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1 THE REQUIRED MATHEMATICS AND ITS APPLICATIONS F. Vn Cuwlt Edito: Mc Stt Fdtion of th Blgin Cmnt Industy B-7 Bussls, Ru Volt 9. i

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3 INTRODUCTION Pvmnt Dsign nd Evlution: Th Rquid Mthmtics nd Its Applictions Kywods: ttbook fo pvmnt ngins, high od mthmticl solutions fo, igid nd flibl pvmnt, mchnistic mthods, pcticl pplictions in th fild of pvmnt ngining Pfc This book is intndd fo Civil Engins nd mo spcific fo Pvmnt Engins, who intstd in th mo dvncd fild of pvmnt ngining md vilbl though th thoy of high mthmtics. In my tnsiv ci s Civil Engin by pofssion, I noticd tht som of my collgus fl unsy whn it coms to high-lvl thoticl nd computtionl wok. Phps this is bcus of th fct tht s soon s thy gdutd, thy confontd with mny nd impotnt pcticl poblms wh divtivs nd intgls do not pp vy usful t fist sight. Howv, t mny occsions, confncs, smins nd oth mtings, I lisd tht Civil Engins min citd bout th mthmticl fundmntls of thi t. Slow but su gw th id of witing ttbook on High Mthmtics concivbl fo pctising Pvmnt Engins. Th pimy objctiv of this book is to sv two puposs: fist, to intoduc th bsic pincipls which must b known by popl dling with pvmnts; nd scondly to psnt th thois nd mthods in pvmnt dsign nd vlution tht my b usd by studnts, dsigns ngining consultnts, highwy nd ipot gncis, nd schs t univsitis. In ddition, som of th nw concpts dvlopd in cnt ys to impov th mthods of pvmnt systms plind. This book is wittn in ltivly simpl wy so tht it my b followd by popl fmili with bsic ngining couss in mthmtics nd pvmnt dsign. This book consists of 6 chpts nd is dividd into two pts. Th fist pt, which includ chpts -9, covs th mthmtics quid by most of th poblms ltd to pvmnt ngining. Th ssumd mthmticl knowldg is tht of high school lvl plus som bsic lmnts of tigonomty nd nlysis. Th scond pt, which includs chpts -6, is concnd with pcticl solutions s fcd by Pvmnt Engins in th ssssmnt of igid nd flibl pvmnts. Th igoous mthmticl solutions psntd in th Appndi, plining on compl vibls. Gtful cknowldgmnt is offd to th Fédétion d l Industi Cimntiè, Blgium (th Fdtion of th Blgin Cmnt Industy fo thi intllctul nd finncil suppot in th pocss of th listion of this book. Thnks is du to my d find, Mc Stt, fo poof-ding nd diting th mnuscipt. His hlpful commnts to th mthmticl nd ditoil contnt highly ppcitd. Fns Vn Cuwlt Bussls, Dcmb 3. iii

