Milton Abramowitz, Irene A. Stegun-Handbook of Mathematical Functions_ with Formulas, Graphs, and Mathematical Tables-National Bureau of Standards (1970)， 资源来自互联网Preface to the Ninth PrintingThe enthusiastic reception accorded the handbook of MathematicalFunctions"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 camefrom the press in 1964, Myron Tribus, the Assistant Secretary of Com-merce for Science and Technology, presented the 100,000th copy of theHandbook to Lee A. DuBridge, then Science advisor to the PresidentToday total distribution is approaching the 150,000 mark at a scarcelydiminished rateThe success of the handbook has not ended our interest in the subjectOn the contrary, we continue our close watch over the growing and changing world of computation and to discuss with outside experts and amongourselves the various proposals for possible extension or supplementationf the formulas, methods and tables that make up the handbook.In keeping with previous policy, a number of errors discovered sincethe last printing have been corrected. Aside from this, the mathematicaltables and accompanying text are unaltered. However, some noteworthychanges have been made in Chapter 2: Physical Constants and conversionFactors, pp. 6-8. The table on page 7 has been revised to give the valuesof physical constants obtained in a recent reevaluation; and pages 6 and 8have been modified to refect changes in definition and nomenclature ofphysical units and in the values adopted for the acceleration due to gravityin the revised potsdam systemThe record of continuing acceptance of the handbook, the praise thathas come from all quarters and the fact that it is one of the most-quotedscientific publications in recent years are evidence that the hope expressedby Dr. Astin in his Preface is being amply fulfilledLEWIS M. BRANSCOMB, DirectorNational bureau of standardsNovember 1970ForewordThis volume is the result of the cooperative effort of many persons and a numberof organizations. The National Bureau of Standards has long been turning outmathematical tables and has had under consideration, for at least 10 years, theproduction 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, butindicated the need for technical advice and financial supportThe mathematics Division of the national research Council has also had anactive interest in tables; since 1943 it has published the quarterly journal, Mathe-matical Tables and Aids to Computation"(MTAC), editorial supervision beingexercised by a Committee of the DivisionSubsequent to the NBs Conference on Tables in 1952 the attention of theNational Science Foundation was drawn to the desirability of financing activity intable production. With its support a 2-day Conference on Tables was called at theMassachusetts Institute of Technology on September 15-16, 1954, to discuss theneeds for tables of various kinds. Twenty-eight persons attended, representingscientists and engineers using tables as well as table producers. This conferencereached consensus on several conclusiong and recommendations which were setforth in the published Report of the Conference. There was general agreementfor example, that the advent of high-speed computing equipment changed thetask of ta ble making but definitely did not remove the need for tables. It wasalso agreed that "an outstanding need is for a handbook of Tables for the OccasionalComputer, with tables of usually encountered functions and a set of formulas andtables for interpolation and other techniques useful to the occasional computerThe Report suggested that the NBS undertake the production of such a Handbookand that the nsf contribute financial assistance. The Conference elected from itsparticipants, the following Committee: P. M. Morse(Chairman), M. abramowitzJ. H. Curtiss, R. W. Hamming, D. H. Lehmer, C. B. Tompkins, J. W. Tukey, tohelp implement these and other recommendationsThe Bureau of Standards undertook to produce the recommended tables and theNational Science Foundation made funds available. To provide technical guidanceto the Mathematics Division of the Bureau, which carried out the work, and to provide the Nsf with independent judgments on grants for the work, the ConferenceCommittee was reconstituted as the Committee on revision of mathematicalTables of the mathematics Division of the national Research Council. This, afterome changes of membership became the Committee which is signing this ForewordThe present volume is evidence that Conferences can sometimes reach conclusionsand that their recommendations sometimes get acted