Mathematical Formula Handbook

1 ContentsIntroduction..1 Bibliography; ..2 ArithmeticandGeometricprogressions;Conve rgenceof series :theratiotest;Convergenceof series :thecomparisontest;Binomialexpansi on;TaylorandMaclaurinSeries;Powerseriesw ithrealvariables;Integerseries; ..3 Scalarproduct; Equationof a line; Equationof a plane; Vectorproduct; Scalartripleproduct;Vectortripleproduct; Non-orthogonalbasis; ..5 Unitmatrices;Products; Transposematrices; Inversematrices;Determinants; 2 2 matrices;Productrules; Orthogonalmatrices;Solvingsetsof linearsimultaneousequations; Hermitianmatrices;Eigenvaluesandeigenvec tors;Commutators;Hermitianalgebra; ..7 Notation; Identities;Grad,Div, CurlandtheLaplacian; Transformationof ..9 Complexnumbers; DeMoivre's theorem; ..10 Relationsbetweensidesandanglesof anyplanetriangle;Relationsbetweensidesan danglesof ..11 Relationsof thefunctions; ..13 Standardforms;Standardsubstitutions; Integrationbyparts;Differentiationof anintegral;Dirac -`function'.

2 16 Diffusion(conduction)equation; Waveequation; Legendre's equation;Bessel's equation;Laplace's equation; ..18 Taylorseriesfortwovariables; Stationarypoints;Changingvariables:thech ainrule;Changingvariablesinsurfaceandvol umeintegrals ..19 Fourierseries; Fourierseriesforotherranges; Fourierseriesforoddandevenfunctions;Comp lexformof Fourierseries; DiscreteFourierseries; Fouriertransforms; Convolutiontheorem;Parseval's theorem; Fouriertransformsintwodimensions; ..24 Findingthezerosof equations; Numericalintegrationof differentialequations;Centraldifferencen otation;Approximatingtoderivatives;Inter polation:Everett's Formula ;Numericalevaluationof de ..25 Rangemethod; Combinationof ..26 MeanandVariance;Probabilitydistributions ;Weightedsumsof randomvariables;Statisticsof a datasamplex1, .. ,xn; Regression(leastsquares tting)IntroductionThisMathematicalFormau laehandbookhasbeenpreparedinresponsetoa requestfromthePhysicsConsultativeCommitt ee,withthehopethatit is tosomeextentmodelledona similardocumentissuedbytheDepartmentofEn gineering,butobviouslyre wasdiscussionastowhetherit shouldalsoincludephysicalformulaesuchasM axwell'sequations,etc.

3 ,buta decisionwastakenagainstthis,partlyontheg roundsthatthebookwouldbecomeundulybulky, butmainlybecause,initspresentform, hasbeenwideconsultationamongthestaff aboutthecontentsofthisdocument,butinevit ablysomeuserswillseekinvainfora , (equallyinevitable)errorswhichare , andcurrentlybyDrDaveGreen,usingtheTEX , , ,HandbookofMathematicalFunctions,Dover, , , ,TableofIntegrals,SeriesandProducts,Acad emicPress, , ,F., TablesofFunctions,Dover, , Osterman,J.,PhysicsHandbook,Chartwell-Br att,Bromley, , ,MathematicalHandbookofFormulasandTables .(Schaum'sOutlineSeries,McGraw-Hill,1968 ).PhysicalConstantsBasedonthe ReviewofParticleProperties , Barnettetal.,1996,PhysicsReviewD,54, p1,and TheFundamentalPhysicalConstants , Cohen&Taylor, 1997,PhysicsToday, BG7.(The guresinparenthesesgivethe1-standard-devi ationuncertaintiesinthelastdigits.)speed oflightina vacuumc2 997 924 58 108ms 1(byde nition)permeabilityofa vacuum 04 10 7Hm 1(byde nition)permittivityofa vacuum 01= 0c2=8 854 187 817.

