CALCULUS OF FINITE DIFFERENCES jordan

  

CALCULUS

  O F

  

FINITE DIFFERENCES

BY

  

INTRODUCTION

B Y

SECOND EDITION

CHELSEA PUBLISHING COMPANY

NEW YORK, N.Y.

  

1950

  

I N T R O D U C T I O N

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 of four outstanding texts.

  The first of these was the treatise by George Boole that

appeared in 1860. I do not by this to underestimate the

valuable contributions of earlier writers on this subject or to

overlook the elaborate work of 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

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 the statistician. It was

most oportune.

  Steffensen’s Interpolation, the third of the four texts to

which 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 Volume 3 of d u C a l c u l D i f f d r e n t i e l e t d u C a l c u l v i

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 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. tion in the Calculus of Finite Differences, The demonstration of

  

Bernoulli’s, or even more so, they should occupy a central posi-

  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

  numbers are as important as

  Stirling’s

  numbers in Mathematical Cal- culus has not yet been fully 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

  The importance of

  formula. Even for

  Newton’s

  Newton’s

  THE AUTHOR’S PREFACE

  The great practical value of

  method of Generating Functions, which last is especially helpful for the resolution of equations of partial differences,

  Laplace’s

  symbo- lical methods, and

  Boole’s

  methods of summation,

  Stirling’s

  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 has been written especially for practical use, with the object of shortening and facilitating the 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

• interpolation more advisable to employ Newton’s expansion than to expand the function into a power series.

a great number of formulae is considerably shortened by using these numbers, and many new formulae are obtained by their aid; for instance, those which express differences by derivatives or vice formulae for the operations and and many others; formulae for the inversion of certain sums, for changing the length of the interval of the differences, for summation of reciprocal powers, etc.

  In this book the functions especially useful in the Calculus of Finite Differences, such as the Factorial, the Binomial Coef- ficient, the Digamma and Trigamma Functions, and the

  Bernoulli

  and Euler Polynomials are fully treated. Moreover two species of polynomials, even more useful, analogous to those of Bernoulli and Euler, have been introduced; these are the Bernoulli nomiali of the second kind and the polynomials

  Some new methods which permit great simplifications, will also be found, such as the method of interpolation without printed differences which reduces the cost and size of tables to a minimum. Though this formula has been especially deduced for Computers working with a calculating machine, it demands no more work of computation, even without this aid, than Everett’s formula, which involves the use of the even dif- ferences. Of course, if a table contains both the odd and the even differences, then interpolation by

  Newton’s

  formula is the shortest way, But there are very few tables which contain the first three differences, and hardly any with more than three, which would make the table too large and too expensive; moreover, the advantage of having the differences is not very . great, provided one works with a machine, as has been shown, even in the case of linear interpolation 133). So the printing of the differences may be considered as superfluous.

  The construction of Tables has been thoroughly treated 126 and 133). This was by no means superfluous, since nearly

  

the existing tables are much too large in comparison with the

  precision they afford. A table ought to be constructed from the point of view of the interpolation formula which is to be

  

number of the decimals in the table, are given, then this deter-

mines the range or the interval of the table. But generally, as

is shown, the range chosen is ten or twenty times too large, or

the interval as much too small; and the table is therefore

table

unnecessarily bulky. If the were reduced to the proper

dimensions, it would be easy and very useful to add another

table for the inverse function. A method of approximation by aid of orthogonal polyno-

mials, which greatly simplifies the operations, is given. Indeed,

the orthogonal polynomials are used only temporarily, and the

so result obtained is expressed by Newton’s formula

that no tables are necessary for giving the numerical values of

the orthogonal polynomials.

  In 143 an exceedingly simple method of graduation ac-

cording to the principle of least squares is given, in which it is

only necessary to compute certain “orthogonal” moments cor-

responding to the data. In the Chapter dealing with the numerical resolution of

equations, stress has been laid on the rule of False Posifion,

which, with the slight modification given 127 and 149, and

Example 1, in enables us to attain the required precision

in a very few steps, so that it is preferable, for the Computer, to

every other method, The Chapters on the Equations of Differences give only those

methods which really lead to practical results. The Equations of

Partial Differences have been especially considered. The method

shown for the determination of the necessary initial conditions

will be found very useful 181). The very seldom used, but

advantageous, way of solving Equations of Partial Differences

by Laplace’s method of Generating -Functions has been dealt with

and somewhat further developed and examples given.

