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Preface’ Science Foundation The present volume is an outgrowth held at Cambridge, Mass., on September National purpose of the meeting was to evaluate of the availability opinion tables would continue to exist. that of a Conference on Mathematical 15-16, 1954, under Tables the auspices of the The in the light of Technology. tables and the Massachusetts Institute the need for mathematical of large scale computing machines. in spite of the increasing use of the new machines It was the consensus of for the basic need Numerical tables of mathematical functions are in continual demand by scien- A greater variety of functions and higher accuracy of tabula- in- tables serve tists and engineers. tion are now required as a result of scientific advances and, especially, creasing use of automatic mainly forpreliminarysurveys For those without easy access to machines, for machine operation. such tables are, of course, indispensable. of problems before programming connection, computers. of the latter the the In Consequently, the Conference version of the classical recognized tables of functions of Jahnke-Emde. that there was a pressing need for a the project, the National Science Foundation to prepare such a volume and established modernized ment of Standards mittee, with Professor Philip M. Morse of the Massachusetts as chairman, to advise ~course of its preparation. of A. Erdelyi, M. C. Gray, N. Metropolis, and J. W. Tukey. Todd, C. B. Tompkins, the staff of the National to the Chairman, In addition requested the National To imple- Bureau an Ad Hoc Advisory Com- of Technology Institute Bureau of Standards the Committee J. B. Rosser, H. C. Thacher, during the consisted Jr., John The primary aim has been to include a maximum of useful information within of a moderately in all fields. large volume, with particular attention An attempt has been made to cover the entire to supplement To carry out the goal set forth by tbe Ad Hoc Committee, the in computation work, as well as by providing tables by including the mathematical to the needs of field of special it has been that numerical methods properties the limits scientists functions. necessary are important which demonstrate the use and extension of the tables. the direction of the late Milton Abramowitz, The Handbook was prepared under Irene A. Stegun. Its success has depended greatly upon Their efforts together with appreciated. The particular the cooperation contributions are acknowledged at appropriate places in the text. Science Foundation for the preparation of the material the cooperation of of the Ad HOC of these and The sponsor- is are greatly and many mathematicians. Committee other ship of gratefully the National recognized. that individuals is hoped It in many cases acquaint will Washington, D.C. this volume will not only meet the needs of all table users but its users with new functions. ALLEN V. ASTIN, L?imctor.

to the Ninth Printing Preface 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 the first copy came from the Assistant Secretary of Com- merce for Science and Technology, presented the 100,OOOth copy of the Handbook to the President. Today, total distribution the 150,000 mark at a scarcely diminished rate. in 1614. Only four and one-half years after the press in 1964, Myron Tribus, then Science Advisor to Lee A. DuBridge, is approaching 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. LEWIS M. BRANSCOMB, Director National Bureau of Standards November 1970

Foreword This volume is the result of the cooperative The National Bureau tables and has had under of Standards consideration, of a compendium like the present one. During the NBS Applied Mathematics Division long been effort of many persons and a number out the on Tables, on May 15, 19.52, Dr. Abramo- turning IO years, has for at a Conference least for such an undertaking, support. but preliminary plans technical advice and financial Division since 1943 it has published of the National Research Council has also had an the quarterly editorial journal, supervision “Mathe- being (MTAC),, of organizations. mathematical production called by witz indicated of t,hat Division mentioned the need for The Mathematics interest in tables; active matical Tables and Aids exercised by a Committee Subsequent to for “that consensus With Institute using on several in tbe published Report tables of various and engineers National Science Foundation table production. Massachusetts needs scientists reached forth for example, task of table making also agreed Computer, tables for The Report and that participants, J. H. Curtiss, help implement that with interpolation suggested that the NSF contribute the following R. W. Hamming, these and other the advent to Computation” of the Division. the NBS Conference was drawn on Tables to the desirability in 1952 its support of Technology a z-day Conference on September kinds. Twenty-eight persons tables as well as table producers. cpnclusions recomlmendations, of of high-speed the Conference. cornputting and but definitely did not remove the need of financing the attention of activity the in on Tables was called at the 15-16, 1954, the attended, to discuss representing This conference which were set There was general agreement, equipment changed the tables”. It was for the Occasional for of Tables and a set of formulas and functions useful the production to the occasional computer”. of such a Handbook need is for a Handbook “an outstanding tables of usually encountered techniques the NBS undertake and other financial assistance. The Conference Committee: P. M. Morse (Chairman), D. H. Lehmer, C. B. Tompkins, recommendations. elected, from M. Abramowitz, J. W. Tukey, its to undertook Division independent The Bureau of Standards to produce National Science Foundation made funds available. to the Mathematics of the Bureau, which vide the NSF with Committee Tables of the Mathematics some changes of membership, The present and that judgments as Division that Conferences sometimes recommendations the Committee reconstituted is evidence became volume their was on grants ffor the work, on Revision of the National Research Council. the Committee which is signing can sometimes get acted on. the recommended To provide carried out the work, tables and the technical guidance and to pro- the Conference of Mathematical This, after this Foreword. reach conclusions V ,/”

