UNITED STATES DEPARTMENT OF COMMERCE Luther H. Hodges, Secretary NATIONAL BUREAU OF STANDARDS A. V. A.tiD, Director

Handbook of Mathematical Functions

With

Formulas, Graphs, and Mathematical Tables

Edited by Milton AhraIIlowitz and Irene A. Stegun

National Bureau of Standards Applied Mathematics Series 55

Issued June 1964

Tenth Printing, December 1972, with corrections

For .ale by the Superiotend .... t of Document.. U.S. Government Printing Omce W .. hington. D.C. 20402 - Price '11.35 dOlDeetio potItpaid, or flO.50 GPO Book.tore

The text relating to physical constants and conversion factors (page 6) has been modified to take into account the newly adopted 8ysteme Interna-tional d'Unites (81).

ERRATA NOTICE The original printing of this Handbook (June 1964) contained

errors that have been corrected in the reprinted editions. These cor-rections are marked with an asterisk (*) for identification. The errors occurred on the following pages: 2-3,6-8,10,15,19-20,25,76,85,91,102, 187, 189-197,218,223,225,233,250,255,260-263, 268,271-273,292,302, 328,332,333-337,362,365,415,423,438-440,443,445,447,449,451,484, 498, 505-506,509-510,543,556,558,562,571,5~5, 599,600,722-723, 739, 742, 744,746,752,756,760-765,774,777-785,790,79~801,822-823,832, 835,844, 886-889,897,914,915, 920,930-931~936,940-941,944-950,953, 960, 963, 989-990, 1010, 1026.

Originally issued June 1964. Second printing, November 1964. Third printing, March 1965. Fourth printing, December 1965. Fifth printing, August 1966. Sixth printing, November 1967. Seventh printing, May 1968. Eighth printing, 1969. Ninth printing, November 1970.

Librnry of Congress Catalog Card Number: 64-60036

n

Preface: The present volume is an outgrowth of a Conference on Mathematical Tables

held at Cambridge, Mass., on September 15-16, 1954, under the auspices of the National Science Foundation and the Massachusetts Institute of Technology. The purpose of the meeting was to evaluate the need for mathematical tables in the light of the availability of large scale computing machines. It was the consensus of opinion that in spite of the increasing use of the new machines the basic need for tables would continue to exist.

Numerical tables of mathematical functions are in continual demand by scien-tists and engineers. A. greater variety of functions and higher accuracy of tabula-tion are now required as a result of scientific advances and, especially, of the in-creasing use of automatic computers. In the latter connection, the tables serve mainly for preliminary surveys of problems before programming for machine operation. For those without easy access to machines, such tables are, of course, indispensable.

Consequently, the Conference recognized that there was a pressing need for a modernized version of the classical tables of functions I[)f Jahnke-Emde. To imple-ment the project, the National Science Foundation requested the National Bureau of Standards to prepare such a volume and established an Ad Hoc Advisory Com-mittee, with Professor Philip M. Morse of the Massachusetts Institute of Technology as chairman, to advise the staff of the National Bureau of Standards during the

-course of its preparation. In addition to the Chairman, the Committee consisted of A. Erdelyi, M. C. Gray, N. Metropolis, J. B. Rosser, H. C. Thacher, Jr., John Todd, C. B. Tompkins, and J. W. Tukey.

The primary aim has been to include a maximum of useful information within the limits of a moderately large volume, with particullar attention to the needs of scientists in all fields. An attempt has been made to cover the entire field of special functions. To carry out the goal set forth by the Ad Hoc Committee, it has been necessary to supplement the tables by including the mathematical properties that are important in computation work, as well as by providing numerical methods which demonstrate the use and extension of the tables.

The Handbook was prepared under the direction of' the late Milton Abramowitz, and Irene A. Stegun. Its success has depended greatly upon the cooperation of many mathematicians. Their efforts together with the cooperation of the Ad Hoc Committee are greatly appreciated. The particular contributions of these and other individuals are acknowledged at appropriate places in the text. The sponsor-ship of the National Science Foundation for the preparation of the material is gratefully recognized.

It is hoped that this volume will not only meet the needs of all table users but will in many cases acquaint its users with new functions.

Ar.LEN V. ASTIN, Director.

Washington, D.C. m

/

Preface to the Ninth Printing The enthusiastic reception accorded the "Handbook of Mathematical

Functions" is little short of unprecedented in the long history of mathe-matical tables that began when John Napier published his tables of loga-rithms in 1614. Only four and one-half years after the first copy came from the press in 1964, Myron Tribus, the Assistant Secretary of Com-merce for Science and Technology, presented the 100,000th copy of the Handbook to Lee A. DuBridge, then Science Advisor to the President. Today, total distribution is approaching the 150,000 mark at a scarcely diminished rate.

The success of the Handbook has not ended our interest in the subject. On the contrary, we continue our close watch over the growing and chang-ing world of computation and to discuss with outside experts and among ourselves the various proposals for possible extension or supplementation of the formulas, methods and tables that make up the Handbook.

In keeping with previous policy, a number of errors discovered since the last printing have been corrected. Aside from this, the mathematical tables and accompanying text are unaltered. However, some noteworthy changes have been made in Chapter 2: Physical Constants and Conversion Factors, pp. 6-8. The table on page 7 has been revised to give the values of physical constants obtained in a recent reevaluation; and pages 6 and 8 have been modified to reflect changes in definition and nomenclature of physical units and in the values adopted for the acceleration due to gravity in the revised Potsdam system.

The record of continuing acceptance of the Handbook, the praise that has come from all quarters, and the fact that it is one of the most-quoted scientific publications in recent years are evidence that the hope expressed by Dr. Astin in his Preface is being amply fulfilled.

November 1970

LEWIS M. BRANSCOMB, Director National Bureau of Standards

Foreword This volume is the resullt of the cooperative effort of many persons and a number

of organizations. The National Bureau of Standards has long been turning out mathematical tables and has had under consideration, for at least 10 years, the production of a compendium like the present one. During a Conference on Tables. called by the NBS Applied Mathematics Division on May 15, 1952, Dr. Abramo-witz of that Division mentioned preliminary plans for such an undertaking, but indicated the need for technical advice and financial support.

The Mathematics Division of the National Research Council has also had an active interest in tables; since 1943 it has published the quarterly journal, "Mathe-matical Tables and Aids to Computation" (MT AC) l' editorial supervision being exercised by a Committee of the Division.

Subsequent to the NBS Conference on Tables in 1952 the attention of the National Science Foundation was drawn to the desirability of financing activity in table production. With its support a 2-day Conference on Tables was called at the Massachusetts Institute of Technology on September 15-16, 1954, to discuss the needs for tables of various kinds. Twenty-eight persons attended, representing scientists and engineers using tables as well as table producers. This conference reached COnsensus on several conclusions and recommendations, which were set forth in the published Report ~f the Conference. There was general agreement, for example, "that the advent of high-speed computing equipment changed the task of table making but definitely did not remove the need for tables". It was also agreed that "an outstanding need is for a Handbook of Tables for the Occasional Computer, with tables of usually encountered functions and a set of formulas and tables for interpolation and other techniques useful to the occasional computer". The Report suggested that the NBS undertake the production of such a Handbook and that the NSF contribute financial assistance. The Conference elected, from its participants, the following Committee: P. M. Morse (Chairman), M. Abramowitz, J. H. Curtiss, R. W. Hamming, D. H. Lehmer, C. B. Tompkins, J. W. Tukey, to help implement these and other recommendations.

The Bureau of Standards undertook to produce the recommended tables and the National Science Foundation made funds available. '1'0 provide technical guidance to the Mathematics Division of the Bureau, which carried out the work, and to pro-vide the NSF with independent judgments on grants for the work, the Conference Committee was reconstituted as the Committee on Revision of Mathematical Tables of the Mathematics Division of the National Research Council. This, after some changes of membership, became the Committee which is signing this Foreword. The present volume is evidence that Conferences can sometimes reach conclusions and that their recommendations sometimes get acted on.

v

/

/

VI FOREWORD

Active work was started at the Bureau in 1956. The overall plan. the selection of authors for the various chapters, and the enthusiasm required to begin the task were contributions of Dr. Abramowitz. Since his untimely death, the effort has continued under the general direction of Irene A. Stegun. The workers at the Bureau and the members of the Committee have had many discussions about content, style and layout. Though many details have had to be argued out as they came up, the basic specifications of the volume have remained the same as were outlined by the Massachusetts Institute of Technology Conference of 1954.

The Committee wishes here to register its commendation of the magnitude and quality of the task carried out by the staff of the NBS Computing Section and their expert collaborators in planning, collecting and editing these Tables, and its appre-ciation of the willingness with which its various suggestions were incorporated into the plans. We hope this resulting volume will be judged by its users to be a worthy memorial to the vision and industry of its chief architect, Milton Abramowitz. We regret he did not live to see its publication.

P. M. MORSE, Chairman. A. ERDELYI M. C. GRAY N. C. METROPOLIS J. B. ROSSER H. C. THACHER. Jr. JOHN TODD 'C. B. TOMPKINS J. W. TUKEY.

Contents Pa.ge

Preface . . rn Foreword . v Introduction IX

1. MathematicaJ Constants . 1 DAVID S. LIEPHAN

2. Physical Constants and Conversion Factors 5 A. G. McNISH

3. Elementa.ry Analytical Methods . . . 9 MILTON ABRAMOWITZ '

4. Elementa.ry Transcenoontal Functions . 65 Logarithmic, ExponentiaI, Circular and Hyperbolic Functions

RUTH ZUCKER 5. Exponential Integral and Related Functions . . .

WALTER GAUTSCHI and WILLIAM F. CAHlLL 6. Gamma Function and Related Functions.

PHlLIP J. UAVIS 7. Error Function and Fresnel In tegrals '.

WALTER GAuTscm 8. Legendre Functions .....

IRENE A. STEGUN 9. Bessel Functions of Integer Order .

F. W J.OLVER 10. Bessel Functions of Fractional Order.

H. A. ANTOSIEWICZ 11. Integrals of Bessel Functions. . . .

y UD ELL L. LUXE 12. Struve Functions and Related Functions .

MILTON ABRAMOWITZ 13. Con.fluent Hypergeometc Functions

Lucy J OAN SU TER 14. Coulomb Wave Functions

MILTON ABRAMOWITZ 15. Hypergeometc Functions

FRITZ OBERHETTINGER 16. Jacobian Ellip~ic Functions and Theta Functions

L. M. MILNE-THOM8oN 17. Elliptic IntegraJ8 . . . . . . . . . . . .

L. M. MILNE-THOM80N 18. Weierstr88S Elliptic and Related Functions .

THOMAS H. SOUTHARD 19. Parabolic Cylinder Functions. . . . . . .

J. C. P. MILLER

227

253

295

331

355

435

479

495

503

537

555

567

587

627

685

VII

-

vm CONTENTS

Page 20. Mathieu Functions . . . . 721

GERTRUDE BLANCH 21. Spberoidal Wave Functions.

ARNOLD N. LOWAN 22. Orthogonal Polynomials . .

URS W. HOCHSTRASSER 23. Bernoulli and Euler Polynomials, Riemann Zet a Function

EMILIE V. HA YNSWORTH and KARL GOLDBERG 24. Combinatorial Analysis . . . . . . . . . . . . . . .

K. GOLDBERG, M. NEWMAN and E. HA YNSWORTH 25. Numerical Interpolation, Differentiation and Integration.

PHILIP J. DAVIS and IVAN POLON8KY 26. Probability Functions . . . . . . . . . . . .

MARVIN ZELEN and NORMAN C. SEVERO 27. Miscella.neous Functions .

lRENE A. STEGUN 28. Scales of Notation. . . .

S. PEA VY and A. SCHOPF 29. Laplace Transforms . Subject Index . . Index of Notations . . .

751

771

803

821

875

925

997

1011

1019 1031 1044

Handbook of Mathematical Functions with

Formulas, Graphs, and Mathematical Tables Edited by Milton Abramowitz and Irene A. Stegun

I. Introduction The present Handbook has been designed to

provide scientific investigators with a compre-hensive and self-contained summary of the mathe-matical functions that arise in physical and engi-neering problems. The well-known Tables of Functions by E. Jahnke and F. Emde has been invaluable to workers in these fields in its many editions! during the past half-century. The present volume extends the work of these authors by giving more extensive and more accurate numerical tables, and by giving larger collections of mathematical properties of the tabulated functions. The number of functions covered has also been increased.

