nth derivate off(t) in the interval bounded by the greatest and least of x, xv x2, ... , xn.
Example. Find an approximate value of Iog10 401 from the following table :
X 
Iog10 x 

40002 
06020 817 

+• 108431 

40104 
•6031 877 
•0136 

+ •108116 

40233 
•6045 824 
 0130 

+ 107869 
40294 6052 404
The divided differences are as shewn. Thus approximately
log 401 = 6020817 + 0098 x 108431 + 0098 x 0004 x 0136 = 6031444,
which is correct to seven places.
The error due to the remainder term is of order
0098 x 0004 x 0133 x 4343 x 2
12] DIVIDED DIFFERENCES 7
where x varies between 40002 and 40294, which is less than 2 in the 10th decimal place. The above value could therefore be affected only by errors of rounding in the seventh place.
13. The Divided Differences are Symmetric Functions of the Arguments. By definition
L XJ ,~, /y, fy> iy
Jj ~~~ tASt JUt Jj
so that we obtain without difficulty
\XX X 1  /^ 4 /to) _ 4. __
I t«/vt'*C'9 I — , k y » ~ , . / \ I /
•*• ** ( T />>ll/y* T*l I'T* T\ ( T T* I i/>*
Idly — *t/i I I«A/ "~~ ^O/ \ 1 / V 1 "~~ 2/ \ 2 **
It is now very easily proved by induction that
l~\\ f ' VTf T* 1* 1 *
i A y I A/A/* a/0 • • • «*/fi I
f /.vX f/^v. \
/W L J\Xl)
~~ ~ . ~t~
(xx1)(xx2)...(xxn) (X1x)(x1x2)...(x1xn)
+ .._. /to)
(xn  x) (xn xj... (xn  xn_r) '
Clearly the interchange of any two of the arguments does not alter the value of the divided difference, which is therefore a symmetric function of its n arguments.
For example [^^2] — [^r*^] ~ [#2X;ril Again [x^xz . . . «n_txwxn+1] = [xnxzx3 . . . xn,^xn+l} , so that
~~ x
n+l
xn ~~ xn {I
131. The divided differences of xn can be obtained as follows : from 13 (1),
1H 1 ^ n
f 1 (Xs ~ Xl) (X8 ~~X2/ • • • to •" ^p
8 DIVIDED DIFFERENCES [131
This last is the coefficient of tn~p in the expansion of
^ _ *.* ._ _____
/u (x,  za) . . . (xt  x^) (1  xtt) (xs  XM) . . . (x.  xp+l)
But this expression is evidently the result of putting into partial fractions the function
and hence * the coefficient of tn~p is the sum of the homogeneous products of degree np of xv x2, ... , xp+l.
Thus [x±x2 . . . xv+I] = S Xj°l x2a* . . . xpa%i l,
where the summation is extended to all positive integers including zero which satisfy the relation al + a2+ ... +aphl — np.
For the divided differences of  we have
x
and this is the value when t = 0 of
1
which is obtained by putting into partial fractions
 1
so that
(1)"
~
1*4. Lagrange's Interpolation Formula. From 13 (1) we have
m fix\
(1) /(*)
.ti\ (xx1)(x
a,)". (xzxn)  ,
where Rn (x) = (xx1)(xx2) ... (x xn) [xx±x2 . . . xn].
*G. Chrystal, Algebra, 2nd edition, (London, 1919), 205.
14] DIVIDED DIFFERENCES 9
This is Lagrange's Interpolation Formula with the remainder term Rn(x). Comparing with 11 (2) we see that this remainder term is the same as the remainder term in Newton's Formula. It follows that Lagrange's Formula has exactly the same range of application as Newton's and yields identical results.
The formula may also be written in a slightly different form. Write (2) <£(z) = (xx1)(xx2)...(xxn).
/Q\ TU^*, ft™\ \.^ J \ S/ T\ / i TJ /,«\
(o) xnen T\X) — / t ~ ~~j~» ~r jLi>n(X).
1 *5. Expression of Divided Differences by means of Deter m i nants. By a wellknown theorem in determinants origin ally due to Vandermonde and generalised by Cauchy, we have
(1)
1