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 forpreliminarysurveys 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 of 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 particular 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 tbe 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. ALLEN V. ASTIN, L?imctor. Washington, D.C. 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,OOOth 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. LEWIS M. BRANSCOMB, Director National Bureau of Standards November 1970 Foreword This volume is the result 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 IO 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, 19.52, Dr. Abramo- witz of t,hat 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” (MTAC),, 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 z-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 cpnclusions and recomlmendations, which were set forth in tbe published Report of the Conference. There was general agreement, for example, “that the advent of high-speed cornputting 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. To 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 ffor 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 t’o 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. ERD~LYI M. C. GRAY N. C. METROPOLIS J. B. ROSSER H. C. THACHER. Jr. JOHN TODD ‘C. B. TOMPKINS J. W. TUKEY. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables Edited by Milton Abramowitz and Irene A. Stegun 1. Introduction The present Handbook has been designed to tional importance. Many numerical examples provide scientific investigators with a compre- are given to illustrate the use of the tables and hensive and self-contained summary of the mathe- also the computation of function values which lie matical functions that arise in physical and engi- outside their range. At the end of the text in neering problems. The well-known Tables of each chapter there is a short bibliography giving Funct.ions by E. Jahnke and F. Emde has been books and papers in which proofs of the mathe- invaluable to workers in these fields in its many matical properties stated in the chapter may be editions’ during the past half-century. The found. Also listed in the bibliographies are the present volume ext,ends the work of these authors more important numerical tables. Comprehen- by giving more extensive and more accurate sive lists of tables are given in the Index men- numerical tables, and by giving larger collections tioned above, and current information on new of mathematical properties of the tabulated tables is to be found in the National Research functions. The number of functions covered has Council quarterly Mathematics of Computation also been increased. (formerly Mathematical Tables and Other Aids The classification of functions and organization to Computation). The ma.thematical notations used in this Hand- of the chapters in this Handbook is similar to book are those commonly adopted in standard that of An Index of Mathematical Tables by texts, particularly Higher Transcendental Func- A. Fletcher, J. C. P. Miller, and L. Rosenhead. tions, Volumes 1-3, by A. ErdBlyi, W. Magnus, In general, the chapters contain numerical tables, F. Oberhettinger and F. G. Tricomi (McGraw- graphs, polynomial or rational approximations Hill, 1953-55). Some alternative notations have for automatic computers, and statements of the also been listed. The introduction of new symbols principal mathematical properties of the tabu- has been kept to a minimum, and an effort has lated functions, particularly those of computa- been made to avoid the use of conflicting notation. 2. Accuracy of the Tables The number of significant figures given in each precision in their interpolates may obtain them table has depended to some extent on the number by use of higher-order interpolation procedures, available in existing tabulations. There has been described below. no attempt to make it uniform throughout the In certain tables many-figured function values Handbook, which would have been a costly and are given at irregular intervals in the argument. laborious undertaking. In most tables at least An example is provided by Table 9.4. The pur- five significant figures have been provided, and pose of these tables is to furnish “key values” for the tabular’ intervals have generally been chosen the checking of programs for automatic computers; to ensure that linear interpolation will yield. four- no question of interpolation arises. or five-figure accuracy, which suffices in most The maximum end-figure error, or “tolerance” physical applications. Users requiring higher in the tables in this Handbook is 6/& of 1 unit everywhere in the case of the elementary func- 1 The most recent, the sixth, with F. Loesch added as cc-author, was tions, and 1 unit in the case of the higher functions published in 1960 by McGraw-Hill, U.S.A., and Teubner, Germany. 