4 10 12F m 1elementarychargee1 602 177 33(49) 10 19 CPlanckconstanth6 626 075 5(40) 10 34J sh=2 h1 054 572 66(63) 10 34J sAvogadro constantNA6 022 136 7(36) 1023mol 1uni edatomicmassconstantmu1 660 540 2(10) 10 27kgmassofelectronme9 109 389 7(54) 10 31kgmassofprotonmp1 672 623 1(10) 10 27kgBohrmagnetoneh=4 me B9 274 015 4(31) 10 24J T 1molargasconstantR8 314 510(70)J K 1mol 1 BoltzmannconstantkB1 380 658(12) 10 23J K 1 Stefan Boltzmannconstant 5 670 51(19) 10 8Wm 2K 4gravitationalconstantG6 672 59(85) 10 11Nm2kg 2 Otherdataaccelerationoffreefallg9 806 65 ms 2(standard valueatsealevel) + (a+d) + (a+2d) + + [a+ (n 1)d] =n2[2a+ (n 1)d] +ar+ar2+ +arn 1=a1 rn1 r, S1=a1 rforjrj<1 (Theseresultsalsoholdforcomplexseries.)C onvergenceofseries:theratiotestSn=u1+u2+ u3+ +unconvergesasn!1iflimn!1 un+1un <1 Convergenceofseries:thecomparisontestIf eachtermina seriesofpositivetermsislessthanthecorres pondingtermina seriesknowntobeconvergent,thenthegivense riesis (1+x)n=1+nx+n(n 1)2!

5 X2+n(n 1)(n 2)3!x3+ Ifnisa positiveintegertheseriesterminatesandisv alidforallx: theterminxrisnCrxror nr wherenCr n!r!(n r)!is thenumberofdifferentwaysinwhichanunorder edsampleofrobjectscanbeselectedfroma nota positiveinteger, theseriesdoesnotterminate:thein niteseriesisconvergentforjxj< (x)is well-behavedinthevicinityofx=athenit hasa Taylorseries,y(x) =y(a+u) =y(a) +udydx+u22!d2ydx2+u33!d3ydx3+ whereu=x aandthedifferentialcoef cientsare evaluatedatx=a. AMaclaurinseriesis a Taylorserieswitha=0,y(x) =y(0) +xdydx+x22!d2ydx2+x33!d3ydx3+ Powerserieswithrealvariablesex=1+x+x22!+ +xnn!+ validforallxln(1+x) =x x22+x33+ + ( 1)n+1xnn+ validfor 1<x 1cosx=eix+e ix2=1 x22!+x44! x66!+ validforallvaluesofxsinx=eix e ix2i=x x33!+x55!+ validforallvaluesofxtanx=x+13x3+215x5+ validfor 2<x< 2tan 1x=x x33+x55 validfor 1 x 1sin 1x=x+12x33+ + validfor 1<x<12 IntegerseriesN 1n=1+2+3+ +N=N(N+1)2N 1n2=12+22+32+ +N2=N(N+1)(2N+1)6N 1n3=13+23+33+ +N3= [1+2+3+ N]2=N2(N+1)241 1( 1)n+1n=1 12+13 14+ =ln 2[seeexpansionofln(1+x)]1 1( 1)n+12n 1=1 13+15 17+ = 4[seeexpansionoftan 1x]1 11n2=1+14+19+116+ = 26N 1n(n+1)(n+2) = + + +N(N+1)(N+2) =N(N+1)(N+2)(N+3)4 Thislastresultis a specialcaseofthemore generalformula,N 1n(n+1)(n+2).