4 Pt : Th Rquid Mthmtics Chpt covs th Lplc diffntil qution; th solution of fo gt sis of pplictions: it psnts th Bssl function of o od, solution of th Lplc qution in isymmtic co-odints tht s usd in gt pts of this book. Th Lplcin with cofficints diffnt fom will b pplid in poblms of nisotopic lsticity, th doubl Lplcin in most of th poblms of multi-ly thoy nd th tndd Lplcin in th poblms of igid pvmnt on Winkl o Pstnk foundtions. Chpt psnts th gmm function, th fctoil function fo non-intgs, quid fo th dfinition of Bssl functions of non-intg od. Chpt 3 givs th gnl solutions of th diffnt Bssl qutions, th B nd Bi functions, nd th modifid fom of th Bssl qution. Bssl functions solutions of th Lplcin in pol o cylind co-odints, usd fo pplictions with il symmty,.g. in multilyd stuctus. Tigonomtic functions solutions of th Lplcin in Ctsin co-odints, pplid in css of bms nd ctngul slbs. Chpt 4 dls with th most impotnt poptis of Bssl functions: - divtivs, functions of hlf od, - symptotic vlus quid to pss boundy conditions tht must min vlid t infinit distncs, - indfinit intgls, qutions btwn Bssl functions of diffnt kind (quid fo intgtions in th compl pln. Chpt 5 psnts th bt function quid fo th solution of dfinit intgls of Bssl functions. Chpt 6 givs th solutions fo sis of impotnt dfinit intgls of Bssl functions; mong oths th Poisson intgl giving n intgl psnttion of ny Bssl function of th fist kind. Chpt 7 psnts th hypgomtic function of Guss quid fo th solution of th infinit intgls of Bssl functions. Chpt 8 psnts th most impotnt infinit intgls of Bssl functions of dict ppliction in pvmnt nlysis, spcilly in multi-ly thoy. Chpt 9 psnts th most impotnt infinit intgls of Bssl functions solvd in th compl pln. Thy ssntilly of ppliction in slb thoy. Pt : Th Applictions Th scond pt focuss on th pplictions in th fild of pvmnt ngining: igid n flibl pvmnts. Chpt givs th bsic solutions (quilibium qutions fo igid pvmnts (thoy of stngth of mtils nd flibl pvmnts (thoy of lsticity.this chpt givs th bsic qutions on continuum mchnics in diffnt co-odint systms. Chpt psnts th intgl tnsfoms, Foui s pnsion, Foui s intgl, Hnkl s intgl, quid fo th pssion of discontinuous functions: th lods in pvmnt pplictions. Chpts to 9 concn mostly igid pvmnts. Chpt givs 3 simpl pplictions on n lstic subgd: bm subjctd to singl lod, bm subjctd to distibutd lod, slb subjctd to singl lod. Chpt 3 givs th complt nlyticl solution fo bm on Pstnk foundtion (both of infinit nd finit tnt. Chpt 4 givs th nlyticl solution fo cicul slb on Pstnk foundtion subjctd to n i-symmtic lod (both of infinit nd finit tnt. Chpt 5 givs th complt nlyticl solution fo ctngul slb on Pstnk foundtion (both of infinit nd finit tnt. Chpt 6 givs th nlyticl solution fo supposition of svl slbs includd th nlysis of th dhsion btwn th slbs. Chpt 7 givs bck-clcultion mthod fo igid pvmnt bsd on Pstnk s thoy. Chpt 8 psnts solution fo th computtion of thml stsss in igid slbs on Pstnk foundtion. Chpt 9 psnts two pcticl tsts of intst with igid slbs: th dimtl tst nd tst fo th dtmintion of k nd G in situ. Chpts to 6 concn mostly flibl pvmnts. Chpts nd psnt th complt thoy fo smi-infinit bodis subjctd to ll sots of lods. Chpt psnts th Boussinsq poblm: stsss iv

5 INTRODUCTION nd displcmnts in smi-infinit body und cicul flibl plt unifomly lodd. It gnliss th solution to diffnt vticl lods (isoltd, igid, ctngul nd to othotopic bodis. Chpt psnts th solution smi-infinit body subjctd to sh stsss: dil nd on-dictionl. Chpt givs th nlyticl solution fo multi-lyd stuctu, includd th poblm of th dhsion btwn th lys, tht of n vntul fid bottom nd tht of n nisotopic subgd. Chpt 3 psnts th numicl pocdu quid fo th solution of multi-lyd stuctu. Chpt 4 psnts th thoy t th bs of th bck-clcultion mthods fo flibl pvmnts. Chpt 5 psnts th numicl pocdu quid fo th bck-clcultion mthod fo flibl pvmnts. Chpt 6 psnts pcticl tst of intst with both typs of od stuctus (igid o flibl multilyd stuctus: th ovlistion tst. Th Appndics givs th bsic thoy of compl numbs, spcilly th intgtion in th compl pln. v

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7 CONTENTS Tbl of Contnts Pg Pfc... iii PART : THE REQUIRED MATHEMATICS... Chpt Th Lplc Eqution.... Intoductoy not.... Divtion of th Lplc qution in pol co-odints fom th Lplc qution in Ctsin co-odints....3 Equtions ltd to th Lplc qution Th Lplcin with cofficints diffnt fom Th doubl Lplcin Th tndd Lplcin Rsolution of th Lplc qution Rsolution by sption of th vibls Rsolution by mns of th chctistic qution (Spigl, Rsolution by mns of indicil qutions... 6 Chpt Th Gmm Function Intoductoy not Hlpful ltions Dfinition of th Gmm Function....4 Vlus of Γ(/ nd Γ(-/... Chpt 3 Th Gnl Solution of th Bssl Eqution Intoductoy not Hlpful ltions Rsolution of th Bssl qution (Bssl functions of th fist kind Rsolution of th Bssl qution fo p n intg (Bssl functions of th scond nd thid kind Fo n intg, J n (- n J -n Bssl functions of th scond kind Bssl functions of th thid kind Th modifid Bssl qution Th b nd bi functions Th k nd ki functions Rsolution of th qution w w Th modifid fom of th Bssl qution...5 Chpt 4 Poptis of th Bssl Functions Intoductoy not Hlpful ltions Divtivs of Bssl functions Divtiv of (t p J p (t Divtiv of (t-pjp(t...9 vii