VI FOREWORD Active work was started at the Bureau for the various chapters, and of Dr. Abramowitz. the general direction required in 1956. The overall plan, the enthusiasm Since his untimely of Irene A. Stegun. the selection task the effort has the of have had many discussions about Though many details have had t’o be argued out as they the same as were death, The workers at of the volume have remained the Committee to begin the and the members style and layout. the basic specifications of authors were contributions continued under Bureau content, came up, outlined by of the task carried out by the staff of the NBS Computing the Massachusetts Institute The Committee wishes here to register in planning, quality expert collaborators ciation of the willingness with which the plans. We hope this resulting memorial We regret he did not live the vision and industry to to see its publication. its commendation of Technology Conference of 1954. of the magnitude and Section and their these Tables, and its appre- into volume will be judged by its users to be a worthy its various suggestions were incorporated and editing collecting of its chief architect, Milton Abramowitz. P. M. MORSE, Chairman. A. ERD~LYI M. C. GRAY N. C. METROPOLIS J. B. ROSSER H. C. THACHER. JOHN TODD ‘C. B. TOMPKINS J. W. TUKEY. Jr.

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables Edited by Milton Abramowitz and Irene A. Stegun 1. Introduction The present Handbook scientific functions problems. the volume ext,ends by E. Jahnke to workers during provide hensive and self-contained matical neering Funct.ions invaluable editions’ present by giving more numerical of mathematical functions. also been increased. The classification the chapters of that of An A. Fletcher, In general, graphs, for automatic principal mathematical lated the chapters or computers, polynomial functions, investigators that arise The well-known with summary in physical has been designed to a compre- of the mathe- and engi- Tables of and F. Emde has been in its many The of these authors accurate larger collections tabulated fields half-century. and more the of in these past the work tables, and by giving extensive properties The number of functions covered has of functions this Handbook of Mathematical in Index J. C. P. Miller, and organization is similar Tables to by and L. Rosenhead. contain numerical rational approximations tables, properties and statements of the tabu- those of computa- the of particularly to illustrate importance. tional are given the computation also range. outside their each chapter there books and papers properties matical found. Also more sive tioned tables Council (formerly to Computation). above, is important lists of listed to be found quarterly Mathematical Many of function At numerical the use of the examples tables and lie values which text in giving proofs of the mathe- in the chapter may be the the is a short bibliography in which stated in the bibliographies the end of are numerical tables. tables are given in and current in information the National Mathematics the Comprehen- Index men- on new Research of Computation Tables and Other Aids notations commonly Higher The ma.thematical are those particularly adopted Transcendental used in this Hand- in standard book Func- texts, 1-3, by A. ErdBlyi, W. Magnus, tions, Volumes 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 the throughout to make it uniform Handbook, which would have been a costly and laborious undertaking. In most tables at least five significant figures 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 recent, in 1960 by McGraw-Hill, was 1 The most 2 The second edition, with L. J. Comrie added as co-author, was published and Scientific Com- the sixth, with F. Loesch added as cc-author, U.S.A., and Teubner, Germany. published in two volumes puting Service Ltd., Great Britain. in 1962 by Addison-Wesley, U.S.A., precision in their by use of higher-order described below. interpolates may obtain them interpolation procedures, In certain tables many-figured function values are given at irregular in the argument. An example 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. intervals The maximum end-figure error, or “tolerance” is 6/& of 1 unit in the tables in this Handbook everywhere 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. in the case of the elementary IX /-

. X INTRODUCTION 3. Auxiliary Functions and Arguments One of the objects of this Handbook is to pro- tables or computing methods which enable functions over tabulated the ranges of real values of their parameters. to achieve this object, functions the original and auxiliary arguments An example will make functions to remove at frequent use has the their to co e with t fi e pro- vide the user to evaluate complete In order been made of auxiliary infinite singularities, infinite cedure clear. part of ranges. The exponential integral of positive argument is given by recludes direct unctions Ei(x)-In P however, in function; this inter- x are well- region. the latter higher function to nine slightly The tabulated For +/(x1-~0> and jP instance, we have the required interpolate. In the present jo=.89717 4302 ji=.89823 7113 p=.527 The most convenient way to evaluate on a desk calculating machine in turn on the keyboard, plications check is reading unity. We obtain the formula is.to set o and ji and carry out t d e multi- a partial dial by then provided and p cumulatively; the multiplier by l-p j.6z,E.‘;9;72;&39717 4302)+.527(.89823 7113) it Since is known of 3 X 10 -6 in the linear result this answer .89773. The maximum possible error is composed of the error committed there is a possible error formula, we round off this in that to