2 The second edition, with L. J. Comrie added as co-author, was published except in a few cases where it has been permitted in two volumes in 1962 by Addison-Wesley, U.S.A., and Scientific Com- puting Service Ltd., Great Britain. to rise to 2 units. IX /- . X INTRODUCTION 3. Auxiliary Functions and Arguments One of the objects of this Handbook is to pro- The logarithmic singularity recludes direct inter- vide tables or computing methods which enable polation near x=0. The Pu nctions Ei(x)-In x the user to evaluate the tabulated functions over and x-liEi(ln x-r], however, are well- complete ranges of real values of their parameters. behaved and readily interpolable in this region. In order to achieve this object, frequent use has Either will do as an auxiliary function; the latter been made of auxiliary functions to remove the was in fact selected as it yields slightly higher infinite part of the original functions at their accuracy when Ei(x) is recovered. The function singularities, and auxiliary arguments to co e with x-‘[Ei(x)-ln x-r] has been tabulated to nine infinite ranges. An example will make t fi e pro- decimals for the range 05x<+. For +<x12, cedure clear. Ei(x) is sufficiently well-behaved to admit direct The exponential integral of positive argument tabulation, but for larger values of x, its expo- is given by nential character predominates. A smoother and more readily interpolable function for large x is xe-“Ei(x); this has been tabulated for 2 <x510. Finally, the range 10 <x_<m is covered by use of the inverse argument x-l. Twenty-one entries of xe-“Ei(x), corresponding to x-l = .l(- .005)0, suf- fice to produce an interpolable table. 4. Interpolation The tables in this Handbook are not provided Let us suppose that we wish to compute the with differences or other aids to interpolation, be- value of xeZ&(x) for x=7.9527 from this table. cause it was felt that the space they require could We describe in turn the application of the methods be better employed by the tabulation of additional of linear interpolation, Lagrange and Aitken, and functions. Admittedly aids could have been given of alternative methods based on differences and without consuming extra space by increasing the Taylor’s series. intervals of tabulation, but this would have con- (1) Linear interpolation. The formula for this flicted with the requirement that linear interpola- process is given by tion is accurate to four or five figures. For applications in which linear interpolation jp= (1 -P)joSPfi is insufficiently accurate it is intended that Lagrange’s formula or Aitken’s method of itera- where jO, ji are consecutive tabular values of the tive linear interpolation3 be used. To help the function, corresponding to arguments x0, x1, re- user, there is a statement at the foot of most tables spectively; p is the given fraction of the argument of the maximum error in a linear interpolate, interval and the number of function values needed in p= (x--x0>/(x1-~0> Lagrange’s formula or Aitken’s method to inter- polate to full tabular accuracy. and jP the required interpolate. In the present As an example, consider the following extract instance, we have from Table 5.1. jo=.89717 4302 ji=.89823 7113 p=.527 zez El (2) ze*El (z) 775 . 89268 7854 d0 . 89823 7113 The most convenient way to evaluate the formula ;:; : 8899348947 69361626 g. I .8999020729 77380868 on a desk calculating machine is.to set o and ji in turn on the keyboard, and carry out t d e multi- E : 8899670187 84733072 i8x: 4 :.9 900212279 6406”935 plications by l-p and p cumulatively; a partial [ 1 check is then provided by the multiplier dial ‘453 reading unity. We obtain The numbers in the square brackets mean that j.6z,E.