   184)

The neglected method of Fourier, Lagrange and Ellis has

been treated in the same way.

  Some formulae of Mathematical Analysis are briefly men-

tioned, with the object of giving as far as possible everything

necessary for the Computer.

  Unfamiliar notations, which make the reading of ma- possible avoided. The principal used are given on pp. xix-xxii. To obviate another difficulty of reading the works on

  Finite Differences, in which nearly every author uses other defi- nitions and notations, these are given, for all the principal authors, in the respective paragraphs in the Bibliographical Notes.

  Though this book has been written as has been said above, especially for the use of the computer, nevertheless it may be considered as an introductory volume to Mathematical Statistics and to the Calculus of Probability.

  I owe a debt of gratitude to my friend and colleague Mr.

  Professor of Mathematics in the University of Buda- A. pest, who read the proofs and made many valuable suggestions; moreover to Mr. Philip Redmond, who kindly revised the text from the point of view of English.

  

CONTENTS.

  

Chapter On Operations.

  1 1. Historical and Bibliographical Notes , . . .

  2 2. Definition of differences . . . . .

  5 3. Operation of displacement . . . . . . .

  6 4. Operation of the mean . . . . . .

  7 .

   5. Symbolical Calculus . , , . . .

  8 , 6. Symbolical methods , , . . . . . .

  14 7. Receding Differences . . . . , . .

  15 8. Central Differences . . . . . . . . .

  18 9. Divided Differences , . . . . . . . . .

  20 10. Generating functions . , . . . . .

  25

   11. General rules to determine generating functions .

  29 12. Expansion of functions into power series . .

   13. Expansion of function by aid of decomposition into partial fractions . . . . . . . , . 34

  40

   14. Expansion of functions by aid of complex integrals ,

  41

   15. Expansion of a function by aid of difference equations

II. Functions important in the Calculus of Finite Differences.

  45 16. The Factorial . . . . . . . . . .

  53 The Gamma function . . , . . . . .

  56 18. Incomplete Gamma function . . , . . . .

  58 19. The Digamma function . . . . , . . .

  60

   20. The Trigamma function . . . . . . . ,

  61 . .

   21. Expansion of into a power series

  62 22. The Binomial coefficient . . . . . . .

   23. Expansion of a function into a series of binomial

  74

coefficients . . . . . . . . . . . xii

  24. Beta-functions . . , . . . . , . , 80

  25. Incomplete Beta-functions . . , . . . 83

  26. Exponential functions . . . . . , . . 87

  27. Trigonometric functions . . . , . . . . 88 28, Alternate functions . . . . . . . . . 92

  29. Functions whose differences or means are equal to zero . . . . . . . . . . , . . 94

  30. Product of two functions. Differences . , . . 94

  31. Product of two functions. Means . . . , . . 98

Chapter Inverse Operation of Differences and Means. Sums.

  32. Indefinite sums . . , . . . . , . . 100 33, Indefinite sum obtained by inversion . . . . 103 34, Indefinite sum obtained by summation by parts . . 105

  108 35. Summation by parts of alternate functions . . .

  36. Indefinite sums determined by difference equations , 109

  37. Differences, sums and means of infinite series . , 110

  111 38. Inverse operation of the mean , . . . . .

  . . 113 39. Other methods of obtaining inverse means .

  40. Sums . . . , , . . , . . . , 116 1 1 7

  41. Sums determined by indefinite sums , , , .

  . 1 2 1 42. Sum of reciprocal factorials by indefinite sums .

  123 43. Sums of exponential and trigonometric functions .

  44. Sums of other Functions , . . . . . . 129

  45. Determination of sums by symbolical formulae , , 131 . 136 46. Determination of sums by generating functions .

  47. Determination of sums by geometrical considerations 138

  48. Determination of sums by the Calculus of Probability 1 4 0

  49. Determination of alternate sums starting from usual sums , . , . , , . . . . , . 140

Chapter IV. Stirling’s Numbers. Expansion of factorials into power series. Stirling’s

  numbers of the 142 first kind , . . . . . .