The classification of functions and organization of the chapters in this Handbook is similar to that of An Index of Mathematical Tables by A. Fletcher, J. C. P. Miller, and L. Rosenhead. 2 In general, the chapters contain numerical tables, graphs, polynomial or rational approximations for automatic computers, and statements of the principal mathematical properties of the tabu-lated functions, particularly those of computa-

tional importance. Many numerical examples are given to illustrate the use of the tables and also the computation of function values which lie outside their range. At the end of the text in each chapter there is a short bibliography giving books and papers in which proofs of the mathe-matical properties stated in the chapter may be found. Also listed in the bibliographies are the more important numerical tables. Comprehen-sive lists of tables are given in the Index men-tioned above, and current information on new tables is to be found in the National Research Council quarterly Mathematics of Computation (formerly Mathematical Tables and Other Aids to Computation).

The ma.thematical notations used in this Hand-book are those commonly adopted in standard texts, particularly Higher Transcendental Func-tions, Volumes 1-3, by A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi (McGraw-Hill, 1953-55). Some alternative notations have also been listed. The introduction of new symbols has been kept to a minimum, and an effort has been made to avoid the use of conflicting notation.

2. Accuracy of the Tables

The number of significant figures given in each table has depended to some extent on the number available in existing tabulations. There has been no attempt to make it uniform throughout the Handbook, which would have been a costly and laborious undertaking. In most tables at least five significant fig'ures have been provided, and the tabular intervals have generally been chosen to ensure that linear interpolation will yield. four-or five-figure accuracy, which suffices in most physical applications. Users requiring higher

1 The most recent, the sixth, with F. Loesch added as co-author, was published In 1960 by McGraw-Hill, U.S.A. and Teubner, Germany.

The second edition, with L. J. Comrie added as co-author, was published In two volumes In 1962 by Addison-Wesley, U.S.A., and Scientific Com-puting Service Ltd., Great Britain.

precision in their interpolates may obtain them by use of higher-order interpolation procedures. described below.

In certain tables many-figured function values are given at irregular intervals in the a!'gument. An exam})le is provided by Table 9.4. The pur-pose of these tables is to furnish "key values" for the checking of programs for automatic computers; no question of interpolation arises.

The ml1ximum end-figure error, or "tolerance" in the tables in this Handbook is %0 of 1 unit everywhere in the case of the elementary func-tions, and 1 unit in the case of the higher functions except in a few cases where it has been permitted to rise to 2 units.

IX

X INTRODUCTION

3. Auxiliary Functions and Arguments

One of the objects of this Handbook is to pro-vide tables or computing methods which enable the user to evaluate the tabulated functions over complete ranges of real values of their parameters. In order to achieve this object, frequent use has been made of auxiliary functions to remove the infinite part of the original functions at their singUlarities, and auxiliary arguments to cope with infinite ranges. An example will make the pro-ced ure clear.

The exponential integral of positive argument is given by

f x eu Ei(x) = -du -",u

The logarithmic singularity precludes direct inter-polation near x=O. The functions Ei(x)-ln x and X-I [Ei (x) -In x-'Y], however, are well-behaved and readily interpolable in this region. Either will do as an auxiliary function; the latter was in fact selected as it yields slightly higher accuracy when Ei(x) is recovered. The function X-I lEi (x) -In x-'Y] has been tabulated to nine decimals for the range O$x~:;!. For !$x$2, Ei(x) is sufficiently well-behaved to admit direct tabullation, but for larger values of x, its expo-nential character predominates. A smoother and more readily interpolable function for 1arge x is xe-XEi(x); this has been tabulated for 2$x$1O. Finally, the range 10 $x $ co is covered by use of the inverse argument X-I. Twenty-one entries of xe-XEi(x) , corresponding to x-1=.1( -.005)0, suf-fice to produce an interpolable table.

4. Interpolation

The tables in this Handbook are not provided with differences or other aids to interpolation, be-cause it was felt that the space they require could be better employed by the tabullation of additional functions. Admittedly aids could have been given without consuming extra space by increasing the intervals of tabulation, but this would have con-flicted with the requirement that linear interpola-tion is accurate to four or five figures.

For applications in which linear interpolation is insufficiently accurate it is intended that Lagrange's formula or Aitken's method of itera-tive linear interpolation 3 be used. To help the user, there is a statement at the foot of most tables of the maMmum error in a linear interpolate, and. the number of function values needed in Lagrange's formula or Aitken's method to inter-polate to full tabular accuracy.

As an example, consider the following extract from Table 5.1.

x xe~El(x) x xe~El(x) 7. 5 .89268 7854 8.0 .89823 7113 7.6 .89384 6312 8.1 .89927 7888 7. 7 .89497 9666 8. 2 .90029 7306 7.8 .89608 8737 8. 3 .90129 60'>3 7.9 .89717 4302 8. 4 .90227 4695

The numbers in the square brackets mean that the maximum error in a linear interpolate is 3 X 10-6, and that to interpolate to the full tabular accuracy five points must be used in Lagrange's and Aitken's methods.

A. C. Aitken, On interpolation by iteration of proportional parts, with out the use of d11ferences, Proc. Edinburgh Math. Soc. 3, 56-76 (1932).

Let us suppose that we wish to compute the value of xeXE1(x) for x=7.9527 from this table. We describe in turn the application of the methods of linear interpolation, Lagrange and Aitken, and of alternative methods based on differences and Taylor's series.

(1) Linear interpolation. The formula for this process is given by

where 10, 11 are consecutive tabular values of the function, corresponding to arguments Xo, Xl> re-spectively; p is the given fraction of the argument interval

p= (x-:l:o) (XI-XO) and 11> the required interpolate. In the present instance, we have

10= .89717 4302 p=.527 The most convenient way to evaluate the formula on a desk calculating machine is. to set fo and 11 in turn on the keyboard, and carry out the multi-plications by 1-p and p cumulatively; a partial check is then provided by the multiplier dial reading unity. We obtain

/527=(1-.527)(.89717 4302)+.527(.89823 7113) = .89773 4403.

Since it is known that there is a possible error of 3 X 10-6 in the linear formula, we round off this result to .89773. The maximum possible error in this answer is composed of the error committed

INTRODUCTION XI

by the last rounding, that is, .4403 X 10-5, plus 3 X 10-6, and so certainly cannot exceed .8 X 10-5

(2) Lagrange's formula. In this example, the relevant formula is the ,S-point one, given by f=A_2(p)}-2+ A_I (p)j -I + Ao(P )}o+ Al (P )}I

+A2(p)f2 Tables of the coefficients Ak(P) are given in chapter 25 for ~he range p=0(.01)1. We evaluate the formula for p=.52, .53 and .54 in turn. Again, in each evalua~ion we accumulate the Ak(p) in the multiplier register since their sum is unity. We now have the following subtable.

x xe>EI(x) 7.952 .89772 9757

10622 7.953 . 89774 0379 -2

10620 7.954 .89775 0999

n x .. y,,=xezEI(x) Yo." YO.I." 0 8.0 .89823 7113 1 7. 9 .89717 4302 . 89773 44034

The numbers in the third and fourth columns are the first and second differences of the values of xe" EI (x) (see below); the smallness of the second difference provides a check on the three interpola-tions. The required value is now obtained by linear interpolation:

}p=.3(.89772 9757)+.7(.89774 0379) =.89773 7192.

In cases where the correct order of the Lagrange polynomial is not known, one of the preliminary mterpolations may have to be performed with polynomials of two or more different orders as a check on their adequacy .

(3) Aitken's method of iterative linear interpola-tion. The scheme for carrying out this process in the present example is as follows:

YO.I.2 YO.I.2.S ... X,,-x .0473

-.0527 2 8. 1 .89927 7888 . 89774 48264 . 89773 71499 .1473 3 7. 8 .89608 8737 4 8. 2 .90029 7306 5

Here 1 IYO yo.,,=--

X,,-Xo y ..

7. 7 .89497 9666

XO-XI X,,-X

1 IYo.I Yo1.,,=x -x Y " 1 0."

1 IYO.I. .... m-I.m YO. I ..... m-I.m ,,=x -x Y

" m 0,1.' ". ",-1.110

2 90220 4 98773 2 35221

Xm-Xl X,,-x

If the quantities Xn-X and Xm-X are used as multipliers when forming the cross-product on a desk machine, their accumulation (xn-x) - (xm-x) in the multiplier register is the divisor to be used at that stage. An extra decimal place is usually carried in the intermediate interpolates to safe-guard against accumulation of rounding errors.

The order in which the tabular values are used is immaterial to some extent, but to achieve the maximum rate of convergence and at the same time minimize accumulation of rounding errors, we begin, as in this example, with the tabular argument nearest to the given argument, then take the nearest of the remaining tabular argu-ments, and so on.

The number of tabular values required to achieve a given precision emerges naturally in the course of the iterations. Thus in the present example six values were used, even though it was known in advance that five would suffice. The extra row confirms the convergence and provides a valuable check.

(4) Difference formulas. We use the central difference notation (chapter 25),

2394 .89773 71938 -.1527 1216 16 89773 71930 .2473 2706 43 30 -.2527

Xo 10

XI /I 8/1/2

82/1 8/a/2 88/a/2

X2 fz a2fz a4fz 8/a/2 88/a/2

Xs la 82/a 817/2

X4 /. Here 8/1/2=/1-/0, 8/a/2=/2-/h ... ,.

'81 = 8/s/2 - 8/112 =/2- 2/1 +/0 a8/3fZ= a2/2- a2/1 =/a- 3fz+ 3/1-/0

84fz = 881512 - 88/S/2 = /. - 4fa + 612 - 4/1 + 10 and so on.

In the present example the relevant part of the difference table is as follows, the differences being written in units of the last decimal place of the function, as is customary. The smallness of ~he high differences provides a check on the functIOn values

x xe"EI(x) 7. 9 . 89717 4302 8. 0 . 89823 7113

a21 -2 2754 -2 2036

841 -34 -39

Applying, for example, Everett's interpolation formula 1,,=(I-p)/0+E2(p)a2/0+E4(p)84/0+ ...

+p/l+ F2(p) 82/ 1 + F,(p) 84/1 +

and taking the numerical values of the interpola-tion coefficients E 2(P) , E,(p) , F2(P) and F,(p) from Table 25.1, we find tIiat

XII INTRODUCTION

109h27 = .473(89717 4302) + .061196(2 2754) - .012(34) + .527(89823 7113) + .063439(2 2036) - .012(39)

=897737193.

We may notice in passing that Everett's formula shows that the error in a linear interpolate is approximately

E2(P)02fo+ Fz(p)02fl "" UE2(p) + Fz(p) ][52fo+52ftl

Since the maximum value of \E2(p)+F2(P) \ in the range O

INTRODUCTION xm

n y,. =xe"EI (x) x,. Xo,,. 0 ,90029 7306 8,2 1 ,89927 7888 8. 1 8. 17083 5712 2 .90129 6033 8.3 8. 17023 1505 3 .89823 7113 8. 0 8. 17113 8043 4 .90227 4695 8. 4 8. 16992 9437 5 ,89717 4302 7,9 8, 17144 0382

:to,!,,. :to ,I ,2,,,

8.17061 9521 2 5948 8. 17062 2244 1 7335 415 2 8142 231

:to ,I ,2,3,,.

8. 17062 2318 265

Y,,-y ,00029 7306

-.00072 2112 .00129 6033

-.00176 2887 ,00227 4695

-, 00282 5Gf!8

result is the same as in the sub tabulation method. case Xo ,1 ,2 ,3 ,4, and Xo,1,2 ,3 ,5. The estimate of the maximum error in this \ discrepancy in the highest interpolates, in this

An indication of the error is also provided by the

6. Bivariate Interpolation Bivariate interpolation is generally most simply

performed as a sequence of univariate interpola-tions, We carry out the interpolation in one direction, by one of the methods already described, for several tabular values of the second argument in the neighborhood of its given value. The interpolates are differenced as a check, and

interpolation is then carried out in the second direction,

An alternative procedure in the case of functions of a complex variable is to use the Taylor's series expansion, provided that successive derivatives of the function can be computed without much difficulty,

7. Generation of Functions from Recurrence Relations Many of the special mathematical functions

which depend on a parameter, called their index, order or degree, satisfy a linear difference equa-tion (or recurrence relation) with respect to this parameter. Examples are furnished by the Le-gendre function Pn(X) , the Bessel function In(x) and the exponential integral En(x), for which we have the respective recurrence relations

(n+ l)P"+1- (2n+ 1)xP .. +nP .. - l =0 2n J,.+I--;J,.+J,.-I=O

Particularly for automatic work, recurrence re-lations provide an important and powerful com-puting tool. If the values of [>n(x) or In(x) are known for two consecutive values of n, or En(x) is known for one value of n, then the function may be computed for other values of n by successive applications of the relation. Since generation is carried out perforce with rounded values, it is vital to know how errors may be propagated in the recurrence process. If the errors do not grow relative to the size of the wanted function, the process is said to be stable. If, however, the relative errors grow and will eventually over-whehn the wanted function, the process is unstable,

It is important to realize that stability may depend on (i) the particular solution of the differ-ence equation being computed; (ii) the values of x or other parameters in the difference equation;

(iii) the direction in which the recurrence is being applied, Examples are as follows,

Stability-increasing n p .. (x), P,:(x) Q .. (x), Q,:(x) (x

XIV INTRODUCTION

8. Acknowledgments The production of this volume has been the

result of the unrelenting efforts of many persons, all of whose contributions have been instrumental in accomplishing the task. The Editor expresses his thanks to each and everyone.