‘;9;72;&39717 4302)+.527(.89823 7113) the maximum error in a linear interpolate is 3X10m6, and that to interpolate to the full tabular accuracy five points must be used in Lagrange’s Since it is known that there is a possible error and Aitken’s methods. of 3 X 10 -6 in the linear formula, we round off this result to .89773. The maximum possible error in 8 A. C. Aitken On inte elation b iteration of roportional parts, with. this answer is composed of the error committed out the use of diherences, ‘B rot Edin i: urgh Math. 8 oc. 3.6676 (1932). INTRODUCTION XI by the last roundingJ that is, .4403X 10m5, plus The numbers in the third and fourth columns are 3 X lo-‘, and so certainly cannot exceed .8X lo-‘. the first and second differences of the values of (2) Lagrange’s formula. In this example, the xezEl(x) (see below) ; the smallness of the second relevant formula is the 5-point one, given by difference provides a check on the three interpola- tions. The required value is now obtained by f=A-,(p)f_z+A-,(p)f-1+Ao(p>fo+A,(p)fi linear interpolation : +A&)fa Tables of the coefficients An(p) are given in chapter fn=.3(.89772 9757)+.7(.89774 0379) 25 for the range p=O(.Ol)l. We evaluate the formula for p=.52, .53 and .54 in turn. Again, = 239773 7192. in each evaluation we accumulate the An(p) in the multiplier register since their sum is unity. We now have the following subtable. In cases where the correct order of the Lagrange polynomial is not known, one of the prelimina x m=&(x) interpolations may have to be performed witT 7.952 .89772 9757 polynomials of two or more different orders as a 10622 check on their adequacy. 7.953 .89774 0379 -2 (3) Aitken’s method of iterative linear interpola- 10620 tion. The scheme for carrying out this process 7.954 .89775 0999 in the present example is as follows: .; & Yn=ze”G@) YoI. Yo. 1, (I Yo, 1.2. I Yo.1.a.s.n X,-X 89823 7113 .0473 1 7.9 : 89717 4302 89773 44034 -. 0527 2 8.1 89927 7888 :89774 48264 .89773 71499 . 1473 3 7.8 : 89608 8737 2 90220 2394 . 89773 71938 -. 1527 4 8.2 . 90029 7306 4 98773 1216 89773 71930 . 2473 5 7.7 . 89497 9666 2 35221 2706 ii 30 -. 2527 Here yo,n=- x.--12 0 YYon x2,0-x-x S2fl safz Yo.1 ,n=-G --1z 1 Yl/Oo..”1 xx,,--xx wa Yo. . . 1 l/0.1. . . ., n-1.98 x,-x 1 1. ., m--l.m.n-- ~n-%n Yo.1. . . -, m-1.n x,-x Here If the quantities Z.-X and x~--5 are used as Sf1l2=f1-f0, 8f3/a=fz-f1, . . . ,, multipliers when forming the cross-product on a a2/1=sf3ia-afiia=fa-2fi+fo desk machine, their accumulation (~~-2) -(x,-x) ~af3~~=~aja-~aj~=fa-3j2+3fi-k in the multiplier register is the divisor to be used 8'fa=~aj~fsla-6~3~2=f4-~f~+~ja-4f~+fo at that stage. An extra decimal place is usually and so on. carried in the intermediate interpolates to safe- In the present example the relevant part of the guard against accumulation of rounding errors. difference table is as follows, the differences being The order in which the tabular values are used written in units of the last decimal lace of the is immaterial to some extent, but to achieve the function, as is customary. The sma B ness of the maximum rate of convergence and at the same high differences provides a check on the function time minimize accumulation of rounding errors, values we begin, as in this example, with the tabular xe=El(x) SY S4f argument nearest to the given argument, then 7:9 .89717 4302 -2 2754 -34 take the nearest of the remaining tabular argu- 8.0 . 89823 7113 -2 2036 -39 ments, and so on. The number of tabular values required to Applying, for example, Everett’s interpolation achieve a given precision emerges naturally in formula the course of the iterations. Thus in the present example six values were used, even though it was j~=(l-P)fo+E2(P)~*jo+E4(P)~4jo+ . . . . known in advance that five would suffice. The +Pfl+F2(P)~afl+F4(P)~4fl+ . * * extra row confirms the convergence and provides a valuable check. and takin the numerical values of the interpola- (4) Difference formulas. We use the central tion toe flf cients Es(p), E4 ), F,(p) and F,(p) difference notation (chapter 25), from Table 25.1, we find t! l at ,,/ XII INTRODUCTION 10Qf.6,= .473(89717 4302) + .061196(2 2754) - .012(34) can be used. We first compute as many of the + .527(89823 7113) + .063439(2 2036) - .012(39) derivatives ftn) (~0) as are significant, and then = 89773 7193. evaluate the series for the given value of 2. An advisable check on the computed values of the We may notice in passing that Everett’s derivatives is to reproduce the adjacent tabular formula shows that the error in a linear interpolate values by evaluating the series for z=zl and x1. is approximately In the present example, we have mPwfo+ F2(P)wl= m(P) + ~2(P)lk?f0+wJ f(x) =xeZEt(x) Since the maximum value of IEz(p)+Fz(p)I in the f’(z)=(l+Z-‘)f(Z)-1 range O<p<l is fd, the maximum error in a linear f”(2)=(1+2-‘)f’(Z)--Z-Qf(2) interpolate is approximately f”‘(X) = (1 -i-z-y’(2) -22~Qf’(5) +22-y(2). With x0=7.9 and x-x0= .0527 our computations are as follows: an extra decimal has been retained in the values of the terms in the series to safeguard (5) Taylor’s series. In cases where the succes- against accumulation of rounding errors. sive derivatives of the tabulated function can be computed fairly easily, Taylor’s expansion p:‘(xo)/k! (x--so) y’k’(x0)/k! i .89717 4302 .89717 4302 ~(x,=~(xo)+(x-x,,~~+(x-xo,~~~ ; - ..0010017143 07666291 -.oo.0o0o0o5 6 63013539 35 +(~-x,)q$+ . . . 