  51. Determination of the Stirling numbers starting from their definition , . . , , . . , . . 145

  52. Resolution of the difference equations . . . . 147

  . .

  69. The operation . , . . . . . , .

  1 7 4 1 7 7 1 7 9

  1 7 3

  166 1 6 8

  1 6 4

  1 5 3 1 5 9 1 6 3

  77. Stirling’s polynomials , . . , . . , .

  75. Changing the origin of the intervals . . . . , 76. Changing the length of the interval . . .

  74. Expansion of the function into a series of powers of x . . , . , . . , , . ,

  73. Expansion of a function into reciprocal factorial series and into reciprocal power series , . , .

  72. Expansion of a function of function by aid of Stirling numbers. Semi-invariants of Thiela .

  The operation Operations and

  68. Expansion a reciprocal factorial into a series of reciprocal powers and vice versa , , . . .

  53. Transformation of a multiple sum “without repeti- tion” into sums without restriction , . . . .

  67. Differences expressed by derivatives , . . .

  64. Formulae containing Stirling numbers of both kinds 65. Inversion of sums and series. Sum equations . . 66, Deduction of certain formulae containing Stirling numbers . . , . , . . . . , , .

  63. Application of the expansion of powers into series of factorials , . . . . . . , . , .

  62. Decomposition of products of prime numbers into factors . . . . . . . . . . . .

  61. Stirling numbers of the second kind obtained by probability . , . , . . . , . . .

  60. Generating functions of the Stirling numbers of the second kind . . . . . . . . . . .

  59. Limits of expressions containing Stirling numbers of the second kind . . . , . . . . . .

  58. Stirling numbers the

  57. Stirling numbers of the first kind obtained by bility

  56. Derivatives expressed by differences

  55. Application of the Stirling numbers of the first kind

  54. Stirling’s numbers expressed by sums. Limits .

  181 1 8 2 1 8 3 1 8 5 189 1 9 2 195 199 200 204 212 216 219 220 224

V. Bernoulli Polynomials and Numbers.

  78. Bernoulli polynomials of the first kind . , . , 79. Particular cases of Bernoulli polynomials . .

  93. Limits of the numbers b,, and of the polynomials

  102. Expansion of the Euler polynomials into a series

  100. Euler’s polynomials . . . . . , . 288 101. Symmetry of the Euler polynomials . . . . 292

  2 3 0 2 3 6 238 2 4 0 242 246 248 2 5 0 252 253 2 6 0 2 6 5 268 269 272 272 275 276 277 277 280 284

  99. Gregory’s summation formula . . . . . .

  98. The Bernoulli series of the second kind . . . .

  97. Expansion of a polynomial into a series of Bernoulli polynomials of the second kind . . . . .

  96. Application of the polynomials

  95. Expansion of the polynomials of second kind , .

  94. Operations on the Bernoulli polynomials of the second kind .

  92. Particular cases of the . . .

  80. Symmetry of the Bernoulli polynomials . . . .

  91. Extrema of the polynomials .

  90. Symmetry of the Bernoulli polynomials of second kind. . . . . , . , . . . . .

  89. Bernoulli polynomials of the second kind

  88. The Maclaurin-Euler summation formula . , .

  87. The Bernoulli series . . . , . , . .

  86. Raabe’s multiplication theorem of the Bernoulli polynomials . . . . . . . . , . ,

  Generating functions , , . . . . . .

  84. Expansion of a polynomial into Bernoulli polynomials 85. Expansion of functions into Bernoulli polynomials.

  83. Application of the Bernoulli polynomials . ,

  82. Expansion of the Bernoulli polynomial into a Fourier series. Limits. Sum of reciprocal power series . .

  81. Operations performed on the Bernoulli polynomial ,

Chapter VI. Euler’s and Boole’s polynomials. Sums of reciprocal powers.

  Operations on the Euler polynomials . , . . 104. The Tangent-coefficients . . . . , , .