onFOREWORDActive work was started at the Bureau in 1956. The overall plan, the selectionof authors for the various chapters, and the enthusiasm required to begin the taskwere contributions of Dr. Abramowitz. Since his untimely death, the effort hascontinued under the general direction of Irene A. Stegun. The workers at theBureau and the members of the Committee have had many discussions aboutcontent, style and layout. Though many details have had to be argued out as theycame up, the basic specifica tions of the volume have remained the same as wereoutlined by the massachusetts Institute of Technology Conference of 1954The committee wishes here to register its commendation of the magnitude andquality of the task carried out by the staff of the NBS Computing Section and theirexpert collaborators in planning, collecting and editing these Tables, and its appre-ciation of the willingness with which its various suggestions were incorporated intothe plans. We hope this resulting volume will be judged by its users to be a worthymemorial to the vision and industry of its chief architect, Milton Abramowitz.We regret he did not live to see its publicationP. M. Morse, ChairmanA. ErdELyiM.C. GRaYN. C. METROPOLISJ. B, rosseRH. C. THACHER, JrOHNODDC. B. TOMPKINSJ. W. TUKEYHandbook of mathematical FunctionswithFormulas Graphs, and Mathematical TablesEdited by Milton Abramowitz and Irene A Stegunl。 IntroductionThe present Handbook has been designed to I tional importance. Many numerical examplesprovide scientific investigators with a compreare given to illustrate the use of the tables andhensive and self-contained summary of the mathe- also the computation of function values which liematical functions that arise in physical and engi- outside their range. At the end of the text inneering problems. The well-known Tables ofeach chapter there is a short bibliography givingFunctions 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 manymatical properties stated in the chapter may beeditions! during the past half-century. The found. Also listed in the bibliographies are thepresent volume extends the work of these authors more important numerical tables. Comprehenby giving more extensive and more accurate sive lists of tables are given in the Index mennumerical tables, and by giving larger collections tioned above, and current information on newof mathematical properties of the tabulated tables is to be found in the National Researchfunctions. The number of functions covered hasCouncil quarterly Mathematics of Computationalso been increased(formerly Mathematical Tables and Other AidsThe classification of functions and organizationto Computationf the chapters in this handbook is similar toThe mathematical notations used in this handthat of An Index of Mathematical Tables bybook are those commonly adopted in standardA. Fletcher.. C. P. Miller, and L. Rosenhead. 2texts, particularly Higher Transcendental FuncIn general, the chapters contain numerical tablestions, Volumes 1-3, by A. Erdelyi, W. MagnusF. Oberhettinger and F.G. Tricomi (McGraw-graphs, polynomial or rational approximationsHill, 1953-55). Some alternative notations havefor automatic computers, and statements of the also been listed. The introduction of new symbolsprincipal mathematical properties of the tabuhas been kept to a minimum, and an effort haslated functions, particularly those of computa- been made to avoid the use of conflicting notation2. Accuracy of the TablesThe number of significant figures given in each precision in their interpolates may obtain themtable has depended to some extent on the numberby use of higher-order interpolation procedures.available in existing tabulations. There has been described belowno attempt to make it uniform throughout theIn certain tables many-figured function valuesHandbook, which would have been a costly and are given at irregular intervals in the argumentlaborious undertaking. In most tables at leastAn example is provided by Table 9.4. The purfive significant figures have been provided, and pose of these tables is to furnish"key values""forthe tabular intervals have generally been chosen the checking of programs for automatic computers;to ensure that linear interpolation will yield fourno question of interpolation arisesor five-figure accuracy, which suffices in mostThe maximum end-figure error,or“ tolerance”physical applications. Users requiring higherin the tables in this Handbook is %o of 1 uniteverywhere in the case of the elementary funcpublished in 1960 by McGraw-Hill, U.S.A and Teubner, GermaP.was1 The most