The Ad Hoc Advisory Committee individually and together were instrumental in establishing the basic tenets that served as a guide in the forma-tion of the entire work. In particular, special thanks are due to Professor Philip M. Morse for his continuous encouragement and support. Professors J. Todd and A. Erdelyi, panel members of the Conferences on Tables and members of the Advisory Committee have maintained an un-diminished interest, offered many suggestions and carefully read all the chapters.

IreneA. Stegun has served effectively as associate editor, sharing in each stage of the planning of the volume. Without her untiring efforts, com-pletion would never have been possible.

Appreciation is expressed for the generous cooperation of publishers and authors in granting permission for the use of their source material. Acknowledgments for tabular material taken wholly or in part from published works are given on the first page of each table. Myrtle R. Kelling-ton corresponded with authors and publishers to obtain formal permission for including their material, maintained uniformity throughout the

bibliographic references and assisted in preparing the introductory material.

Valuable assistance in the preparation, checking and editing of the tabular material was received from Ruth E. Capuano, Elizabeth F. Godefroy, David S. Liepman, Kermit Nelson, Bertha H. Walter and Ruth Zucker.

Equally important has been the untiring cooperation, assistance, and patience of the members of the NBS staff in handling the myriad of detail necessarily attending the publication of a volume of this magnitude. Especially appreciated have been the helpful discussions and services from the members of the Office of Techni-cal Information in the areas of editorial format, graphic art layout, printing detail, preprinting reproduction needs, as well as attention to pro-motional detail and financial support. In addition, the clerical and typing staff of the Applied Mathe-matics Division merit commendation for their efficient and patient production of manuscript copy involving complicated technical notation.

Finally, the continued support of Dr. E. W. Cannon, chief of the Applied Mathematics Division, and the advice of Dr. F. L. Alt, assistant chief, as well as of the many mathematicians in the Division, is gratefully acknowledged.

M. ABRAMOWITZ.

I. Mathematical Constants

DAVID S. LIEPMAN 1

Contents

Table 1.1. Mathematical Constants

-fii, n prime < 100, 20S . . . . Some roots of 2, 3, 5, 10, 100, 1000, e, 20S.

en, n=l(l)lO, 25S . en .. , n=l(l)lO, 208. e', e'Y, 20S .

In n, loglo 71, n=2(1)1O, primes

2 MATHEMATICAL CONSTANTS

TABLE 1. 1. MATHEMATICAL CONSTANTS n(prime) -Vn

2 1. 4142 13562 37309 50488 101/2 3. 1622 77660 16837 93320 3 1. 7320 50807 56887 72935 101/3 2. 1544 34690 03188 37219 5 2. 2360 67977 49978 96964 101li 1. 7782 79410 03892 28012 7 2.6457 51311 06459 05905 101/5 1. 5848 93192 46111 34853 11 3. 3166 24790 35539 98491 1001/3 4.6415 88833 61277 88924 13 3. 6055 51275 46398 92931 1001/5 2.5118 86431 50958 01112 17 4. 1231 05625 61766 05498 10001/4 5. 6234 13251 90349 08040 19 4. 3588 98943 54067 35522 1000115 3. 9810 71705 53497 25077 23 4. 7958 31523 31271 95416 21/3 1.2599 21049 89487 31648 29 5.3851 64807 13450 40313 31/3 1. 4422 49570 30740 83823 31 5. 5677 64362 83002 19221 21/i 1. 1892 07115 00272 10667 37 6. 0827 62530 29821 96890 31/i 1. 3160 74012 95249 24608 41 6.4031 24237 43284 86865 2-1/2 (- 1) 7.0710 67811 86547 52440 43 6. 5574 38524 30200 06523 3-1/2 (- 1) 5.7735 02691 89625 76451 47 6.8556 54600 40104 41249 5-1/2 (- 1) 4.4721 35954 99957 93928 53 7.2801 09889 28051 82711

59 7.6811 45747 86860 81758 61 7.8102 49675 90665 43941 eT/2 4. 8104 77380 96535 16555 67 8.1853 52771 87244 99700 eT/4 2. 1932 80050 73801 54566 71 8. 4261 49773 17635 86306 e-T/2 (- 1) 2.0787 95763 50761 90855 73 8. 5440 03745 31753 11679 e-T/4 (- 1) 4.5593 81277 65996 23677 79 8. 8881 94417 31558 88501 el /2 1.6487 21270 70012 81468 83 9.1104 33579 14429 88819 e-l /2 (- 1) 6.0653 06597 12633 42360 89 9.4339 81132 05660 38113 el /3 1.3956 1242') 08608 95286 97 9. 8488 57801 79610 47217 e-l /3 (- I) 7. 1653 13105 73789 25043

n eft n e- ft 1 2. 7182 81828 45904 52353 60287 1 (- 1) 3.6787 94411 71442 32159 55238 2 7. 3890 56098 93065 02272 30427 2 (- 1) 1. 3533 52832 36612 69189 39995 3 1) 2. 0085 53692 31876 67740 92853 3 (- 2) 4.9787 06836 78639 42979 34242 4 ( 1) 5.4598 15003 31442 39078 11026 4 (- 2) 1. 8315 63888 87341 80293 71802 5 ( 2) 1. 4841 31591 02576 60342 11156 5 (- 3) 6.7379 46999 08546 70966 36048 . 6 ( 2) 4. 0342 87934 92735 12260 83872 6 (- 3) 2.4787 52176 66635 84230 45167 7 ( 3) 1.0966 33158 42845 85992 63720 7 (- 4) 9.1188 19655 54516 20800 31361 8 ( 3) 2. 9809 57987 04172 82747 43592 8 (- 4) 3.3546 26279 02511 83882 13891 9 ( 3) 8. 1030 83927 57538 40077 09997 9 (- 4) 1.2340 98040 86679 54949 76367

10 ( 4) 2. 2026 46579 48067 16516 95790 10 {. - 5) 4.5399 92976 24848 51535 59152 n eftT n e-n1r 1 ( 1) 2.3140 69263 27792 69006 1 (- 2) 4.3213 91826 37722 49774 2 ( 2) 5.3549 16555 24764 73650 2 (- 3) 1.8674 42731 70798 88144 3 ( 4) 1. 2391 64780 79166 97482 3 (- 5) 8.0699 51757 03045 99239 4 ( 5) 2.8675 13131 36653 29975 4 (- 6) 3.4873 42356 20899 54918 5 ( 6) 6.6356 23999 34113 42333 5 (- 7) 1. 5070 17275 39006 46107 6 ( 8) 1. 5355 29353 95446 69392 6 (- 9) 6.5124 12136 07990 07282 7 ( 9) 3. 5533 21280 84704 43597 7 (-10) 2.8142 68457 48555 27211 8 (10) 8.2226 31558 55949 95275 8 ( -11) 1. 2161 55670 94093 08397 9 (12) 1. 9027 73895 29216 12917 9 (-13) 5. 2554 85176 00644 85552 10 (13) 4. 4031 50586 06320 29011 10 (-14) 2. 2711 01068 32409 38387 e' ( 1) 1. 5154 26224 14792 64190 e-' (- 2) 6.5988 03584 53125 37077 e~ 1. 7810 72417 99019 79852 e-~ (- 1) 5. 6145 94835 66885 16982 n In n n IoglO n 2 O. 6931 47180 55994 53094 172321 2 (-1) 3.0102 99956 63981 19521 37389 3 1. 0986 12288 66810 96913 952452 3 (-1) 4.7712 12547 HiG62 43729 50279 4 1. 3862 94361 11989 06188 344642 4 (-1) 6.0205 99913 27962 39042 74778 5 1. 6094 37912 43410 03746 007593 5 (-1) 6.9897 00043 36018 80478 62611 6 1. 7917 59469 22805 50008 124774 6 (-1) 7.7815 12503 83643 63250 87668 7 1. 9459 10149 05531 33051 053527 7 (-1) 8.4509 80400 14256 83071 22163 8 2. 0794 41541 67983 59282 516964 8 (-1) 9.0308 99869 91943 58564 12167 9 2. 1972 24577 33621 93827 904905 9 (-1) 9.5424 25094 39324 87459 00558 10 2. 3025 85092 99404 56840 179915 10 1. 0000 00000 00000 00000 00000 11 2.3978 95272 79837 05440 619436 11 1. 0413 92685 15822 50407 50200 13 2. 5649 49357 46153 67360 534874 13 1. 1139 43352 30683 67692 06505 17 2.8332 13344 05621 60802 495346 17 1. 2304 48921 37827 39285 40170 19 2. 9444 38979 16644 04600 090274 19 1. 2787 53600 95282 89615 36333 23 3.1354 94215 92914 96908 067528 23 1. 3617 27836 01759 28788 67777 29 3.3672 95829 98647 40271 832720 29 1.4623 97997 89895 60873 32847 31 3.4339 87204 48514 62459 291643 31 1. 4913 61693 83427 26796 66704 37 3.6109 17912 64422 44443 680957 37 1. 5682 01724 06699 49968 08451 41 3. 7135 72066 70430 78038 667634 41 1. 6127 83856 71973 54945 09412 43 3. 7612 00115 69356 24234 728425 43 1.6334 68455 57958 65264 05088

See page II.

MATHEMATICAL CONSTANTS 3 TABLE 1.1. MATHEMATICAL CONSTANTS-Continued

n In n n log lO n 47 3. 8501 47601 71005 85868 209507 47 1. 6720 97857 93571 74644 14219 53 3.9702 91913 55212 18341 444691 53 1. 7242 75869 60078 90456 32992 59 4. 0775 37443 90571 94506 160504 59 1. 7708 52011 64214 41902 60656 61 4.1108 73864 17331 12487 513891 61 1. 7853 29835 01076 70338 85749 67 4. 2046 92619 39096 60596 700720 67 1. 8260 74802 70082 64341 49132 71 4. 2626 79877 04131 54213 294545 71 1. 8512 58348 71907 52860 92829 73 4. 2904 59441 14839 11290 921089 73 1. 8633 22860 12045 59010 74387 79 4. 3694 47852 46702 14941 729455 79 1. 8976 27091 290-14 14279 94821 83 4.4188 40607 79659 79234 754722 83 1. 9190 78092 37607 39038 32760 89 4. 4886 36369 73213 98383 178155 89 1. 9493 90006 64491 27847 23543 97 4. 5747 10978 50338 28221 167216 97 1. 9867 71734 :~624 48517 84362 In1l" 1. 1447 29885 84940 01741 43427 log101l" (-1) 4.9714 98726 94133 85435 12683 In-y'2; ( -1) 9. 1893 85332 04672 74178 03296 loglOe ( -1) 4. 3429 44819 03251 82765 11289

n n In 10 n n1l" 1 2. 3025 85092 99404 56840 17991 1 a. 1415 92653 58979 32384 62643 2 4. 6051 70185 98809 13680 35983 2 13.2831 85307 17958 64769 25287 3 6.9077 55278 98213 70520 53974 3 n. 4247 77960 76937 97153 87930 4 9.2103 40371 97618 27360 71966 4 ( 1) 1. 2566 37061 43591 72953 85057 5 1) 1. 1512 92546 49702 28420 08996 5 ( 1) 1. 5707 96326 79489 66192 31322 6 1) 1.3815 51055 79642 74104 10795 6 ( 1) 1. 8849 55592 15387 59430 77586 7 1) 1.6118 09565 09583 19788 12594 7 ( 1) 2. 1991 14857 51285 52669 23850 8 1) 1. 8420 68074 39523 65472 14393 8 ( 1 ) 2. 5132 74122 87183 45907 70115 9 1) 2.0723 26583 69464 11156 16192 9 ( 1) 2.8274 33388 23081 39146 16379 n 11"" n 11"-" 1 3.1415 92653 58979 32384 62643 1 (-1) 3.1830 98861 83790 67153 77675 2 9. 8696 04401 08935 86!88 34491 2 (-1) L 0132 11836 42337 77144 38795 3 1) 3. 1006 27668 02998 20175 47632 3 (-2) 3.2251 53443 31994 89184 42205 4 1) 9.7409 09103 40024 37236 44033 4 (-2) 1. 0265 98225 46843 35189 15278 5 2) 3. 0601 96847 85281 45326 27413 5 (-3) 3.2677 63643 05338 54726 28250 6 2) 9. 6138 91935 75304 43703 02194 6 (-3) 1. 0401 61473 2958.5 22960 89838 7 3) 3.0202 93227 77679 20675 14206 7 (-4) 3.3109 36801 77566 76432 59528 8 3) 9.4885 31016 07057 40071 28576 8 (-4) 1. 0539 03916 534~13 66633 17287 9 4) 2. 9809 09933 34462 11666 50940 9 (-5) 3.3546 80357 20886 91287 39854 10 4) 9. 3648 04747 60830 20973 71669 10 (-5) 1.0678 27922 68615 33662 04078

11"/2 1. 5707 96326 79489 66192 31322 311"/2 4.7123 88980 38468 98576 93965 11"/3 1. 0471 97551 19659 77461 54214 411"/3 4. 1887 90204 78639 09846 16858 7r/4 ( -1) 7. 8539 81633 97448 30961 56608 11"(2)1/2 4. 4428 82938 158:-16 62470 15881 * 11"1/2 1. 7724 53850 90551 60272 98167 11"-1/2 (-1) 5.6418 95835 47756 28694 80795 11"1/3 1. 4645 91887 56152 32630 20143 11"-1/3 (-1) 6.8278 40632 552!l5 68146 70208 11"1/4 1. 3313 35363 80038 97127 97535 11"-1/4 (-1) 7.5112 55444 64942 48285 87030 r2/3 2.1450 29397 11102 56000 77444 11"-2/3 (-1) 4.6619 40770 35411 61438 19885 ,..