INTRODUCTION XI is, that by the 3 X lo-‘, roundingJ last and so certainly formula. this example, is the 5-point one, given by .4403X 10m5, plus cannot exceed .8X lo-‘. the (2) Lagrange’s formula relevant f=A-,(p)f_z+A-,(p)f-1+Ao(p>fo+A,(p)fi In the range p=O(.Ol)l. for p=.52, Tables of the coefficients An(p) are given 25 for formula in each evaluation we accumulate multiplier now have register since the following their sum subtable. .53 and .54 in We evaluate +A&)fa in chapter the Again, in the is unity. We turn. the An(p) x 7.952 7.953 7.954 m=&(x) 9757 .89772 .89774 0379 .89775 0999 10622 10620 -2 in the third and fourth columns are first and second differences of the values of (see below) ; the smallness of the second The numbers the xezEl(x) difference provides a check on the three tions. linear is now obtained required : interpolation interpola- value The by fn=.3(.89772 9757)+.7(.89774 0379) = 239773 7192. In cases where the correct order of the Lagrange polynomial interpolations polynomials check on their adequacy. is not known, one of the prelimina may have to be performed wit T of two or more different orders as a (3) Aitken’s method of iterative for carrying The scheme tion. in the present example linear out is as follows: interpola- this process .; 1 2 3 4 5 & 7.9 8.1 7.8 8.2 7.7 : Yn=ze”G@) 7113 89823 89717 4302 7888 89927 8737 : 89608 7306 . 90029 . 89497 9666 Yo. I 89773 :89774 44034 48264 2 90220 4 98773 2 35221 Yo. 1, (I Yo, 1.2. I Yo.1.a.s.n .89773 71499 2394 1216 2706 . 89773 71938 ii 89773 71930 30 X,-X .0473 0527 . 1473 1527 . 2473 2527 -. -. -. Here yo,n=- 1 Yo x.--20 Yn 20-x x,-x Yo.1 Yo. 1 Yo.1 ,n=- G--z1 l/O.” . . ., m--l.m.n-- 1. x,-x x,-x 1 ~n-%n . l/0.1. Yo.1. . . . ., n-1.98 -, m-1.n x,-x x,-x 1 and x~--5 are used as on a the cross-product If the quantities when Z.-X forming their accumulation multipliers desk machine, in the multiplier at that stage. An extra decimal place intermediate carried interpolates of rounding guard against accumulation is the divisor register (~~-2) the in -(x,-x) to be used is usually to safe- errors. in which to some extent, but the tabular rate of convergence and at accumulation this example, with of rounding the the given argument, tabular to the nearest of the remaining values are used to achieve the the same errors, tabular then argu- The order is immaterial maximum time minimize we begin, as in argument take ments, and so on. of The number nearest values required emerges naturally tabular achieve a given precision the course of the iterations. example six values were used, even though known in advance extra row confirms check. a valuable to in in the present it was that five would suffice. The the convergence and provides Thus (4) Difference difference notation formulas. We use the central (chapter 25), S2fl safz wa Here Sf1l2=f1-f0, 8f3/a=fz-f1, . . . ,, a2/1=sf3ia-afiia=fa-2fi+fo ~af3~~=~aja-~aj~=fa-3j2+3fi-k 8'fa=~aj~fsla-6~3~2=f4-~f~+~ja-4f~+fo and so on. In table is as follows, the present example difference written function, high differences provides a check on the function values the relevant part of the the differences being lace of the The sma ness of the in units of the as is customary. last decimal B xe=El(x) 7:9 8.0 .89717 . 89823 4302 7113 SY 2754 2036 -2 -2 S4f -34 -39 Applying, formula for example, Everett’s interpolation j~=(l-P)fo+E2(P)~*jo+E4(P)~4jo+ . +Pfl+F2(P)~afl+F4(P)~4fl+ . . . . * * the numerical and tion from Table 25.1, we find takin toe flf cients Es(p), E4 !l t at values of the interpola- and F,(p) ), F,(p) ,,/

INTRODUCTION XII 10Qf.6,= .473(89717 + .527(89823 7193. = 89773 4302) + 7113) .061196(2 + .063439(2 2754) - 2036) .012(34) - .012(39) We may formula shows is approximately notice that in passing that in a linear Everett’s interpolate the error mPwfo+ F2(P)wl= m(P) + ~2(P)lk?f0+wJ Since the maximum range O

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