6 (n+r) =N(N+1)(N+2)..(N+r)(N+r+1)r+ (ikz) =exp(ikrcos ) =1 l=0(2l+1)iljl(kr)Pl(cos ),wherePl(cos )are Legendre polynomials(seesection11)andjl(kr)are sphericalBesselfunctions,de nedbyjl( ) =r 2 Jl+1=2( ),withJl(x)theBesselfunctionoforderl(see section11). ,j,kare orthonormalvectorsandA=Axi+Ayj+AzkthenjA j2=A2x+A2y+A2z. [Orthonormalvectors orthogonalunitvectors.]ScalarproductA B=jAj jBjcos where is theanglebetweenthevectors=AxBx+AyBy+AzBz = [AxAyAz]24 BxByBz35 Scalarmultiplicationis commutative:A B=B lineApointr (x,y,z)liesona linepassingthrougha pointaandparalleltovectorbifr=a+ bwith a planeApointr (x,y,z)is ona planeif either(a)r bd=jdj, wheredis thenormalfromtheorigintotheplane,or(b)xX +yY+zZ=1 whereX,Y,Zare B=njAj jBjsin , where is theanglebetweenthevectorsandnis a unitvectornormaltotheplanecontainingAand BinthedirectionforwhichA,B,nforma Bindeterminantform ijkAxAyAzBxByBz A Binmatrixform240 AzAyAz0 Ax AyAx03524 BxByBz35 Vectormultiplicationis notcommutative:A B= B B C=A B C= AxAyAzBxByBzCxCyCz = A C B, (B C) = (A C)B (A B)C,(A B) C= (A C)B (B C)ANon-orthogonalbasisA=A1e1+A2e2+A3e3A1 = 0 Awhere 0=e2 e3e1 (e2 e3).

7 3a b=aibi(a b)i="i jkajbkwhere"123=1;"i jk= "ik j"i jk"klm= il jm im a square matrixwithalldiagonalelementsequaltoonea ndalloff-diagonalelementszero, ,(I)i j= i j. IfAis a square matrixofordern, thenAI=I A=A. AlsoI=I sometimeswrittenasInif a(n l)matrixandBis a(l m)thentheproductABis de nedby(AB)i j=l k=1 AikBk jIngeneralAB6= a matrix,thentransposematrixATis suchthat(AT)i j= (A) a square matrixwithnon-zero determinant,thenitsinverseA 1is suchthatAA 1=A 1A=I.(A 1)i j=transposeofcofactorofAi jjAjwhere thecofactorofAi jis( 1)i+ a square matrixthenthedeterminantofA,jAj( detA)is de nedbyjAj= i,j,k,.. i ..where thenumberofthesuf xesis 2 matricesIfA= abcd then,jAj=ad bcAT= acbd A 1=1jAj d b ca Productrules(AB..N)T=NT..BTAT(AB..N) 1=N 1..B 1A 1(ifindividualinversesexist)jAB..Nj=jAj jBj..jNj(ifindividualmatricesare square)OrthogonalmatricesAnorthogonalmat rixQis a square matrixwhosecolumnsqiforma ,Q 1=QT,jQj= 1,QTis square thenAx=bhasa uniquesolutionx=A 1bifA 1exists, ,ifjAj6= square thenAx=0 hasa non-trivialsolutionif andonlyifjAj= oneinwhichAhasmrowsandncolumns,wherem(th enumberofequations)isgreaterthann(thenum berofvariables).

8 Thebestsolutionx(inthesensethatit minimizestheerrorjAx bj) isthesolutionofthenequationsATAx=ATb. If thecolumnsofAare orthonormalvectorsthenx= (A )T, whereA isa matrixeachofwhosecomponentsisthecomplexc onjugateofthecorrespondingcomponentsofA. IfA=AythenAis calleda iandeigenvectorsuiofann nmatrixAare thesolutionsoftheequationAu= u. Theeigenvaluesare thezerosofthepolynomialofdegreen,Pn( ) =jA Ij. IfAis Hermitianthentheeigenvalues iare realandtheeigenvectorsuiare Ij=0 is i i,alsojAj= i a symmetricmatrix, is thediagonalmatrixwhosediagonalelementsar e theeigenvaluesofS, andUis thematrixwhosecolumnsare thenormalizedeigenvectorsofA, thenUTSU= andS=U anapproximationtoaneigenvectorofAthenxTA x=(xTx)(Rayleigh'squotient) [A,B] AB BA[A,B]= [B,A][A,B]y= [By,Ay][A+B,C] = [A,C] + [B,C][AB,C]=A[B,C] + [A,C]B[A,[B,C]] + [B,[C,A]] + [C,[A,B]] =0 Hermitianalgebraby= (b 1,b 2, ..)MatrixformOperatorformBra-ketformHerm iticityb A c= (A b) cZ O =Z(O ) h jOj iEigenvalues, realAui= (i)uiO i= (i) iOjii= ijiiOrthogonalityui uj=0Z i j=0hijji=0(i6=j)Completenessb= iui(ui b) = i i Z i = ijii hij iRayleigh RitzLowesteigenvalue 0 b A bb b 0 Z O Z h jOj ih j i6 Paulispinmatrices x= 0110 , y= 0 ii0 , z= 100 1 x y=i z, y z=i x, z x=i y, x x= y y= z z= is a scalarfunctionofa = (x,y,z); incylindricalpolarcoordinates = ( ,',z); insphericalpolarcoordinates = (r, ,'); incaseswithradialsymmetry = (r).