2. Physical Constants and Conversion Factors A. G. McNISH 1

Contents

Table 2.1. Common Units and Conversion Factors Table 2.2. Names and Conversion Factors for Electric and Magnetic

Page 6

Units. . . . . . . . . . . . 6 Table 2.3. Adjusted Values of Constants . . . . . . . . . . . .. 7 TaMe 2.4. Miscellaneous Conversion Factors. . . . . . . . . . .. 8 Table 2.5. Conversion Factors for Customary U.S. Units to Metric

Units. . . . . . 8 Table 2.6. Geodetic Constants. . . . . . . . . . 8

1 National Bureau of Standards. 5

2. Physical Constants and Conversion Factors

The tables in this chapter supply some of the more commonly needed physical co.n-stants and conversio.n factors.

All scientific measurements in the fields of mechanics and heat are based upon four in-ternational arbitrarily adopted units, the magnitudes of which are fixed by four agreed on standards:

Length-the meter-fixed by the vacuum wavelength of radiation corresponding to the transition 2P\O-5D~ of krypto.n 86

(1 meter -1650763.73"-). Mass - the kilo.gram - fixed by the interna-

tional kilogram at Sevres, France. Time-the second-fixed as 1/31,556,925.9747

of the tropical year 1900 at 12" ephemeris time, or the duration of 9,192,631,770 cycles of the hyperfine transitio.n frequency o.f cesi-um 133.

Temperature-the degree-fixed on a ther-modynamic basis by taking the temperature for the triple point of natural water as 273.16 oK. (The Celsius scale is obtained by adding - 273.15 to the Kelvin scale.)

Other units are defined in terms of them by assigning the value unity to. the pro.Po.rtio.n-ality constant in each defining equatio.n. The entire system, including electricity units, is called the Systeme International d'Unites (SI). Taking the 1/100 part of the meter as the unit o.f length and the 1/1000 part of the kilogram as the unit of mass, similarly, gives

rise to the CGS system, often used in physics and chemistry.

Table 2.1. Common Units and Conversion Factors

Quantity SI CGS 81 unit/ name name CGS unit

-

Force, F newton dyne 10 Energy, W joule erg 107 Power P watt .................... 10'

The SI unit of electric current is the ampere defined by the equation 2r ml 1lt/41T = F giving the force in vacuo per unit length between two infinitely long parallel conductors of in-finitesimal cross-section. If F is in newtons, and r.n has the numerical value 477 x 10- 7, then II and I ~ are in amperes. The custom-ary equations define the other electric and magnetic units o.f SI such as the volt, ohm, farad, henry, etc. The force between elec-tric charges in a vacuum in this system is given by Q1Q2/41Trer2=F. re having the nu-merical value 1Q7/41TC2 where c is the speed of light in meters per second (r e = 8.854 X 10-12).

The CGS unrationalized system is obtained by deleting 41T in the denominators in these equations and expressing F in dynes, and r in centimeters. Setting rill equal to. unity de-fines the CGS unrationalized electromagnetic system (emu), re then taking the numerical value o.f 1/c2 Setting re equal to. unity de-fines the CGS unrationalized electrostatic system (esu), r m then taking the numerical value of 1/c2

Table 2.2. Names and Conversion Factors for Electric and Magnetic Units

Quantity 81 emu esu 81 unit/ SI unit/ name name name emu unit esu unit

Current ampere abampere stat ampere 10-1 ",3 X 109 Charge coulomb abcoulomb statcoulomb 10-1 -3X 109 Potential volt abvolt statvolt 108 "-'(1/3) X 10-2 Resistance ohm abohm statohm 109 -(1/9) X 10-11 Inductance henry centimeter

--------------

109 -(1/9) X 10-11 Capacitance farad

--------------

centimeter 10-9 "",,9X 1011 Magnetizing force amp. turns! oersted

--------------

4". X 10-3* -3X 109*

meter Magnetomotive force amp. turns gilbert

--------------

4". X 10-1* _3/106* Magnetic flux weber maxwell

--------------

108 -(1/3) X 10-2 Magnetic flux density tesla gauss

--------------

104 ",(1/3) X 10-6 Electric displacement

---------------------------- --------------

lO-s* ",3 X 105*

Example: If the value assigned to a current is 100 amperes its value in abamperes is lOOX 10-1 = 10. *Divide this number by 41T if unrationalized system is involved; other numbers are unchanged.

6

PHYSICAL CONSTANTS AND CONVERSION FACTORS 7

The values of constants given in Table 2.3 are based on an adjustment by Taylor, Parker, and Langenberg, Rev. Mod. Phys. 41, p.375 (1969). They are being considered for adoption by the Task Group on Fundamental Con-stants of the Committee on Data for Science and Technology, International Council of Scientific Unions. The uncer-tainties given are standard errors estimated from the experimental data included in the adjustment. Where appli-cable, values are based on the unified scale of atomic masses in which the atomic mass unit (u) is defined as 1112 of the mass of the atom of the 12C nuclide.

Table 2.3. Adjusted Values of Constants

Constant Symbol

Speed of light in vacuum ................ C Elementary charge ............................ e

Avogadro constant .......... .. .. .. .. ........ N. Atomic mass unit ............................ u Electron rest mass ...... .. .................. m.

Proton rest mass .... .. ...................... .. m,.

Neutron rest mass ..................... _... m. I

Faraday constant ............................ i F Planck constant .............................. .. 1 h

,X ' Fine structure constant ................ .. i II

, 1/0. Charge to mass rlltio for electron.. elm.

Quantum-charge ratio ......... _........... hIe

Compton wavelength of electron .... >'c >.0/2,..

Compton wavelength of proton .... >.0 >'0 . /2,..

Rydberg constant .............................. Reo Bohr radius ...................................... "-Electron radius ................ ................ r. Gyromagnetic ratio of proton ........ "'(

"'(/2 .... (u~~~;re.~~.~ .. ~~~ .. ~~~~~~~~~ .... { ~: /2,. Bohr magneton .. .......... ...................... POB Nuclear 'rnagneton ............................ POll Proton moment ...... .. .. .. .................... ..

(uncorrected for diamagnetism, H,O) ................ ................................ po' .lPOll

Gas constant ...................................... R Normal volume perfect gas ............ V. Boltzmann constant .......................... k First radiation constant (8,..hc) .... c, Second radiation constant .............. Ct Stefan-Boltzmann constant .......... .. tI Gravitational constant ........... __ ..... G

Value

2.9979260 1.6021917 4.803250 6.022169 1.660531 9.109558 5.485930 1.672614 1.00727661 1.674920 1.00866620 9.648670 2.892699 6.626196 1.0545919 7.297361 1.3703602 1.7588028 6.272769 4.135708 1.3796234 2.4263096 3.861592 1.3214409 2.103139 1.09737312 6.2917716 2.817939 2.6761966 4.257707 2.6751270 4.267597 9.274096 6.060951 1.4106203 2.792782

2.792709 8.31434 2.24136 1.380622 4.992579 1.438833 5.66961 6.673 2

Unit Uncer- -----------------,---------------------tainty Systeme International

:I: (SI) 10 X108 mls

70 10-19 C 21 ................ ................ ...... .. 40 IOU mo\-I 11 10-21 kg 64 10-sl kg 34 I 10-4 U 11 ' 10-11 kg

8 100 u I 11 ! 10-27 kg

10 100 u 54 , 10' Clmol 16 ' ...................................... .. 50 10-34 J. s 80 10-34 J. s 11 10-s ...................... .. 21 10' ...................... .. 54 1011 C/kg 16 .................. .................... .. 14 10-15 J. sIC 46 .................. ....... _ ............ . 74 10-12 m 12 10-IS m 90 10-11 m 14 10-18 m 11 lor m-I 81 10-11 m 13 10-15 m 82 I 108 rad s- IT- I 13 101 HZ/T 82 108 rad g-IT-I 13 101 Hz/T 65 10-24 J/T 60 I 10-21 J IT 99 10-28 J/T 17 100 ...................... ..

17 35 39 59 38 61 96 31

100 ........ ...... ........ .. 100 J. K-I mol- I 10-1 m3/mol 10-23 J/K 10-24 Jm 10-' m K 10-8 W. m-'K-4 10-11 N. mJ/kg'

Centimeter-gram-second (CGS)

X1010 cm/s 10-20 cml/igi / t 10-10 emS/2g1/2s-1 t 1023 mol-I

10-1 ~ g 10-sa g 10-4 u 10-24 g 100 u 10-24 g 100 u lOS 1014 10-21 10-21 10-3 10' 101 1017 10-1 10-11 10-10 10-11 10-13 10-14 105 10-& 10-13 10 103 104 103 10-11 10-'4 10-13 100

100 101 104 10-18 10-15 100 10-5 10-a

em 1/2gl/2mol-U em3/'gl/ls- l mol-1 t erg s erg s

em l / l /gl / 2 eml/Ig-I/ts-l t em8/2g1/2g-1 em1 / 2g 1 / 2 t em em cm em cm- I

cm em rad s-IG-I g-IG-I rad. s- lG-1 g-lG-I erg/G erg/G erg/G

erg K-I mol-I cm 3/mol erg/K erg em emK erg. cm-ls-1 K -4 dyn cm'/g'

tBased on 1 std. dev; applies to last digits ~ preceding column. Electromagnetic system. tElectrostatic system.

8 PHYSICAL CONSTANTS AND CONVERSION FAC'l'OK::>

Table 2.4. MiscellaneOus Conversion Factors

Standard gravity, go Standard atmospheric pressure, Po

1 thermodynamic calorie,! calc 1 IT calorie2, cal. 1 liter, I 1 angstrom unit, A 1 bar

1 gal

1 astronomical unit, AU 1 light year 1 parsec

= 9.806 65 meters per second per second* = 1.01325 X 105 newtons per square meter* = 1.01325 X 108 dynes per square centimeter* = 4.1840 joules* = 4.1868 joules* = 10-8 cubic meter* = 10-'0 meter* = 105 newtons per square meter* = 108 dynes per square centimeter* = 10-2 meter per second per second = 1 centimeter per second per second* = 1.496 X IOU meters = 9.46 X 10'5 meters = 3.08 X 1016 meters = 3.26 light years

1 curie, the quantity of radioactive material undergoing 3.7 X 1010 disintegrations per second*. 1 roentgen, the exposure of x- or gamma radiation which' produces together with its secondaries

2.082 X 109 electron-ion pairs in 0.0.01 293 gram of air. The index of refraction of the atmosphere for radio waves of frequency less than 3 X 1010 Hz

is given by (n - 1)106 = (77.6/t) (p + 4810e/t) , where n is the refractive index; t, temperature in kelvins; p , total pressure in millibars; e, water vapor partial pressure in millibars.

Factors for converting the customary United Geodetic constants for the international States units to units of the metric system are (Hayford) spheroid are given in Table 2.6. given in Table 2.5. The gravity values are on the basis of the re-Table 2.5. Factors for Converting Customary

U.S. Units to SI Units

1 yard 1 foot 1 inch 1 statute mile 1 nautical mile (inter-

national) 1 pound (avdp.) 1 oz. (avdp.) 1 pound force 1 slug 1 poundal 1 foot pound Temperature

(Fahrenheit) 1 British thermal unit"

0.914 4 meter* 0.304 8 meter* 0.0254 meter* 1609.344 meters 1 852 meters .