  300 302 303 306 307 311 313 315 317 320 321 322 323 325

  129. Everett’s formula . . . . , . . . . 376 130. Inverse interpolation by Everett’s formula .

  126. Inverse interpolation by Newton’s formula . 366 127. Interpolation by the Gauss series . , . . . 368 128. The Bessel and the Stirling series . , . . . 373

  357 Construction of Tables . . . . . . , “360

  . . , . 125. Interpolation by aid of Newton’s Formula and

  123. Expansion of a Function into a series of polynomials 355 124. The Newton series .

  

Chapter VII. Expansion of Functions, interpolation,

Construction of Tables.

  338 347

  330 335

  296 298

  Euler numbers . . , , . . . . . . Limits of the Euler polynomials and numbers .

  . . . 122. Sum of alternate reciprocal powers by the i)‘,(x) function . . . . . . . . . . .

  . 120. Sum of a rational fraction . . , . . . . 121. Sum of reciprocal powers. Sum of .

  119. Sum of by aid of the trigamma function .

  116. Expansion of a function into Boole polynomials . 117. Boole’s second summation formula . , . . , 118. Sums of reciprocal powers. Sum of l/x by aid of the digamma function . . . . , . , .

  114. Operations on the Boole polynomials. Differences . 115. Expansion of the Boole polynomials into a series of Bernoulli polynomials of the second kind . . .

  . . Boole’s first summation formula . . . , , Boole’s polynomials . . . . . . . .

  Expansion of a polynomial into a series of Euler polynomials . . . . . . . . . . 110. Multiplication theorem of the Euler polynomials . 111. Expansion of a function into an Euler series .

  108. Application of the Euler polynomials .

  , Expansion of the Euler polynomials into Fourier series , . . , . . . . . . . ,

  . . 381 131. Lagrange’s interpolation formula . , . . . 385

  xvi 132. Interpolation formula without printed differences. 390 Remarks on the construction of tables . 133. Inverse Interpolation by aid of the formula of 411

preceding paragraph . , . . . . . ,

417

  , 134. Precision of the interpolation formulae . . . 420 135. General Problem of Interpolation . . , , .

Chapter VIII. Approximation and Graduation.

   136. Approximation according to the principle of 422

moments . . . . . . . . . . .

426 137. Examples of function chosen , . . . , . 138. Expansion of a Function into a series of Legendre’s 434

polynomials . , . . . . . . . .

139. Orthogonal polynomials with respect to 426 . . . , . , . . . , . . . , 140. Mathematical properties of the orthogonal poly- 442

nomials . , . . . . , . . . .

  

Expansion of a function into a series of orthogonal

447

polynomials . . . , . , . . . .

142. Approximation of a function given for . . . , 451

iv-l . . . . . . . . . . . .

456 143. Graduation by the method of least squares . . 460 144. Computation of the binomial moments . . , . 463 . . . . . . , . Fourier series , 146. Approximation by trigonometric functions of dis- 465

continuous variables . . . , . . , .

467 147. polynomials , . , . . . , . 473 148. G. polynomials . , , . , . . , .

  Numerical Chapter IX. Numerical resolution of equations.

integration.

  . . . 486 149. Method of False Position. Regula . . 489 150. The Newton-Raphson method . . . , . 492 Method of Iteration , , 494 152. Daniel Bernoulli’s method 496 . . . 153. The Ch’in-Vieta-Horner method 154. Root-squaring method. Dandelin,’ Lobs

  xvii

  155. Numerical Integration , , , . . , , . 512 156. Hardy and Weddle’s formulae . . . . . . 516 157. The Gauss-Legendre method . . . , . , 517

  158. Tchebichef’s formula . . . . . . , . 519 159. Numerical integration of functions expanded into a series of their differences . , , , , .

  524

  , 160. Numerical resolution of differential equations .

  527

Chapter X. Functions of several independent variables.

  161. Functions of two variables ,

  530

  162. Interpolation in a double 532 163. Functions of three variables . . . , . , 541

Chapter XI. Difference Equations.

  164. Genesis of difference equations . . . . . .

  543

  165. Homogeneous linear difference equations, constant coefficients . . . . . . . . . . .

  545 .

