HandbookofMathematicalFunctions with Formulas,Graphs,andMathematicalTables EditedbyMiltonAbramowitzandIreneA.Stegun 1.Introduction ThepresentHandbookhasbeendesignedto providescientificinvestigatorswithacompre- hensiveandself-containedsummaryofthemathe- maticalfunctionsthatariseinphysicalandengi- neeringproblems.Thewell-knownTablesof Funct.ionsbyE.JahnkeandF.Emdehasbeen invaluabletoworkersinthesefieldsinitsmany editionsduringthepasthalf-century.The presentvolumeext,endstheworkoftheseauthors bygivingmoreextensiveandmoreaccurate numericaltables,andbygivinglargercollections ofmathematicalpropertiesofthetabulated functions.Thenumberoffunctionscoveredhas alsobeenincreased. Theclassificationoffunctionsandorganization ofthechaptersinthisHandbookissimilarto thatofAnIndexofMathematicalTablesby A.Fletcher,J.C.P.Miller,andL.Rosenhead. Ingeneral,thechapterscontainnumericaltables, graphs,polynomialorrationalapproximations forautomaticcomputers,andstatementsofthe principalmathematicalpropertiesofthetabu- latedfunctions,particularlythoseofcomputa- tionalimportance.Manynumericalexamples aregiventoillustratetheuseofthetablesand alsothecomputationoffunctionvalueswhichlie outsidetheirrange.Attheendofthetextin eachchapterthereisashortbibliographygiving booksandpapersinwhichproofsofthemathe- maticalpropertiesstatedinthechaptermaybe found.Alsolistedinthebibliographiesarethe moreimportantnumericaltables.Comprehen- sivelistsoftablesaregivenintheIndexmen- tionedabove,andcurrentinformationonnew tablesistobefoundintheNationalResearch CouncilquarterlyMathematicsofComputation (formerlyMathematicalTablesandOtherAids toComputation). Thema.thematicalnotationsusedinthisHand- bookarethosecommonlyadoptedinstandard texts,particularlyHigherTranscendentalFunc- tions,Volumes1-3,byA.ErdBlyi,W.Magnus, F.OberhettingerandF.G.Tricomi(McGraw- Hill,1953-55).Somealternativenotationshave also been listed.Theintroductionofnew symbols hasbeenkepttoaminimum,andanefforthas been madetoavoidtheuse ofconflictingnotation. 2.AccuracyoftheTables Thenumberofsignificantfiguresgivenineach tablehas dependedtosome extentonthenumber availableinexistingtabulations.Therehas been noattempttomakeituniformthroughoutthe Handbook,whichwouldhavebeenacostlyand laboriousundertaking.Inmosttablesatleast fivesignificantfigureshavebeenprovided,and thetabularintervalshavegenerallybeenchosen toensure thatlinearinterpolationwillyield.four- orfive-figureaccuracy,whichsufficesinmost physicalapplications.Usersrequiringhigher 1 Themostrecent,thesixth,withF.Loeschaddedascc-author,was publishedin1960byMcGraw-Hill,U.S.A.,andTeubner,Germany. 2 Thesecondedition,withL.J.Comrieaddedas co-author,waspublished intwovolumesin1962byAddison-Wesley,U.S.A.,andScientificCom- putingServiceLtd.,GreatBritain. precisionintheirinterpolatesmayobtainthem byuseofhigher-orderinterpolationprocedures, describedbelow. Incertaintablesmany-figuredfunctionvalues aregivenatirregularintervalsintheargument. AnexampleisprovidedbyTable9.4.Thepur- pose ofthesetablesistofurnishkeyvaluesfor thecheckingofprogramsforautomaticcomputers; noquestionofinterpolationarises. Themaximumend-figureerror,ortolerance inthetablesinthisHandbookis6/& of1unit everywhereinthecase oftheelementaryfunc- tions,and1 unitinthecase ofthehigherfunctions exceptina fewcases whereithas beenpermitted toriseto2units. IX/- . X INTRODUCTION 3.AuxiliaryFunctionsandArguments OneoftheobjectsofthisHandbookistopro- videtablesorcomputingmethodswhichenable theusertoevaluatethetabulatedfunctionsover completerangesofrealvaluesoftheirparameters. Inordertoachievethisobject,frequentusehas beenmadeofauxiliaryfunctionstoremovethe infinitepartoftheoriginalfunctionsattheir singularities,andauxiliaryargumentstocoe with infiniteranges.Anexamplewillmaket fi epro- cedureclear. Theexponentialintegralofpositiveargument isgivenby Thelogarithmicsingularityrecludesdirectinter- polationnearx=0.TheunctionsEi(x)-InxP andx-liEi(lnx-r],however,arewell- behavedandreadilyinterpolableinthisregion. Eitherwilldoasanauxiliaryfunction;thelatter wasinfactselectedasityieldsslightlyhigher accuracywhenEi(x)isrecovered.Thefunction x-[Ei(x)-lnx-r]hasbeentabulatedtonine decimalsfortherange05x (n#-1) 3.3.15 S $&In)ax+b) 3.3.26 3.3.27 Thefollowingformulasareusefulforevaluating S P(x)dx (ux+fJx+cy whereP(x)isapolynomialand n>lisaninteger. 3.3.16 S dx2 (ax2+br+c)=(4c&c-bz)~ (b2--4ucO) 3.3.18 -2 =2az+b (P-4ac=O) 3.3.19 SS 3.3.20 S c+dx (a+bx$c+dxj=kbc In- II a+bx (ad#bc) 3.3.21___ S dx1 =-arctanE! u2+b2;C2ubU 3.3.22 S lnb2+b2x21 3.3.23 S 3.3.24 S (x2;1-a2)2=&arctan~+20~(xf+U2) 3.3.25 S IntegralsofIrrationalAlgebraicFunctions S dx-d(a+bx)12 t(u+bx)(c+dx)112 =h2arctan Cb(c+dx)1 W (b>O,dO) 3.3.29 S dx (a+bx)P(c+dx)=[d(bc~ud)]1~2 arctan~~~~-J(d(u&-bc)O) 3.3.30 ELEMENTARYANALYTICALMETHODS 17 Ifzn=un+ivn,then~~+l:=u,+,+iv~+~where 3.7.23u,+~=xu~-~v,;v,+,=xv,+yu, 9?zand92arecalledharmonicpolynomials. 3.7.24 3.7.25 Roots 3.7.26z*=&=rte+rs=r+cos@+irisin$0 If--?r