0.45359237 kilogram * 0.02834952 kilogram 4.448 22 newtons 14.593 9 kilograms 0.138255 newtons 1.355 82 joules 32 + (9/5) Celsius

temperature* 1055 joules

I Used principally by chemists. I Used principally by engineers.

vised Potsdam value. They are about 14 parts per million smaller than previous values. They ;ire calculated for the surface of the geoid by the international formula.

Table 2.6. Geodetic Constants a = 6378388 m; f = 1/297; b = 6356912 m

Length of Length of Latitude l' of l' of ~

longitude latitude

Meters Meters mlsl 0 1855.398 1842.925 9.780350

15 1792.580 1844.170 9.783800 30 1608.174 1847.580 9.793238 45 1314.175 1852.256 9.806154 60 930.047 1856.951 9.819099 75 481.725 1860.401 9.828593 90 0 1861.666 9.832072

3 Various definitions are given for the British thenna 1 unit. This represents a rounded mean value differing from none of the more important definitions by more than 3 in 104

.. Exact value.

3. Elementary Analytical Methods MILTON ABRAMOWITZ 1

Contents Page

Elementary Analytical Methods. . . . . . . . . . . . . . . .. 10 3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and

Geometric Progressions; Arithmetic, Geometric, Harmonic and Generalized Means . . . . . . . 10

3.2. Inequalities. . . . . . . . . . . . . . 10 3.3. Rules for Differentiation and Integration. 11 3.4. Limits, Maxima and Minima 13 3.5. Absolute and Relative Errors. . 14 3.6. Infinite Series. . . . . . . . . 14 3.7. Complex Numbers and Functions 16 3.8. Algebraic Equations. . . . . . 17 3.9. Successive Approximation Methods 18

3.10. Theorems on Continued Fractions. 19

Numerical Methods ......... . 3.11. Use and Extension of the Tables 3.12. Computing Techniques.

References. . . . . . . . .

Table 3.1. Powers and Roots nk , k= 1 (1) 10, 24, 1/2, 1/3, 1/4, 1/5

n=2(1)999, Exact or lOS

19 19 19

23

24

The author acknowledges the assistance of Peter J. O'Hara and Kermit C. Nelson in the preparation and checking of the table of powers and roots.

J National Bureau of Standards. (Deceased.)

9

3. Elementary Analytical Methods

3.1. Binomial Theorem and Binomial Coeffi-cients; Arithmetic and Geometric Progres-sions; Arithmetic, Geometric, Harmonic and Generalized Means

Binomial Theorem 3.1.1

(a+b)n=an+G) an-1b+(;) an - 2b2

+G) an- 3b3+ ... +bn (n a positive integer)

Binomial Coefficients (see chapter 24) 3.1.2

( n)_ C n(n-l) ... (n-k+l) * k -n k k! n!

(n-k)!k!

3.1.3 (~)=(n~k)=(-l)k e-~-I) 3.1.4 (ntl)=(~)+(k~l) 3.1.5 G)=(:)=l 3.1.6 1+(~)+(;)+ ... +(:)=2n 3.1.7 I-G)+G)- ... +(-I)n (:)=0

Table of Binomial Coefficients (~) 3.1.8

~k ~ 0 1 2 3 4 5 6 7 8 9 10 11 12

L ___ 1 1 2 ____ 1 2 1 3 ____ 1 3 3 1 4 ____ 1 4 6 4 1 5 ____ 1 5 10 10 5 1 6 ____ 1 6 15 20 15 6 1 7 ____ 1 7 21 35 35 21 7 1 8 ____ 1 8 28 56 70 56 28 8 1 9 ____ 1 9 36 84 126 126 84 36 9 1 10 ____ 1 10 45 120 210 252 210 120 45 10 1 lL ___ 1 11 55 165 330 462 462 330 165 55 11 1 12 ____ 1 12 66 220 495 792 924 792 495 220 66 12 1

For a more extensive table see chapter 24. See page II.

10

3.1.9 Sum of Arithmetic Progression to n Terms

a+(a+d) + (a+2d) + ... +(a+(n-l)d)

3.1.1I

3.1.14

3.1.15

1 n =na+2 n(n-l)d=2 (a+l),

last term in series=l=a+(n-l)d Sum of Geometric Progression to n Terms

lim 8n=a!(I-r) n-t'"

Arithmetic Mean of n Quantities A

Geometric Mean of n Quantities G

Harmonic Mean of n Quantities H

Generalized Mean

M(t)= - ~ a~ ( 1 n )1/1 n k-I M(t)=O(t

ELEMENTARY ANALYTICAL METHODS 11 3.2.3 min. a< Gl, q>1 p q

3.2.8 ~ lakbkl~(~ laklpylp(~ IbkluY/u; equality holds if and only if Ibkl =clakl p- I (c=con-stant>O). If p=q=2 we get

Cauchy's Inequality 3.2.9

[ n J2 n n (;i akbk ~~ a~ (;i

c constant).

HOlder's Inequality for In tegrals

If ~+!=1, p>l, q>1 p q 3.2.10

ib If(x) g(x) IdXl~[ib I.f(x) IPdx JIll' [ib Ig(xWdx J I/u

equality holds if and only if Ig(x)l=clf(x)lp-1 (c=constant>O).

If p=q=2 we get

3.2.11 Schwarz's Inequality

[ib f(x)g(x)dx J ~ i b [f(x)2dx i b [g(x)2dx

Minkowski's Inequality for Sums

If p>1 and ak, bk>O for all ~:, 3.2.12

equality holds if and only if bk=cak (c=con-stant>O).

Minkowski's Inequality i':()r Integrals

It p>l,

3.2.13

(i b )1/1' (ib )1/1' a If(x) +g(:r)!pdx ~ a If(x)!Pdx ( fb )1/1' + Ja Ig(x) IPdx

equality holds if and only if g(x) =cf(x) (c=con-stant>O).

3.3. Rules for Differentiation and Integration Derivatives

3.3.1

3.3.2

3.3.3

3.3.4

3.3.5

3.3.6

d du dx (cu)=c dx' c constant

.!i (u+v)=du+ dv dx dx dx

d dv du - (uv)=u -+v-dx dx dx

d vdu/dx-'Iwv/dx dx (u/v) v2

d du d:o - u(v)=---dx dv dx

d "(V du dV) - (u")=u - -+In u-dx u dx dx

Leibniz's Theorem for Differentiation of an Integral

3.3.7

d fb(C) .1_ f(x, c)dx Uti ate)

f b(C) 0 . db da = ;;;- f(x,c)dx+ f(b,c) d-- f(a,c) d-ate) vC C C

12 ELEMENTARY ANALYTICAL METHODS Leibniz's Theorem for Differentiation of a Product The following formulas are useful for evaluating

3.3.8

3.3.9

3.3.10

3.3.ll

dX=lf-l dy dx

dJlx =_[dJly dy -3 (d2Y)1 (dy)-5 d1f dJ! dx dx2 dx

Integration by Parts

J P(x)dx . (ax2+bx+e)n where P(x) IS a polynomial and n>l is an integer.

3.3.16

I dx 2 2ax+b (ar+bx+e) (4ac-b2)t arctan (4ac-b2)! (b2-4ac0) -2 3.3.18

=2ax+b 3.3.19

I xdx 1 2 b I dx ar+bx+e 2a In lax +bx+cl-2a ax2+bx+e 3.3.12 J udv=uv-J vdu 3.3.20

f dx 1 1 \e+dx\ 3.3.13 J UVdx=(J udX) v-J (JUdX) ~ dx (a+bx)(e+dx)= ad-be n a+bx

(ad r!i be)

3.3.21

Integrals of Rational Algebraic Functions 3.3.22

(Integration constants are omitted)

3 3 14 I( +b)nd (ax+b)n+1 ax x a(n+l) (n;e-1) 3.3.15 3.3.25

Integrals of Irrational Algebraic Functions

3.3.26 f dx 2 [-d(a+ bX)]1/2 [(a+bx) (e+dx)]l/2 (-bd)1/2 arctan b(e+dx) 3.3.27 -1 . (2bdX+ ad+ be) = ( _ bd) 1/2 arcsm be-ad 3.3.28 2 = (bd)llzln j[bd(a+ bX)]l/2+ b(c+dx)l/ZI

3.3.29 f dx 2 [d(a+bx)]IIZ (a+bx)1/2(e+dx) [d(be-ad)]l/2 arctan (be-ad) (d(ad-be)

3.3.31

f [(a+bx)(e+dx)]l/2dx (ad-be)~~b(e+dX) [(a+bx) (e+dx)]l/2

(ad-be)2 f dx 8bd [(a+ bx) (e+dx)]l!2

3.3.32

f[ e+dxJI/2 1 a+bx dX="b [(a+bx) (e+dx)]1/2

3.3.33

3.3.34

3.3.35

3.3.36

3.3.37

3.3.38

(ad-be) f dx 2b [(a+bx) (e+dx),12

_ - 1/2 . h (2ax+b) -a arCSlll (4ac-b2)1/2

(a>0,4ac>b2) =a-1/2 In l2ax+bl(a>O, b2=4ac)

=-(-a) - 1/2 &rcsin (2ax+b) (b2-4ac) 1/2 (a4ac, l2ax+blb2-4ac)"2)

f dx x(ar +bx+e)1/2 f dt (a+bt+et2) 112 where t=l/x 3.3.39

f xcix (ar+bx+e} 1/2 . I

.. ' i-! (ax2+bx+e)I~.l-~ f dx !-- 2a (ax2+bx+e) 1/2

3.3.40 f (x2~a2)t In /x+(x2a2)i/ 3.3.41

3.3.43

3.3.44

3.3.45

3.3.46

3.3.47

3.3.48

f dx 1 a ( 2 2) 1 arccos -x x -a a x

. x-a arCSlll-

a

f (x-a) a2 x-a (2ax-x2)idx=-- (2ax-x2)i+- arcsin -2 2 a 3.3.49

f dx (ax2+b) (er+d)t 1 x(ad-be)i

[b(ad-be)]i a!.ctan [b(ex2+ d)]t

3.3.50

1 In l[b(ex2+d)]t+ x(be-ad) t l . 2[b(be-ad)]i [b(ex2+d)]t-x(be-ad)t

(be>ad)

3.4. Limits, Maxima and Minima

Indeterminate Forms (L'Hospital's ,ule)

3.4.1 Let }(x) and g(x) be differentiable on an interval a~x

14 ELEMENTARY ANALYTlCAL METHODS MaxiJna and Minima

3.4.2 (1) Functions oj One Variable The function y-j(x) has a maximum ut X=Xo

if 1'txo)=O and j"(Xo)

ELEMENTARY ANALYTICAL METHODS

3.6.9 a(a-1) 2 a(a-1)(a-2) (1+X)a=1+aX+~ X + 3! r+ ... ,

3.6.10 (1+x)-1=1-x+x2-x3+x'- .. .

3.6.11 X X2 r 5x 7X5 21x6

(1+x)t=1+2-g+16-128+256-1024+' ..

3.6.14

( )-1- ' _! ~x2_14r ~ { l+x -1 3 x+9 . 81 +243 x 91 6 728 6

-729 x +6561 x - . ..

Asymptotic ExpansioDl!

CD

15

(-l

16 ELEMENTARY ANALYTICAL METHODS,

Reversion of Series

3.6.25 Given y=ax+bx2+cxa+dx'+ex~+fx6+gx7 +

then x=Ay+ By2+0?t+Dy'+E1f+Fy6+GyT+ .. .

where aA=l aBB=-b a60=2b2-ac a7D=5abc-a2d-5b3

aOE=6a2bd+3a2c2+ 14b'-ase-21ab2c allF=7a8be+7ascd+84absc-a'f

-28a2bc2-42b6_28a2b2d aI3G=8a'bf+8a'ce+4a'd2+ 120a2bBd

+ 180a2b2c2'+ 132b6-a~g-36aab2e -72a3bcd-12a3cB-3,30ab'c

KUlDmer's Transformation of Series ..

3.6.26 Let '5: a k=8 be a given convergent series and t=O

.. L: Ck=C be a given convergent series with known taO

sum C such that lim ~=>'~O. t~ .. Ck

Then

Euler's Transformation of Series ..

3.6.27 If '5: (-l)tak=ao-al +~- ... is a con-t=O

vergent series with sum 8 then

Euler-Maclaurin Summation Formula 3.6.28

~jk= i" j(k)dk-~ (f(O)+j(n) J+ 1; (f'(n)-j'(O)J __ 1 [f"'(n)-j'II(0)J+_1- [fVl(n)-fVl(O)J

720 30240

120!600 (f(VIn(n)-fV11l(0)J+ . . .