  166. Characteristic equations with multiple roots , 549 167. Negative roots . , . , . , , . , 552

  168. Complex roots . . . . . , . , . . 554 169. Complete linear equations of differences with constant coefficients . . , . . , , .

  557

  170. Determination of the particular solution in the general case , . , , . . , , .

  564 . . .

  171. Method of the arbitrary constants . 569 172. Resolution of linear equations of differences by aid of generating functions . . . . . . , 572

  173. Homogeneous linear equations of the first order with variable coefficients . . , , , . .

  576

  174. Laplace’s method for solving linear homogeneous difference equations with variable coefficients . ,

  579

  175. Complete linear equations of differences of the first order with variable coefficients . . . . . 583 176. Reducible linear equations of differences with riable coefficients , . . . , . . . , 584

  177. Linear equations of differences whose coefficients are polynomials of x, solved by the method of generating functions . . , . . . , .

  586

  178, Andre’s method for solving difference equations

  587

  179. Sum equations which are reducible to equations of . differences . 599

  180. Simultaneous linear’ equations differences with constant coefficients . , . , . . , . 600

Chapter XIII. Equations of Partial Differences.

  181. Introduction

  604

  182. Resolution of of differences with constant coefficients by Laplace’s method of generating . . , . , , . . 607

  183. Boole’s symbolical method for solving equations of 616 partial differences . . . . . . 184. Method of Fourier, Lagrange, and Ellis for solving equations of partial differences . . . . . . 619 185. Homogeneous linear equations of mixed differences

  632

  186. Difference equations of three independent variables 633 187, Differences equations of four independent variables 638

  1

NOTATIONS AND DEFINITIONS

  . 232 coefficient of the Bernoulli . . 427 arithmetical mean of x . . , . . , , . . 163

  , binomial moment of order , . 233 B,, , Bernoulli numbers . . . . . . , 341 function . . , . . , . . . 80 Beta function . . . . . . . . 8 3 incomplete Beta function , , , .

  3 9 5 numbers (interpolation) . . , , . . 449 numbers (approximation) . . . . , 5 8 C, Euler’s constant . . . . . . . , 150 C numbers . . . , . . , . . . 1 7 1 numbers . , . . , . . . , , 388 Cotes numbers (only in 131 and 155) . . 395 C,(x), (interpolation) . , . . , . . 453 numbers (approximation) . . . . 454 numbers (approximation) . , ,

  3 , derivative . . . . . . . . .

  2 difference, interval one . . . . .

  2 difference, interval

  h . . , . .

  ,

  2 difference with respect to interval

  h

  , 15 central difference . . . . . , .

  18 divided difference . . . . . . 1 0 1 inverse difference . . . . . .

  58 F(x), digamma function . . . . ,

  6 E, operation of displacement . . . . Euler numbers . . . , .

  E,,

  , 288 . Euler polynomial of degree .

  E,(x),

  , 290 coefficient of the Euler polynomial , .

   , tangent-coefficient . E , , ( S ) , coefficients in in

Bernoulli polynomial of degree . . . . ,

function, (Czuber, Jahnke) , , . . . . .

generating function . . . , . . . , .

  6

  moment . . . , . . ,

   Stirling numbers of the second kind . . . . mean binomial

   , , , Stirling numbers of the first kind . . . . .

   Stirling polynomials (only in 77) . . .

psi-function . . . . . , . . .

operation . . . . , , . , . . . indefinite sum or inverse difference . . , definition of , . . . . . . . . _S sum of reciprocal powers of degree . sum of reciprocal powers of degree m

  57 n,

(except in 77) . . . . . . .

  15 111 163 163 221

  83 394 385 512

  

G,(x), polynomial of degree . , . . , ,

gamma function . . . . . . . . . .

incomplete gamma function . , , . .

, numbers, (graduation) . . . . , . , .

polynomial of degree , . . .

incomplete gamma function divided by the cor-

responding complete function . . , . . .

incomplete Beta function divided by the cor-

responding complete function . . , . , .

, (interpolation) , . . . . . . . . . .

  56

  56 467

  53

  21 473

  

Poisson’s probability function . , . , ,

Bernoulli polynomial of the second kind of degree

298 378 231 403

  

Lagrange polynomial of degree , , . , .