9 Ais a vectorfunctionwhosecomponentsare scalarfunctionsofthepositioncoordinates: inCartesiancoordinatesA=iAx+jAy+kAz, whereAx,Ay,Azare independentfunctionsofx,y, (`del') i x+j y+k z 266666664 x y z377777775grad =r ,divA=r A,curlA=r AIdentitiesgrad( 1+ 2) grad 1+grad 2div(A1+A2) divA1+divA2grad( 1 2) 1grad 2+ 2grad 1curl(A1+A2) curlA1+curlA2div( A) divA+ (grad ) A,curl( A) curlA+ (grad ) Adiv(A1 A2) A2 curlA1 A1 curlA2curl(A1 A2) A1divA2 A2divA1+ (A2 grad)A1 (A1 grad)A2div(curlA) 0,curl(grad ) 0curl(curlA) grad(divA) div(gradA) grad(divA) r2 Agrad(A1 A2) A1 (curlA2) + (A1 grad)A2+A2 (curlA1) + (A2 grad)A17 Grad,Div, CurlandtheLaplacianCartesianCoordinatesC ylindricalCoordinatesSphericalCoordinate sConversiontoCartesianCoordinatesx= cos'y= sin'z=zx=rcos'sin y=rsin'sin z=rcos VectorAAxi+Ayj+AzkA b +A'b'+AzbzArbr+A b +A'b'Gradientr xi+ yj+ zk b +1 'b'+ zbz rbr+1r b +1rsin 'b'Divergencer A Ax x+ Ay y+ Az z1 ( A ) +1 A' '+ Az z1r2 (r2Ar)

10 R+1rsin A sin +1rsin A' 'Curlr A ijk x y zAxAyAz 1 b b'1 bz ' zA A'Az 1r2sin br1rsin b 1rb' r 'ArrA rA'sin Laplacianr2 2 x2+ 2 y2+ 2 z21 +1 2 2 '2+ 2 z21r2 r r2 r +1r2sin sin +1r2sin2 2 '2 TransformationofintegralsL=thedistanceal ongsomecurve`C'inspaceandis measuredfromsome surfacearea =a volumecontainedbya speci edsurfacebt=theunittangenttoCatthepointP bn=theunitoutward pointingnormalA=somevectorfunctiondL=the vectorelementofcurve(=btdL)dS=thevectore lementofsurface(=bndS)ThenZCA btdL=ZCA dLandwhenA=r ZC(r ) dL=ZCd Gauss's Theorem(DivergenceTheorem)WhenSde nesa closedregionhavinga volume Z (r A)d =ZS(A bn)dS=ZSA dSalsoZ (r )d =ZS dSZ (r A)d =ZS(bn A)dS8 Stokes's TheoremWhenCis closedandboundstheopensurfaceS,ZS(r A) dS=ZCA dLalsoZS(bn r )dS=ZC dLGreen's TheoremZS r dS=Z r ( r )d =Z r2 + (r ) (r ) d Green's SecondTheoremZ ( r2 r2 )d =ZS[ (r ) (r )] +iy=r(cos +i sin ) =rei( +2n ), where i2= 1 andnis anarbitraryinteger.