3.7. Complex Numbers and Functions

Cartesian Form

3.7.1 z=x+iy

3.7.2

3.7.3

Polar Form

3.7.4 Argument: arg z=arctan (y/x)=8 (other notations for arg z .are am z and ph z) . 3.7.5

3.7.6

3.7.7

3.7.8

3.7.9

3.7.10

3.7.11

3.7.12

3.7.14

3.7.15

3.7.16

3.7.17

3.7.18

3.7.19

3.7.20

Real Part: x=!JIz=r cos 8

Imaginary Part: y=.h=r sin 8 Complex Conjugate of :IS

z=x-iy

-

arg z=-arg z

Multiplication and Division

Powers

=r" cos n8+ir" sin n8 (n=O , 1,2, ... )

z2=x2-y2+i(2xy) z3=r-3xy2+i(3ry-?t)

z'=x4-6:r;2y2+y'+i(4ry-4:z;1f) 3.7.21 z6=z6-10ry2+5~+i(5x'y-lOx21f+1f)

3.7.22

z"=[x"-G) X"-2y2+(~) x"-V- ... J +i r(~) X"-ly_(~) x,,-ayB+ ... J,

(n=l , 2, . .. )

ELEMENTARY ANALYTICAL METHODS 17

3.7.23 Un+l=XUn-Yvn; Vn+l=XVn+yun f!JI zn and .f zn are called harmonic polynomials.

3.7.24

3.7.25

1 z x-iy -Z=rzr=" X2+y2 1 _ zn _ -I n zn-lzI2n-(Z )

Roots

3.7.26 zl=.Jz=rlelI8=rl cos !8+iri sin !8

If -1/'< 8 ~ 1/' this is the principal root. The other root has the opposite sign. The principal root is given by

3.7.27 zi=[!(r+x)]ii[!(1'-x)jt=uiv where 2uv=y and where the ambiguous sign is taken to be the same as the sign of y.

3.7.28 zlln=1'l/net6In, (principal root if -1/'0, one real root and a pair of complex conjugate roots,

Let

then

t+r=O, all roots real and at least two are equal,

q3+1'2

18 ELEMENTARY ANALYTICAL METHODS

If all roots of the cubic equation are real, use the value of !ll which gives real coefficients in tlll'

*quadratic equation and select signs so that if

z'+a3z3+~z2+alZ+ao= (Z2+ PIZ+ q:t) (z2+P3Z+q2), then

PI+Pa=a3,P1P2+ql+q2=~,Plq2+p2q)=al' qlq2=aO' If Z" Z2, Za, Z4 are the roots,

3.9. Successive Approximation Methods

General Comments

3.9.1 Let X=XI be an approximation to x=c where !m=O and both XI and ~ are in the interval a5:x5:b. We define

X"H = x" + c,.f(x,,) (n= 1,2, ... ). Then, if 1'(x)'?,O and the constants COl are

negative and bounded, the sequence Xn converges monotonically to the root ~.

If c,,=c=constant

ELEMENTARY ANALYTICAL METHODS 19 3.10. Theorems on Continued Fractions

Definitions 3.10.1

(1) Let

If the number of terms is finite, .f is called. It terlllinating continued fraction. If the number of terms is infinite, f is called an infinite continued fraction and the terminating fraction

1 An b + al ~ an n=Bn=o b,+b2+"'bn is called the nth convergent of j. (2) If lim ABn exists, the infinite continued frac-

n~Q) n

tion 1 is said to be convergent. If at= 1 and the bt are integers there is always convergence.

Theorems

(1) If at and bt are positive then 12711271+1' (2) If1n=~:'

An= bnAn-1 +anAn-2 Bn=bnBn-,+anBn-2

where A_I=I, Ao=bo, B_,=O, Bo=1.

(3)

(4) 11 AnBn_I-An_IBn=(-I)n-1 II ak k=' (5) For every n~ 0,

*

1 =bo+ Clal ClC2a2 C2C3a3 Cn-ICnan.

71 c,b, + c2b2+ C3b3+ . .. crtbn (6) l+b2+b2b3+ ... +b2ba .. bn

1 b2 b3 bn =1- b2+1- /)3+1- ... -bn+l

J:..+J:..+ ... +1.=_1_ ~_ . . . U~_I u 1 U2 un U I - UI+U2- -Un-I+Un

1 X x2 xn - __ +_ '" + (-1)71 __ ao aoal aOala2 aOalla2' . an

2.0

1.8

1.6

1.4

1.2

1.0

.8

.6

.4

.2

o

1 aox alx an-I X ao+ al-x+ n2-x+ ... +an-x

~---~~===-------.---.--n::;:o

_________ ~-_"': _____ l _______ "_. __ ~_~ 1 1 I I I I I I 1 I I

o .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0

FIGURE 3.1. y'=xn 1 1

n=O, 5' 2' 1, 2, 5.

Numerical Methods 3.11. Use and Extension of the Tables

Example 1. Computt; ;rIg and X47 for x=29 using Table 3.1.

= (1.45071 4598.1013) (4.20707 2333.1014) =6.10326 1248.1027

X47 = (X24 )2/X

= (1.25184 9008.1036)2/29 =5.403882547.1068

Example 2. Compute X- 3/4 for x=9.19826.

(9.19826)1/4= (919.826/100)114= (919.826)1/4/101

Linear interpolation in Table 3.1 gives (919.826 )114 ~ 5.507144.

By Newton's method for fourth roots with N=919.826,

~ L5~;~7~!~)3+3(5.507144)]=5.50714 3845 Repetition yields the same result. Thus,

x1l4=5.50714 3845/101=1.74151 1796, X-3/4=J~t/X= .18933 05683.

3.12. Computing Techniques

Example 3. Solve the quadratic equation x2-18.2x+.056 given the coefllcients as 18.2.1,

See page II.

*

20 ELEMENTARY ANALYTICAL METHODS

.056 .001. From 3.8.1 the solution is x= H18.~ ((18.~)2_4(.05fi) P)

=H18.2.[:331.01GP)=H18.2 18.1939) = 18.1969, .OOJ

The smaller root may be obtained more accurately from

.052/18.1969= .0031 .0001. Example 4. Compute (-3+.0076i)1.

From 3.7.26, (-3+.0076i)!=u+iv where

Thus r= [( -3)2+ (.0076)2]!= (9.00005776)!=3 .00000 9627 v=[3.00000 9~27-(-3)J!=1.73205 2196

y .0076 u=2v 2(1.732052196) .00219 392926

We note that the principal square root has been computed.

Example 6. Solve the quartic equation ;r4-2.37752 4922x3+6.07350 5741x2

-11.17938 023x+9.05265 5259=0. Hesolution Into Quadratic Factors

(x2 + PIX + ql) (x2 + P2X + q2) by Inverse Interpolation

Starting with the trial value ql = 1 we compute successively

ql ao al-aaql p2=aa-pl y(ql) =ql + q2+ PIP2 q2=- PI=--ql q2-ql -a2

1 9.053 -1. 093 -1. 284 5.383 2 4. 526 -2.543 .165 .032 2. 2 4.115 -3.106 .729 -2.023

ql q2 I PI

Example 5. Solve the cubic equation x3-I8.lx -34.8=0.

To use Newton's method we first form the table of f(x) =x3-18.1x-34.8

x I(x) 4 -43.2 5 .3 6 72.6 7 181.5

We obtain by linear inverse interpolation: 0-(-.3)

xo=5+72 .6_ (_ .3) 5.004. Using Newton's method,f'(x)=3x2-18.1 we get

XI ~Xo- f(xo)/f' (xo) ~5 004- (- .07215 9936) ~5 00526

. 57.020048 . .

Repetition yields xI=5.00526 5097. Dividing f(x) by x-5.00526 5097 gives x2+5.00526 5097x +6.95267869 the zeros of which are -2.502632549 .83036 800i.

We seek that value of 1.1 for which y(1.I) =0. Inverse interpolation in y(q\) gives y(1.\) ",,0 for q\ ~2.003. Then,

ql q2 PI P2 y(ql) ----

2. 003 4. 520 -2.550 . 172 .011

Inverse interpolation between 1.1=2.2 and ql= 2.003 gives q\ =2.0041, and thus,

P2 y(q\)

2. 0041 4. 51706 7640 - 2. 55259 257 .17506 765 .00078 552 2. 0042 4 .. 51684 2260 - 2. 55282 851 .17530 358 .00001 655 2. 0043 4. 51661 6903 - 2. 55306 447 . 17553 955 -.00075 263

Inverse interpolation gives ql=2.00420 2152, nnd we get finnlly,

ql q2 PI P2 y(ql)

2. 00420 2152 4. 51683 7410 - 2. 55283 358 . 17530 8659 -. 00000 0011

See page Il.

ELEMENTARY ANALYTICAL METHODS 21 Double Precision Multiplication and Division on a

Desk Calculator

Example 7. Multiply M=20243 9745971664 32102 by m=69732 8242843662 95023 on a 1OX1OX20 desk calculating machine.

Let M o=20243 97459,1\'[1=71664 32102, mo= 69732 82428, ml =43662 95023. Then Mm= Momo1020 + (Moml+Mlmo) 101O +Mlml'

(1) Multiply Mlml=31290 75681 96300 28346 and record the digits 96300 28346 appearing in positions 1 to 10 of the product dial.

(2) Transfer the digits 31290 75681 from posi-tions 11 to 20 of the product dial to positions 1 to 10 of the product dial.

(3) Multiply cumulatively Mlmo+Moml +31290 75681=58812 67160 12663 25894 and record the digits 12663 25894 in positions 1 to 10.

(4) Transfer the digits 58812 67160 from posi-tions 11 to 20 to positions 1 to 10.

(5) Multiply cumulatively Momo+58812 67160 = 14116 69523 40138 17612. The results as oh-tained are shown below,

9630028346 1266325894

14116695234013817612 1411669523401381761212663 25894 963C~ 28346

If the product Mm is wanted to 20 digits, only the result obtained in step 5 need be recorded. Further, if the allowable error in the 20th place is a unit, the operation MImI may be omitted. When either of the factors M or m contains less than 20 digits it is convenient to position the numbers as if they both had 20 digits. This multiplication process may be extended to any higher accuracy desired.

Example 8. Divide N = 14116 69523 40138 17612 by d=20243 97459 71664 32102.

Method (1 )-linear interpolation.

N/20243 97459.1010 =.69732 82430 90519 39054 N/20243 97460.1010=.69732 82427 46057 26941

Difference=3 44462 12113.

Difference X.71664 32102=24685 644028.10-20 (note this is an 11 X 10 multiplication). -

Quotient= (69732 82430 90519 39054-246856 4402~).10-20

= .69732 82428 43662 95022

There is an error of 3 units in the 20th place due to neglect of the contribution from second differ-ences.

Method (2)-If Nand d are numbers each not more than 19 digits let N=NI +No109, d=dl+ do109 where No and do contain 10 digits and NI and dl not more than 9 digits. Then

Here

N=14116 69523 401381761, d=20243, 97459 71664 3210

N o= 14116 69523, do=20243 97459, dl=716643210

(1) NodI = 1011663378421888830 (product dial). (2) (Nodl)/do=49973 55504 (quotient dial). (3) N- (Nodl)/do= 14116 69622 90164 62106

(product dial). (4) [N - (Nodl)/dol Ido 109 = .69n2 82428 = first 10

digits of quotient in quotient dial. Remainder =r=08839 11654, in positions 1 to 10 of product dial.

(5) r/(do109)=.43662 95021O- IO=next9 digits of quotient. N/d= .69732 82428 13662 9502. This method may he modified to give the quotient of 20 digit numbers. Method (1) may be extended to quotients of numbers containing more than 20 digits by employing higher order interpolation.

Example 9. Sum the series S=I-!+t-i + ... to 5D using the Euler transform.

The sum of the first 8 terms is .634524 to 6D. If un=l/n WP, get

n Un !J.Un !J.2Un !13un !14un 9 .111111

-11111 10 .100000 2020

-9091 -505 11 .090909 1515 156

-7576 -349 12 .083333 1166

-6410 13 .076923 From 3.6.27 we then obtain

S=.634524+.1l~111 (-.011111)+.002020 22 23

( -- .000505) + .000156 24 26

= .634524+ '()55556+ .002778+ .000253 +.000032+.000005

=.693148

(S=ln 2=.6931472 to 7D).

22 ELEMENTARY ANALYTICAL METHODS

. f'" sin x Example 10. Evaluate the mtegral -- dx o x

=~ to 4D using the Euler transform.