Langrange polynomial , . . , . , . .

operation of the mean , , . . . , . . . central mean . . , , . . . . . ,

inverse mean . . . . . . . , , . .

power-moment of degree n . . . . , . .

factorial-moment of degree . . . . . .

numbers . . . . . . . . , , ,

  , 265 . 224 . 342 . 199 . 101 , 117 . 244 . 142 , 168 . 448

   mean orthogonal moments . . . . . . . 450 operation , , . . , , , , . , . , 195 U,(x), orthogonal polynomial of degree . . . , 439

(x),, , factorial of degree . . . . . , . . , 45

generalised factorial of degree . . . . 45

   . . . , . . . . , . 53

   I

, binomial coefficient of degree . . . . . . 64

  X I , generalised binomial coefficient . . . , . 70 Boole polynomial of degree’ . . . . . . 317

CHAPTER I. O N O PERATIONS . Historical and Bibliographical Notes. The most

  important conception of Mathematical Analysis is that of the function. If to a given value of x a certain value of y correspond, we say that y is a function of the independent variable x.

  Two sorts of functions are to be distinguished. First, func- tions in which the variable x may take every possible value in a given interval; that is, the variable is continuous. These func- tions belong to the domain of Infinitesimal Calculus. Secondly, functions in which the variable x takes only the given values _{. . then the variable is discontinuous. To such} functions the methods of Infinitesimal Calculus are not applicable.

  The Calculus of Finite Differences deals especially with such functions, but it may be applied to both categories.

  The origin of this Calculus may be ascribed to Brook (London , but the real

  Taylor’s

  founder of the theory was Jacob Stirling, who in his Differentialis (London , 1730) solved very advanced questions, and gave useful methods, introducing the famous Stirling numbers; these, though hitherto neglected, will form the back- bone of the Calculus of Finite Differences.

  The first treatise on this Calculus is contained in Leonhardo

   Institutiones Calculi Differentialis

  rialis Scientiarum Petropolitanae, 1755. See also Opera Omnia, Series I. Vol. X. 1913) in which he was the first to introduce the symbol for the differences, which is universally used now.

  From the early works on this subject the interesting article

  2

  written by Charles Bossut, should be mentioned, also,

F. S. Lacroix’s

  des differences et series” Paris, 1800.

2. Definition of the differences, A function f(x) is given

  

Newton’s general interpolation formula is based on these dif-

  D.

  for x x,,, . . . , x,,. In the general case these values are not equidistant. To deal with such functions, the “Divided Dif-

  G. Interpolation und Quadratur, Leipzig. 1932.

   B. Scarborough, Numerical Mathematical Analysis, Baltimore, 1930.

   F. Sfeffensen,

Interpolation, London, 1927.

   E. Berlin, 1924.

  E. T. and G. Robinson, Calculus of Observations, London, 1924.

  Lehrbuch der Leipzig, 1904.

   Leipzig, 18%.

  ferences. The Calculus, when working with divided differences, is always complicated, The real advantages of the theory of Finite Differences are shown only if the values of the variable x are equidistant; that is if

  A. A.

   The most important treatises on the Calculus of Differences are the following: George A treatise on the Calculus of Finite Differences, Cambridge, 1860.

  ferences” have been introduced, We shall see them later 9).

  

A

  The symbol is not complete; in fact the independent va- riable and its increment should also be indicated. For instance thus:

   f ( x + h ) - f ( x ) .

  In this case, the first difference of f(x) will be defined by the increment of f(x) corresponding to a given increment h of the variable x. Therefore, denoting the first difference by we shall have

   h where h is independent of i.

  This must be done every time if there is any danger of a misunderstanding, and therefore must be considered as an abbreviation of the symbol above.

  3

  Often the independent variable is obvious, but not the increment; then we shall write A, omitting h only in the case _h of h If the increment of is equal to one, then the formulae of the Calculus are much simplified. Since it is always possible to introduce into the function f(x) a new variable whose increment is equal to one, we shall generally do so. For instance if and the increment of is h, then we put from this it follows that that is, will increase by one if increases by h. Therefore, starting from f(x) we find

  

f(x) F(i)

  and operate on putting finally into the results obtained

   instead of

  We shall call second difference of f(x) the difference of its first difference. Denoting it by (x) we have = = =

f (x+h) + f(x).