3 .00012 1987 ..oao9o7o7o3 07011974 9 5. Inverse Interpolation With linear interpolation there is no difference The desired z is therefore in principle between direct and inverse interpola- tion. In cases where the linear formula rovides z=zQ+p(z,--2,,)=8.1+.708357(.1)=8.17083 57 an insufficiently accurate answer, two met fl ods are available. We may interpolate directly, for To estimate the possible error in this answer, example, by Lagrange’s formula to prepare a new we recall that the maximum error of direct linear table at a fine interval in the neighborhood of the interpolation in this table is Aj=3X lOwe. An approximate value, and then apply accurate approximate value for dj/dx is the ratio of the inverse linear interpolation to the subtabulated first difference to the argument interval (chapter values. Alternatively, we may use Aitken’s 25), in this case .OlO. Hence the maximum error method or even possibly the Taylor’s series in x is approximately 3XlO-e/(.OlO), that is, .0003. method, with the roles of function and argument (ii) Subtabulation method. To improve the interchanged. ap roximate value of x just obtained, we inter- It is important to realize that the accuracy of po Pa te directly for p=.70, .7l and .72 with the aid an inverse interpolate may be very different from of Lagrange’s 5-point formula, that of a direct interpolate. This is particularly true in regions where the function is slowly X xe=El (x) 6 QQ varying, for example, near a maximum or mini- 8. 170 . 89999 -.-_ 1 0151 mum. The maximum precision attainable in an 8.171 . 90000 3834 -2 inverse interpolate can be estimated with the aid of 1 0149 the formula 8. 172 90001 3983 AxmAj/df Inverse linear interpolation in the new table dx gives in which Aj is the maximum possible error in the function values. Example. Given xe”Ei(z) = .9, find 2 from the Hencex=8.17062 23. table on page X. (i) Inverse linear interpolation. The formula An estimate of the maximum error in this result for v is is df~1x10-8_1x10-7 Ajl z .OlO In the present example, we have .9 - .89927 7888 72 2112 (iii) Aitken’s method. This is carried out in the ‘=.90029 7306- .89927 7888=101=‘708357’ same manner as in direct interpolation. INTRODUCTION XIII n yn=xeeZE1(x) 2, Z0.n QJ.98 a.1 2.n zo.l.2.3.74 l/n-u 0 . 90029 7306 8. 2 .00029 7306 4 : 8990912279 67083838 88.. 31 88.. 1177008233 51751025 8. 1706,l 9521 -. . 0000017229 26101323 3 . 89823 7113 8. 0 8. 17113 8043 2 5948 8. 17062 2244 -. 00176 2887 % : 9809272177 44360925 87.. 49 88.. 1167919424 90433872 21 87134325 421351 8. 17062 2321685 -. .0002022782 45G69958 The estimate of the maximum error in this discrepancy in the highest interpolates, in this result is the same as in the subtabulation method. case xo .I ,2. 3 A, and ZLI .2. 8 .s. An indication of the error is also provided by the I 6. Bivariate Interpolation Bivariate interpolation is generally most simply interpolation is then carried out in the second performed as a sequence of univariate interpola- direction. tions. We carry out the interpolation in one An alternative procedure in the case of functions direction, by one of the methods already described, of a complex variable is to use the Taylor’s series for several tabular values of the second argument expansion, provided that successive derivatives in the neighborhood of its given value. The of the function can be computed without much interpolates are differenced as a check, and difficulty. 7. Generation of Functions from Recurrence Relations Many of the special mathematical functions (iii) the direction in which the recurrence is being which depend on a parameter, called their index, applied. Examples are as follows. order or degree, satisfy a linear difference equa- Stability-increasing n tion (or recurrence relation) with respect to this parameter. Examples are furnished by the Le- Pm(x), p:(2) gendre function P,(z), the Bessel function Jn(z) Qnb), Q:(x) (x<l) and the exponential integral E,(x), for which we y&9, KG) have the respective recurrence relations J-n-&), z-t44 &Cd (n<d J n+*-~Jn+J.