recent the sixth, with F. Loesch added as co-authortions, and 1 unit in the case of the higher functionsa The second edition, with L.J. Comrie added as co- author was publishedexcept in a few cases where it has been permittedtwo volumes in 1962 by addison-Wesley, U.S.A and Scientific Computing Service Ltd, Great Britainto rise to 2 unitsINTRODUCTION3. Auxiliary Functions and ArgumentsOne of the objects of this Handbook is to pro-I The logarithmic singularity precludes direct intervide tables or computing methods which enable polation near x=0. The functions Ei(=)In xhe user to evaluate the tabulated functions over and -iEi(r)hr], however, are wellcomplele ranges of real values of their parameters.behaved and readily interpolable in this regionIn order to achieve this object, frequent use hasEither will do as an auxiliary function; the latterbeen made of auxiliary functions to remove thewas in fact selected as it yields slightly higherinfinite part of the original functions at theiraccuracy when Ei(a)is recovered. The functionsingularities,and auxiliary arguments to cope with x"[Ei(a)-In r-v] has been tabulated to nineinfinite ranges. An example will make the procedure cleardecimals for the range0≤x≤.Forb≤x≤2,Ei(e)is sufficiently well-behaved to admit directThe exponential integral of positive argumenttabulation, but for larger values of t, its expo18 given bynential character predominates. A smoother andBi(r)=dumore readily interpolable function for large x ise-2Ei(x); this has been tabulated for2≤x≤10y+In t+l1!28.3i+Finl, the range10≤x≤ co is covered by use ofthe inverse argument r-. Twenty-one entries of1!,2!,3!re-Ei(a), corresponding to x-=1(.005)0, suf-fice to produce an interpolable table4. InterpolationThe tables in this Handbook are not providedLet us suppose that we wish to compute thewith differences or other aids to interpolation, be-i value of xe E,(a)for x=7.9527 from this table.cause it was felt that the space they require could Wee describe in turn the application of the methodsbe better employed by the tabulation of additionalof linear interpolation, lagrange and Aitken, andfunctions. Admittedly aids could have been given of alternative methods based on differences andwithout consuming extra space by increasing the Taylors seriesintervals of tabulation, but this would have con-(1Linear interpolation. The formula for thisficted with the requirement that linear interpola-rocessis given bytion is accurate to four or five figuresFor applications in which linear interpolationfp=(1-)+fis insufficiently accurate it is intended thatLagrange's formula or Aitken,s method of itera-i where fo, fi are consecutive tabular values of thetive linear interpolation be used. To help the function, corresponding to arguments To,31,re-user, there is a statement at the foot of most tables spectively; p is the given fraction of the argumentof the maximum error in a linear interpolate,i intervaland the number of function values needed in=(x-x0)/(x1-31o)Lagrange's formula or Aitken's method to inter-olate to full tabular accuracyand f, the required interpolate. In the presentan example, consider the follo wing extract instance, we havefrom Table 5.1f6=.8971743021=898237113p=527ce"Ei(a)xe=Elc7.5.8926878548.08982371137,6The most convenient way to evaluate the formula8938463128992778887.8949796668.2.900297306on a desk calculating machine is, to set fo and fi7.88960887378.3.9012960°3in turn on the keybeard, and carry outthe multi-7.9.8971743028.4、902274695plications by 1-p and p cumulatively; a partial6)3check is then provided by the multiplier dialreading unity. We obtThe numbers in the square brackets mean that I 5s27 =(1-.527)(89717 4302)+.527(.89823 7113)the maximum error in a linear interpolate,is=897734403.3X10- and that to interpolate to the full tabularaccuracy five points must be used in Lagrange'sSince it is known that there is a possible errorand Aitken's methodsof 3 x10-in the linear formula, we round off this3A. C. Aitken, On interpolation by iteration of proportional parts, with-result to. 89773. The maximum possible error inout the use of differences, Proe. Edinburgh Math. Soc. 3, 66-76(1932)this answer is composed of the error committedINTRODUCTIONby the last rounding, that is, .4403X10-, plusThe numbers in the third and fourth columns are3x10- and so certainly cannot exceed 8x10-. the first and second differences of the values of(2) Lagrange's formula. In this example, thece E1(a)(see below); the smallness of the secondrelevant formula is the 5-point one, given bydifference provides a check on the three interpolaf=A-2(?)f-2+A-1(P)f-1+A(p)f+A1(?tions. The required value is now obtained by+A2(y)f2linear interpolationTables of the coefficients A(p) are given in chapter25 for the range p=0(.01)1. We evaluate thef=3(897729757)+7(897740379)eformula for p=52,. 