-dx=~ -dx i '" sin x '" i(k+ll" sin x o x k=O kif X = flf sin (lc1r+t) dt= (_I)k f" sin t dt.

k=Q Jo k7r+t k=Q Jo k7r+t Evaluating the integrals in the last sum by numerical integration we get

k i" sint dt o k,r+t

0 1. 85194

1 .43379

2 .25661

3 .18260 6. 6.2 A3 A4

4 . 14180 -2587

5 . 11593 799 -1788 -321

6 .09805 478 153 -1310 -168

7 .08495 310 -1000

8 .07495

The sum to k=3 is 1.49216. Applying the Euler transform to the remainder we obtain

1 1 1 2 (.14180)-2"2 (-.02587)+23 (.00799) 1 1

-2"4 (-.00321)+25 (.00153) = .07090+ .00647 + .00100+ .00020

+.00005 =.07862

We obtain the value of the integral as 1.57078 as compared with 1.57080.

'" 11"2. Example 11. Sum the serIes ~ lc- 2=-6 usmg

k=l the Euler-Maclaurin summation formula.

From 3.6.28 we have for n= l, bo=O, bn =2n-1, A_l=1, B_1=0, Ao=O, Bo=1. For n~ 1

[An]=1 An-lAn-2 1 Bn Bn- 1Bn- 2

[2n-1 ] A (n-l)2x2 B:=O

[~:J=I 0 1 II.: 1=1 .2 I Al 1 0 1 B 1=2 [~:]=I .2 0 11.:41=1 .6 I ~:=.197368 1 1 3.04 [A3] I .6

.2

11.:61=1

3.032 I ~:=.197396 Ba = 3.04 1 15.36

= = -= 197396 [A4] 13.032 .6 11 7 11 21 .440 I A4 B4 15.36 3.04 .36 108.6144 B4 . Note that in carrying out the recurrence method for computing continued fractions the numerators An and the denominators Bn must be used as originally computed. The numerators and de-nominators obtained by reducing An/Bn to lower terms must not be used.

ELEMENTARY ANALYTICAL METHODS 23

References Texts

[3.1] R. A. Buckingham, Numerical methods (Pitman Publishing Corp., Xew York, N.Y., 1957).

[3.2] T. Fort, Finite differences (Clarendon Press, Oxford, England, 1948).

[3.3] L. Fox, The use and construction of mathematical tables, Mathematical Tables, vol. 1, National Physical Laboratory (Her Majesty's Stationery Office, London, England, 1956).

[3.4] G. H. Hardy, A course of pure mathematics, 9th ed. (Cambridge Univ. Press, Cambridge, England, and The Macmillan Co., New York, N.Y., 1947).

[3.5) D. R. Hartree, Numerical analysis (Clarendon Press, Oxford, England, 1952).

[3.6) F. B. Hildebrand, Introduction to numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1956).

[3.7] A. S. Householder, Principles of numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1953).

[3.8] L. V. Kantorowitsch and V. I. Krylow, Naherungs-methoden der Hiiheren Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, Germany, 1956; translated from Russian, Moscow, U.S.S.R., 1952) .

[3.9] K. Knopp, Theory and application of infinite series (Btackie and Son, Ltd., London, England, 1951).

[3.10] Z. Kopal, Numerical analysis (John Wiley & Sons, Inc., New York, N.Y., 1955).

[3.11J G. Kowalewski, Interpolation und geniiherte Quad-ratur (B. G. Teubner, Leipzig, Germany, 1932).

[3.12] K. S. Kunz, Numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1957).

[3.13] C. Lanczos, Applied analysis (Prentice-Hall, Inc., . Englewood Cliffs, N.J., 1956).

[3.14] I. M. Longman, Note on a method for computing infinite integrals of oscillatory functions, Proc. Cambridge Philos. Soc. 52, 764 (1956).

[3.15] S. E. Mikeladze, Numerical methods of mathe-matical analysis (Russian) (Gos. Izdat. Tehn.-Teor. Lit., Moscow, U.S.S.R., 1953).

[3.16] W. E. Milne, Numerical calculus (Princeton Univ. Press, Princeton, N.J., 1949).

[3.17] L. M. Milne-Thomson, The calculus of finite differ-ences (Macmillan and Co., Ltd., London, England, 1951).

[3.18] H. Mineur, Techniques de calcul numerique (Librairie Poly technique Ch. Beranger, Paris, France, 1952).

[3.19] National Physical Laboratory, Modern computing methods, Notes on Applied! Science No. 16 (Her Majesty's Stationery Office, London, England, 1957).

[3.20] J. B. Rosser, Transformations to speed the con-vergence of series, J. Research NBS 46, 56-64 (1951).

[3.21] J. B. Scarborough, NumericfLI mathematical anal-ysis, 3d ed. (The Johns Hopkins Press, Baltimore, Md.; Oxford Univ. Pres~l, London, England, 1955).

[3.22] J. F. Steffensen, Interpolation (Chelsea Publishing Co., New York, N.Y., 1950).

[3.23] H. S. Wall, Analytic theory of continued fractions (D. Van Nostrand Co., Inc., New York, N.Y., 1948).

[3.24] E. T. Whittaker and G. Robiinson, The calculus of observations, 4th ed. (BtsLckie and Son, Ltd., London, England, 1944).

[3.25] R. Zurmiihl, Praktische Mathematik (Springer-Verlag, Berlin, Germany, 1953).

Mathematical Tables and Collec::tions of Formulas

[3.26] E. P. Adams, Smithsonian mathematical formulae and tables of elliptic functions, 3d reprint (The Smithsonian Institution, Wa.shington, D.C., 1957).

[3.27] L. J. Comrie, Barlow's tables of squares, cubes, square roots, cube roots a.nd reciprocals of all integers up to 12,500 (Chemical Publishing Co., Inc., New York, N.Y., 19M) .

[3.28] H. B. Dwight, Tables of integrals and other mathe-matical data, 3d ed. (The Macmillan Co., New York, N.Y., 1957).

[3.29] Gt. Britain H.M. Nautical Almanac Office, Inter-polation and allied tables (Her Majesty's Sta-tionery Office, London, England, 1956).

[3.30] B. O. Peirce, A short table of integrals, 4th ed. (Ginn and Co., Boston, Mass., 1956).

[3.31] G. Schulz, Formelsammlung zur praktischen Mathe-matik (de Gruyter and Co., Berlin, Germany, 1945).

..::-"

24 ELEMENTARY ANALYTICAL METHODS

Table 3.1 POWERS AND ROOTS nk

k 1 See Examples 1-5 for use n!= 2 3 4 2 ",2= 4 9 16 3 of the table. ",3= 8 27 64 4 ",4~ 16 81 256 5 ",5= 32 243 1024 6 Floating decimal notation: ",6= 64 729 4096 7 ",77' 128 2187 16384 8 91=34867 84401 ",8= 256 6561 65536 9 ",9= 512 19683 2 62144

10 = (9)3.4867 84401 ",10= 1024 59049 10 48576 24 '1124= 167 77216 (11) 2.8242 95365 (14) 2.8147 49767

1/2 '111/2= 1.4142 13562 1. 7320 50808 2.0000 00000 1/3 '111/3= 1. 2599 21050 1.4422 49570 1. 5874 01052 1/4 '111/4= 1.18CJ2 07115 1.3160 74013 1. 4142 13562 1/5 '111/5= 1. 1486 98355 1.2457 30940 1. 3195 07911

1 5 6 7 8 9 2 25 36 49 64 81 3 125 216 343 512 729 4 625 1296 2401 4096 6561 5 3125 7776 16807 32768 59049 6 15625 46656 1 17649 2 62144 5 31441 7 78125 2 79936 8 23543 20 97152 47 82969 8 3 90625 16 79610 57 64801 167 77216 430 46721 9 19 53125 100 77696 403 53607 1342 17728 3874 20489

10 CJ7 65625 604 66176 2824 75249 9)1.0737 41824 CJ) 3. 4867 84401 24 (16)5.960464478 (18)4.7383 81338 (20)1.9158 12314 (21) 4. 7223 66483 (22) 7. 9766 44308

1/2 2.2360 67977 2.4494 89743 2.6457 51311 2.8284 27125 3. 0000 00000 1/3 1. 7099 75947 1. 8171 205CJ3 1. 9129 31183 2.0000 00000 2. 0800 83823 1/4 1. 4953 48781 1. 5650 84580 1. 6265 76562 1. 6817 92831 1. 7320 50808 1/5 1. 3797 29662 1.4309 69081 1. 4757 73162 1. 5157 16567 1.5518 45574

1 10 11 12 13 14 2 100 121 144 169 196 3 1000 1331 1728 2197 2744 4 10000 14641 20736 28561 38416 5 1 00000 1 61051 2 48832 3 71293 5 37824 6 10 00000 17 71561 29 85984 48 26809 75 29536 7 100 00000 194 8717l 358 31808 627 48517 1054 13504 8 1000 00000 2143 58881 4299 81696 8157 30721 ~ 9r.4757 89056 9 ( 9l1.0000 00000 ( 9)2.3579 47691 ~ 9)5.1597 80352 ! 10) 1.0604 49937 10 2.0661 04678

10 (10 1. 0000 00000 (10)2.5937 42460 10) 6. 1917 36422 11) 1. 3785 84918 11 2.8925 46550 24 (24)1.0000 00000 (24) 9.8497 32676 (25) 7. 9496 84720 (26) 5. 4280 07704 (27)3.2141 99700

1/2 3.1622 17660 3.316624790 3.4641 01615 3.6055 51275 3. 7416 57387 1/3 2.1544 34690 2.2239 80091 2.2894 28485 2. 3513 34688 2.4101 42264 1/4 1.7782 79410 1. 8211 60287 1. 8612 09718 1. 8988 28922 1.9343 36420 1/5 1. 5848 93192 1. 6153 94266 1. 6437 51830 1. 6702 77652 1. 6952 18203

1 15 16 17 18 19 2 225 256 289 324 361 3 3375 4096 4913 5832 6859 4 50625 65536 83521 1 04976 1 30321 5 7 59375 10 48576 14 19857 18 89568 24 76099 6 113 90625 167 77216 241 37569 340 12224 470 45881 7 1708 59375 2684 35456 4103 38673 6122 20032 8938 71739 8 ~ 9l2.5628 90625 ( 9l4.2949 67296 ! 9l6.9757 57441 ! 10)1.1019 96058 ~ 10~ 1.6983 56304 9 10 3.8443 35938 (10 6.8719 47674 11 1.1858 78765 lll1. 9835 92904 11 3.2268 76978

10 11) 5. 7665 03906 (12)1.099511628 (12) 2.0159 93900 12 3.5704 67227 (12 6.1310 66258 24 (28)1.6834 11220 (28)7.9228 16251 (29) 3. 3944 86713 (30) 1. 3382 58845 (30) 4. 8987 62931

1/2 3.8729 83346 4. 0000 00000 4.1231 05626 4.2426 40687 4.3588 98944 1/3 2.4662 12074 2.5198 42100 2,5712 81591 2.6207 41394 2.6684 01649 1/4 1. 96 79 89671 2.0000 00000 2.0305 43185 2.0597 67144 2. 0877 97630 1/5 1. 7187 71928 1.7411 01127 1. 7623 40348 1. 7826 02458 1. 8019 83127

1 20 21 22 23 24 2 400 441 484 529 576 3 8000 9261 10648 12167 13824 4 1 60000 1 94481 2 34256 2 79841 3 31776 5 32 00000 40 84101 51 53632 64 36343 79 62624 6 640 00000 857 66121 1133 79904 1480 35889 1911 02CJ76 7 ~ 9) 1. 2800 00000 ( 9) 1. 8010 88541 1 9l2.4943 57888 ( 9) 3.4048 25447 ! 914.58M 71424 8 10)2.5600 00000 (10) 3. 7822 85936 10 5.4875 87354 {10)7.8310 98528 11 1. 1007 53142 9 (11)5.1200 00000 (11)7.9428 00466 12l1.2072 69218 12) 1. 8011 52661 12 2.6418 07540

10 (13) 1. 0240 00000 (13)1.6679 88098 (13 2.6559 92279 13) 4.1426 51121 13 6. 3403 38097 24 (31) 1. 6777 21600 (31)5.410819838 (32) 1. 6525 10926 (32) 4.8025 07040 (33)1.3337 35777

1/2 4.4721 35955 4.5825 75695 4.6904 15760 4. 7958 31523 4. 8989 79486 1/3 2. 7144 17617 2.7589 24176 2.8020 39331 2.8438 66980 2. 8844 99141 1/4 2. 1147 42527 2.140695143 2. 1657 36771 2.1899 38703 2.2133 63839 1/5 1.8205 64203 1. 8384 16287 1. 8556 00736 1. 8721 71231 1. 8881 75023

k 1 2 3 4 5 6 7 8 9

10 24

1/2 1/3 1/4 1/5

1 2 3 4 5 6 7 8 9

10 24

1/2 1/3 1/4 1/5

1 2 3 4 5 6 7 8 9

10 24

1/2 1/3 1/4 1/5

1 2 3 4 5 6 7 8 9

10 24

1/2 1/3 1/4 1/5

1 2 3 4 5 6 7 8 9

10 24

1/2 1/3 1/4 1/5

25 625

15625 3 90625

97 65625 2441 40625

l 9! 6. 1035 15625 11 1.5258 78906 12 3.8146 97266 (13 9.536" 43164 (33)3. 5527 13679