-l=O Stability-decreasing 7t P”(X), P.,(z) @<l) nE,+,+xE,,=e-=. Qnh), Q:(x) Particularly for automatic work, recurrence re- J&4, Z.@) lations provide an important and powerful com- Jn+Hcd , Zn+&) puting tool. If the values of P&r) or Jn(z) are Em(z) (n >r) known for two consecutive values of n, or E',(z) F,,(t, p) (Coulomb wave function) is known for one value of n, then the function may be computed for other values of n by successive Illustrations of the generation of functions from applications of the relation. Since generation is their recurrence relations are given in the pertinent carried out perforce with rounded values, it is chapters. It is also shown that even in cases vital to know how errors may be propagated in where the recurrence process is unstable, it may the recurrence process. If the errors do not grow still be used when the starting values are known relative to the size of the wanted function, the to sufficient accuracy. process is said to be stable. If, however, the Mention must also be made here of a refinement, relative errors grow and will eventually over- due to J. C. P. Miller, which enables a recurrence whelm the wanted function, the process is unstable. process which is stable for decreasing n to be It is important to realize that st,ability may applied without any knowledge of starting values depend on (i) the particular solution of the differ- for large n. Miller’s algorithm, which is well- ence equation being computed; (ii) the values of suited to automatic work, is described in 19.28, x or other parameters in the difference equation; Example 1. XIV INTRODUCTION 8. Acknowledgments The production of this volume has been the bibliographic references and assisted in preparing result of the unrelenting efforts of many persons, the introductory material. all of whose contributions have been instrumental Valuable assistance in the preparation, checkin in accomplishing the task. The Editor expresses and editing of the tabular material was receive IFi his thanks to each and every one. from Ruth E. Capuano, Elizabeth F. Godefroy, The Ad Hoc Advisory Committee individually David S. Liepman, Kermit Nelson, Bertha H. and together were instrumental in establishing Walter and Ruth Zucker. the basic tenets that served as a guide in the forma- Equally important has been the untiring tion of the entire work. In particular, special cooperation, assistance, and patience of the thanks are due to Professor Philip M. Morse for members of the NBS staff in handling the myriad his continuous encouragement and support. of detail necessarily attending the publication Professors J. Todd and A. Erdelyi, panel members of a volume of this magnitude. Especially of the Conferences on Tables and members of the appreciated have been the helpful discussions and Advisory Committee have maintained an un- services from the members of the Office of Techni- diminished interest, offered many suggestions and cal Information in the areas of editorial format, carefully read all the chapters. graphic art layout, printing detail, preprinting Irene A. Stegun has served eff ectively as associate reproduction needs, as well as attention to pro- editor, sharing in each stage of the planning of motional detail and financial support. In addition, the volume. Without her untiring efforts, com- the clerical and typing stafI of the Applied Mathe- pletion would never have been possible. matics Division merit commendation for their Appreciation is expressed for the generous efficient and patient production of manuscript cooperation of publishers and authors in granting copy involving complicated technical notation. permission for the use of their source material. Finally, the continued support of Dr. E. W. Acknowledgments for tabular material taken Cannon, chief of the Applied Mathematics wholly or in part from published works are iven Division, and the advice of Dr. F. L. Alt, assistant on the first page of each table. Myrtle R. Ke Yliin g- chief, as well as of the many mathematicians in ton corresponded with authors and publishers the Division, is gratefully acknowledged. to obtain formal permission for including their material, maintained uniformity throughout the M. ABRAMOWITZ.