53 and 54 in turn. Again=,897737192.in each evaluation we accumulate the Ak(p) in themultiplier register since their sum is unity. Wenow have the following subtableIn cases where the correct order of the lagrangeezer(a)polynomial is not known, one of the preliminaryinterpolations may have to be performed with7.952897729757polynomials of two or more different orders as a106227.9538977403792check on their adequacy10620(3)Aitken's method of iterative linear interpola7.954897750999tion. The scheme for carrying out this processin the present example is as follows:B1(z)08.0.898237113047317.9.897174302.8977344034052728.1.899277888.8977448264.8977371499147337.8.8960887372902202394.8977371938152748.2.9002973064987731216168977371930247357.7.894979666235221270643302527Heere8f1/22f1n-Toly nyo. 11bfy0,In--ET1lyo, n6fn-1,mm30,1·,-1,nHereIf the quantities Ex- and m-a are used asbf1/=f1-f,6f32=f2-f1,multipliers when forming the cross-product on a61=8f3/2-81=2-2f1+f0desk machine their accumulation(En-r-amr83/=82-82f1=3-82+3f1-fin the multiplier register is the divisor to be used6y2=852-853=-4+6-41+J0at that stage. An extra decimal place is usually and so oncarried in the intermediate interpolates to safIn the present example the relevant part of theguard against accumulation of rounding errorsdifference table is as follows, the differences beingThe order in which the tabular values are used written in units of the last decimal place of theis immaterial to some extent, but to achieve the I function, as is customary. The smallness of themaximum rate of convergence and at the same high differences provides a check on the functiontime minimize accumulation of rounding errors,valueswe begin, as in this example, with the tabularargument nearest to the given argument, thereel(r)en7.9.897174302-2275434take the nearest of the remaining tabular argu-8.0.8982371132203639ments, and so onThe number of tabular values required toapplying, for example, Everett,s interpolationachieve a given precision emerges naturally in formulathe course of the iterations. Thus in the presentexample six values were used, even though it was fp=(1-p)+E(p)83f0+E4(p)8fo+known in advance that five would suffice. The?f1+F2()8f1+F4(m)61+extra row confirms the convergence and providesa valuable checkand taking the numerical values of the interpola(4)Difference formulas. We use the centrall tion coeffcients E(p), E4(p), Fa(p)and F(p)difference notation (chapter 25)from Table 25.1. we find thatXIIINTRODUCTION10.=473(897174302)+,061196(22754)-012(34)can be used. We rst compute as many of the+.527(898237113)+.063439(22036),012(39)derivatives f(R(oo)as are significant, and then=897737193evaluate the series for the given value of xAn advisable check on the computed values of theWe may notice in passing that Everett's derivatives is to reprformula shows that the error in a linear interpolateoduce the adjacent tabularvalues by evaluating the series for r=a-i and wnIs approximatelyE2(P)83f0+F2(p)8≈E2(p)+F2()][62f0+821]In the present example, we haveSince the maximum value of E2(p)+F2(p)l in the∫(x)=ce-E1(x)range Orknown for two consecutive values of n, or ena)Fn(n, p)(Coulomb wave function)is known for one value of T, then the function maybe computed for other values of n by successiveIllustrations of the generation of functions fromapplications of the relation. Since generation is their recurrence relations are given in the pertinentcarried out perforce with rounded values, it is chapters. It is also shown that even in casesvital to know how errors may be propagated inwhere the recurrence process is unsta ble, it maythe recurrence process. If the errors do not grow still be used when the starting values are knownrelative to the size of the wanted function theto sufficient accuracy.process is said to be stabie. 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 recurrencewhelm the wanted function, the process is unstable. process which is stable for decreasing n to beIt is important to realize that stability may applied without any knowledge of starting valuesdepend on (i) the particular solution of the differ- for large n. Miller's algorithm, which is wellence equation being computed;(i)the values of suited to automatic work, is described in 19.28r or other parameters in the difference equation; I Example 1