5.0000 00000 2.9240 17738 2.2360 67977 1. 9036 53939

30 900

27000 8 10000

243 00000 7290 00000

! 10! 2.1870 00000 11 6.5610 00000 13 1. 9683 00000 14 5.9049 00000 (35) 2. 8242 95365

5.4772 25575 3.1072 32506 2.3403" 47319 1. 9743 50486

35 1225

42875 15 00625

525 21875 911. 8382 65625

106.433929688 12 2.2518 75391 13 7.8815 63867 15) 2. 7585 47354

(37) 1. 1419 13124 5.9160 79783 3.271066310 2.4322 99279 2.0361 68005

40 1600

64000 25 60000

1024 00000

! 914.0960 00000 11 1. 6384 00000 12 6.5536 00000 14 2.6214 40000 (16) 1. 0485 76000 (38) 2. 8147 49767

6.3245 55320 3.419951893 2. 5148 66859 2.0912 79105

45 2025

91125 41 00625

1845 28125 ( 9) 8.3037 65625 ( 11)3. 7366 94531 ( 1311. 6815 12539 (147.5668 06426 (16 3.4050 62892 (39) 4. 7544 50505

6.7082 03932 3.5568 93304 2. 5900 20064 2.1411 27368

1

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS n k Tahle 3.1

26 676

17576 4 56976

118 81376 3089 15776

! 918.0318 10176 11 2. 0882 70646 12 5.4295 03679 14 1. 4116 70957 (33) 9.1066 85770

5. 0990 19514 2.9624 96068 2.2581 00864 1.9186 45192

31 961

29791 9 23521

286 29151 8875 03681

! 10l2. 7512 61411 11 8.5289 10374 13 2. 6439 62216 14 8. 1962 82870 (35) 6. 2041 26610

5.5677 64363 3.1413 80652 2. 3596 11 062 1. 9873 40755

36 1296

46656 16 79616

604 66176 9l2.1767 82336

10 7.8364 16410 12 2. 8211 09907 14 1. 0155 99567 15) 3. 6561 58440

(37) 2.2452 25771 6. 0000 00000 3. 3019 27249 2.4494 89743 2.0476 72511

41 1681

68921 28 25761

1158 56201

! 9) 4. 7501 04241

11) 1. 9475 42739 12)7.984925229 14)3.2738 19344

(16) 1. 3422 65931 (38) 5.0911 10945

6.4031 24237 3. 4482 17240 2.5304 39534 2.1016 32478

27 729

19683 5 31441

143 48907 3874 20489

ll011. 0460 35320 11 2.8242 95365 12 7.6255 97485 (14 2.0589 11321 ( 34) 2. 2528 39954

5.196152423 3.0000 00000 2.2795 07057 1.9331 82045

32 1024

32768 10 48576

335 54432 9 1.0737 41824

10 3.4359 73837 12 1. 0995 11628 13 3.5184 37209 15 1. 1258 99907

(36) 1. 3292 27996 5. 6568 54249 3; 1748 02104 2.3784 14230 2.0000 00000

37 1369

50653 18 74161

693 43957 9 2.5657 26409

10 9.4931 87713 12 3.5124 79454 14 1.2996 17398 15 4.8085 84372

( 37) 4. 3335 25711 6.0827 62530 3.3322 21852 2. 4663 25715 2. 0589 24137

42 1764

74088 31 11696

1306 91232 ( 9l5.4890 31744 (11 2.3053 93332 (12 9.6826 51996 (14) 4. 0667 13838 (16) 1. 7080 19812 (38) 9.0718 49315

6.4807 40698 3.4760 26645 2. 5457 29895 2.1117 85765

46 47 2116 2209

97336 1 03823 44 17456 48 79681

2059 62976 2293 45007 9j9.4742 96896 (10 1.0779 21533

11 4.3581 76572 ! 11 5.0662 31205 13 2.0047 61223 13 2.3811 28666 14) 9.2219 01627 15 1.1191 30473 16) 4. 2420 74748 16 5.2599 13224

(39) 8.0572 70802 (40) 1. 3500 46075 6.7823 29983 6.8556 54600 3.5830 47871 3.6088 26080 2. 6042 90687 2. 6183 30499 2.1505 60013 2.1598 30012

28 784

21952 6 14656

172 10368 4818 90304

! 10! 1.3492 92851 11 3.1780 19983 13 1.0578 45595 14 2.9619 67667 (34) 5.3925 32264

5.2915 02622 3. 0365 88972 2. 3003 26634 1.9472 94361

33 1089

35937 11 85921

391 35393 911. 2914" 67969

10 4.2618 44298 12 1. 4064 08618 13 4. 6411 48440 15)1.5315 78985

( 36) 2. 7818 55434 5.7445 62647 3.2075 34330 2. 3967 81727 2.0123 46617

38 1444

54872 20 85136

792 35168 ( 9 3.0109 36384

! 11 1. 1441 55826 12 4.3477 92138 14 1. 6521 61013 15 6.2782 11848 (37) 8. 2187 60383

6.1644 14003 3. 3619 75407 2.4828 23796 2. 0699 35054

43 1849

79507 34 18801

1470 08443 ( 9) 6. 3213 63049 (11) 2.7181 86111 (13) 1. 1688 20028 (14) 5.0259 26119 (16) 2.1611 48231 ( 39) 1. 596 7 72093

6.5574 38524 3.5033 98060 2.5607 49602 2.1217 47461

48 2304

1 10592 53 08416

2548 03968 (10) 1. 2230 59046 (11) 5. 8706 83423 (13) 2. 8179 28043 (15) 1. 3526 05461 (16) 6. 4925 06211 (40) 2. 2376 37322

29 841

24389 7 07281 ,~05 11149 5~148 23321

! lOll. n49 87631 11 5. 0024 64130 13 1. 4~,07 14598 14 4.2070 72333 (35) 1. 2~j18 49008

5.3851 64807 3.0723 16826 2. 3,~05 95787 1.9610 09057

34 1156

39304 13 36336

454 35424

! 9)1.5448 04416 10l5. 2~,23 35014 12 1.7B57 93905 ( 13l6. 0716 99277 (15 2.0643 77754 (36) 5. 6950 03680

5.8309 51895 3.2396 11801 2. 4lL47 36403 2.0;!43 97459

39 1521

59319 23 13441

902 24199 9) 3. 5lL87 43761

11) 1. 3723 10067 12l5. 3S20 09260 14 2. OB72 83612 15 8.1404 06085

( 38) 1. 5:l30 29700 6.2449 97998 3. 3912 11443 2.4989 99399 2.0B07 16549

44 1936

85184 37 48096

1649 16224 9 7.2%3 13856

11 3.1927 78097 13 1. 4048 22363 14 6.lIH2 18395 16 2. 7197 36094

( 39) 2. 7724 53276 6.6332 49581 3.5303 48335 2.5755 09577 2. D15 25513

49 2401

17649 57 64801

21324 75249 (10) 1. 31341 28720 ( 11) 6. 71322 30728 (l3l3. 3:?32 93057 (15 1.6:184 13598 (16 7.9792 26630 (40) 3.6703 36822

6.9282 03230 7.0000 00000 3.6342 41186 3.6!;93 05710 2.6321 48026 2.6457 51311 2.1689 43542 2.1779 06425

n2[(-~)3J 1 1 n3[(-i)lJ n4[(-~)9J 1 n3 [( -~) 7J 'l'he numbers in square brackets at the bottom of the page mean that the maximum

error In n linear Interpolate Is a X lO-p (p In parenthesel!), and that to Interpolate to the full tabular accuracy m points must be used In Lagrange's and Altkens methods for the respective functions n'/r.

See page II.

25

26 ELEMENTARY ANALYTICAL METHODS Table 3.1 POWERS AND ROOTS n k

k 1 50 51 52 53 54 2 2500 2601 2704 2809 2916 3 1 25000 1 32651 1 40608 1 48877 1 57464 4 62 50000 67 65201 73 11616 78 90481 85 03056 5 3125 00000 3450 25251 3802 04032 4181 95493 4591 65024 6

r

O 1.5625 00000 lOll. 7596 28780 r 1. 9770 60966 10 2.2164 36113 ( 10 2.4794 91130 7 11 7.8125 00000 11 8.9741 06779 12 1. 0280 71703 12 1. 1747 11140 ! 12 1. 3389 25210 8 13 3.9062 50000 13 4.5767 94457 13 5. 3459 72853 13 6.2259 69041 13 7.2301 96134 9 15 1. 9531 25000 15l2.3341 65173 15 2.7799 05884 15 3.2997 63592 15 3.9043 05912 10 ( 16 9.7656 25000 17 1. 1904 24238 17 1.4455 51059 17 1. 7488 74704 (17 2.1083 25193 24 ( 40) 5.9604 64478 (40) 9. 5870 33090 (41) 1. 5278 48342 (41) 2. 4133 53110 (41) 3.7796 38253

1/2 7.0710 67812 7.1414 28429 7.2111 02551 7. 2801 09889 7.3484 69228 1/3 3. 6840 31499 3.7084 29769 3.7325 11157 3.7562 85754 3.7797 63150 1/4 2.6591 47948 2.6723 45118 2. 6853 49614 2.6981 67876 2.7108 06011 1/5 2.1867 24148 2.1954 01897 2.2039 44575 2.2123 56822 2.2206 43035

1 55 56 57 58 59 2 3025 3136 3249 3364 3481 3 1 66375 1 75616 1 85193 1 95112 2 05379 4 91 50625 98 34496 105 56001 113 16496 121 17361 5 5032 84375 5507 31776 6016 92057 6563 56768 7149 24299 Q (lOr' 7680 64063

'"1'" .M. '"'' 10 3.4296 44725 10 3.8068 69254 ro 4.2180 53364 7 ! 12 1. 5224 35234 12 1. 7270 94850 12 1. 9548 97493 12 2.2079 84168 12 2.4886 51485 8 13 8.3733 93789 13 9.6717 31157 14 1.1142 91571 14 1. 2806 30817 14 1. 4683 04376 9 15 4. 6053 66584 15 5. 4161 69448 15 6. 3514 61955 15 7.427658740 15 8. 6629 95819 10 ( 17) 2. 5329 51621 17 3.0330 54891 17 3.6203 33315 17 4. 3080 42069 (17 5.1111 67533 24 (41) 5.8708 98173 (41) 9.0471 67858 (42) 1.3835 55344 (42) 2.1002 54121 (42) 3. 1655 43453

1/2 7.4161 98487 7.4833 14774 7.5498 34435 7.6157 73106 7.6811 45748 1/3 3.8029 52461 3.8258 62366 3.8485 01131 3. 8708 76641 3.8929 96416 1/4 2.7232 69815 2. 7355 64800 2.7476 96205 2. 7596 69021 2.7714 88002 1/5 2.2288 07384 2.2368 53829 2.2447 86134 2.2526 07878 2.2603 22470

1 60 61 62 63 64 2 3600 3721 3844 3969 4096 3 2 16000 2 26981 2 38328 2 50047 2 62144 4 129 60000 138 45841 147 76336 157 52961 167 77216 5 7776 00000 8445 96301 9161 32832 9924 36543 ! 16 1. 0737 41824 6 ! 10) 4. 6656 00000 10 5.1520 37436

ro 5.6800 23558 10 6.2523 50221 6.8719 47674

7 1212.7993 60000 12 3. 1427 42836 12 3.5216 14606 12 3. 9389 80639 12 4. 3980 46511 8 14 1. 6796 16000 14 1. 9170 73130 14 2.1834 01056 14 2.4815 57803 ~14 2.8147 49767 9 16 1. 0077 69600 16 1. 1694 14609 16 1. 3537 08655 16 1. 5633 81416 16 1. 8014 39851

10 (17 6.0%6 17600 17 7.1334 29117 (17 8. 3929 93659 17 9. 8493 02919 ( 18 1. 1529 21505 24 ( 42) 4. 7383 81338 (42) 7.0455 68477 (43) 1. 0408 79722 (43) 1. 5281 75339 (43) 2. 2300 74520

1/2 7.7459 66692 7.8102 49676 7.874007874 7.9372 53933 8.0000 00000 1/3 3. 9148 67641 3.9364 97183 3.9578 91610 3.9790 57208 4. 0000 00000 1/4 2.7831 57684 2.7946 82393 2.8060 66263 2.8173 13247 2.8284 27125 1/5 2.2679 33155 2.2754 430