In the same manner the n-th difference of f(x) will be defined by

   = = . Remark. In Infinitesimal Calculus the first derivative of a

  function f(x) generally denoted by or more briefly by

   (x) (if there can be no misunderstanding), is given by lim .

  Moreover it is shown that the derivative of f(x) is l i m If a function of continuous variable is given, we may determine the derivatives and the integral of the function by using the methods of Infinitesimal Calculus. From the point of

  4 view of the Calculus of Finite Differences these functions are treated exactly in the same manner as those of a discontinuous WC variable; may determine the differences, and the sum of the h function; but the increment and the beginning of the intervals must be given, For instance, log x may be given by a table from to where Generally we write the values of the function in the first column of a table, the first differences in the second column, the third, and so on. the second differences in If we begin the first column with then we shall write

the first difference (a) in the line between f(a) and f

the second difference will be put into the row between and so on. We have. _h a n d f

  I 2h)

  I It should be noted that proceeding in this way, the expressions with the same argument are put in a descending line; and that the arguments in each horizontal line are decrea- sing. The reason is that the notation u s e d a b o v e f o r the differences is nof symmetric with respect to the argument.

  5 Differences of functions with negative arguments. If we have _h f ( x ) - f ( x ) then according to our definition

Q f ( - x ) f ( - x - h ) - f ( - x ) .

this it follows that f ( - x ) that is, the of the function is diminished by h.

  In the same manner we should obtain = = f ( x ) , This formula will be very useful in the following.

  Difference of a sum. It is easy to show that

=

• moreover if C is a constant that According to these the difference of a polynomial

   . . . + . . , + 3, Operation of displacement.2 An important operation Boole denoted this operation by D; but since D is now universally used as a symbol for derivation, it had to be changed. De la in his Cours Tome II, p. denoted

operation by Pseudodelta We have adopted here since this is generally

used in England. So for instance in

   Britannica, the edition, 1910, in W. F. Sheppard, Differences (Calculus of., Vol. VIII 223.

  E. and G. Robinson, cit. p. 4.

   E. cit. 1. p. 4.

cit. 1, p. 31.

  L. M. Milne-Thomson.

  This operation has already been considered by L. F. A. [Du des derivations, Strasbourg, he called it an operation of

  

F. proposed for this operation the symbol which was also used

b y c i t .

  

C. Jordan, in his Cours [Second edition tom. I,

also introduced this operation and deduced several formulae by aid of

symbolical methods. His notation was

f nh).

  6 was introduced into the Calculus of Finite Differences by cit. 1. the operation of displacement. This consists, f(x) being given, in increasing the variable by h. Denoting the operation by we have

  

This symbol must also be considered as abbreviation of

The operation will be defined by E f ( x + h ) f and in the same way

  

E [ f ( x )

It is easy to extend this operation to negative indices of so that we have f ( x ) f ( x - h ) f ( x ) f ( x - n h ) .

4. Operation of the Mean. The operation the mean introduced by Sheppard cit. 2) corresponds to the system of Central Differences which we shall see later. We shall denote the operation corresponding to the system of differences considered in 2, by Its definition is

  

= +

7 The operation will be defined in the same manner by

   f(x) f(x) f(x) + Of course the notation is an abbreviation of --- Returning to our table of 2, we may write the first column between f(a) and the number (a); between

   Sheppard denoted the central difference by and the corresponding mean by Since corresponds to it is logical that should central

correspond to Thiele introduced for the central mean the symbol

which has also been adopted by Sfeffensen cit. 1, p, cit. 1. denoted our mean by the symbol Pseudodelta This also h a s b e e n a d o p t e d b y c i t . W e h a v e s e e n that other authors have already used the symbol for the operation of

  7 and the number and so on. In the second column we put (a) between and

continuing in this manner we shall obtain for instance the

following lines of our table

5. Symbolical Calculus. It is easy to show that the con-

  

sidered operations represented by the symbols , , and

are distributive; for instance that we have

• + + = +

  Moreover they are commutative, for instance f ( x ) = = a n d

=