8 4.3.3 Divtiv of J p (t Bssl functions of hlf od Vlus of J / (t, J -/ (t, J 3/ (t, J -3/ (t Asymptotic vlus Asymptotic vlus fo Jp nd J-p Asymptotic vlus fo Y p nd Y -p Asymptotic vlus fo H ( p nd H ( p Asymptotic vlus fo I p nd I -p Asymptotic vlu fo K n Asymptotic vlus fo b nd bi Asymptotic vlus fo k nd ki Indfinit intgls of Bssl functions Fundmntl ltions Th intgl n J (td Rltions btwn Bssl functions of diffnt kind Bssl functions with gumnt t Rltions btwn th th kinds of Bssl functions Bssl functions of puly imginy gumnt...39 Chpt 5 Th Bt Function Intoductoy not Hlpful ltions Dfinition of th bt function Rltion btwn bt nd gmm functions Th dupliction fomul fo gmm functions...4 Chpt 6 Dfinit intgls of Bssl functions Hlpful ltions Ggnbu s intgl Sonin s fist finit intgl Sonin s scond finit intgl Poisson s intgl...48 Chpt 7 Th hypgomtic typ of sis Intoductoy not Hlpful ltions Dfinition Poptis of th multipl poduct (α m A ltion fo ( - b - m m-n A ltion fo (bm m / m A ltion fo Γ(-n Th thom of Vndmond Th poduct of two Bssl functions with th sm gumnt Th hypgomtic sis of Guss F [,b;c;] Elmnty poptis Intgl psnttion of th hypgomtic function Vlu of F[,b;c;] fo Convgnc of th sis F[,b;c;] Th poduct of two Bssl functions with diffnt gumnts...56 viii

9 CONTENTS Chpt 8 Infinit Intgls of Bssl Functions Intoductoy not Usful ltions Th intgl -t J (btt - dt Rsolution of th intgl Pticul vlu Th intgl -t cos(btdt Th intgl -t sin(btdt Th intgl -t cos(btsin(bt dt Th intgl -t sin(bt/tdt Th intgl -t sin(bttdt Th intgl -t J (btj (ctt - dt Tnsfomtion of th intgl Th intgl -t sin(btsin(ctt - dt Th intgl -t sin(btsin(ctdt Th intgl -m sin(btsin(cs/(tsdsdt Th intgl m -m sin(btsin(cs/(tsdsdt Th intgl -t J (btj (ctt λ- dt Th discontinuous intgl J (tj (btt -λ dt Rsolution of th intgl Th intgl J (tj (tt -λ dt Th intgl J (tj -k- (btdt Pticul solutions of th intgl J (tj (btt -λ dt Pticul solution of th intgl J (tj (bt/t λ dt...73 Chpt 9 Bssl functions in th compl pln Intoductoy not Hlpful ltions Poof of Γ(Γ(- /sin( Th Hnkl s contou intgl fo /Γ( Th intgl psnttion of J ( Th intgl psnttion of I ( Th intgl psnttion of K ( Th intgl psnttion of K v ( Rsolution of J (/( k d Rsolution of ρ- J (bj (/( k d Rsolution of sin(bcos(//( 4 k 4 d...9 PART : THE APPLICATIONS...93 Chpt Lplc Eqution in Pvmnt Engining Equilibium qution fo bms in pu bnding Sign convntions Assumptions Bnding momnt nd bnding stss Th dius of cuvtu Equilibium...96 i

10 . Equilibium qution fo bnt plts Bnding momnt nd sh focs Equilibium Comptibility qution fo homognous, lstic, isotopic body submittd to focs pplid on its sufc Pincipl of quilibium Th pincipl of continuity Th pincipl of lsticity Stss potntil....4 Comptibility qution fo homognous, lstic, nisotopic body submittd to focs pplid on its sufc Bsic qutions of continuum mchnics in diffnt co-odint systms Pln pol co-odints Ai-symmtic Cylindicl Co-odints Non symmtic cylindicl co-odints Ctsin volum co-odints Ai-symmtic Cylindicl Co-odints fo n othotopic body...9 Chpt Th Intgl Tnsfoms.... Intoductoy not.... Hlpful ltions....3 Th Foui pnsion (Spigl, Dfinitions Poof of th Foui pnsion Empl Th Foui intgl Dfinition Poof of Foui s intgl thom Empl Th Hnkl s tnsfom Dfinition Empl Appliction of th discontinuous intgl of Wb nd Schfhitlin...8 Chpt Simpl Applictions of Bms nd Slbs on n Elstic Subgd.... Th lstic subgd.... Th bm on n lstic subgd subjctd to n isoltd lod: 4 w/ 4 Cw Th bm on n lstic subgd subjctd to distibutd lod: 4 w/ 4 Cw Th infinit slb subjctd to n isoltd lod: w Cw...5 Chpt 3 Th Bm Subjctd to Distibutd Lod nd Rsting on Pstnk Foundtion Th bsic diffntil qutions Cs of bm of infinit lngth Solution of th diffntil qution Appliction Cs of bm of finit lngth with f dg Solution # Appliction Solution # Appliction...37

11 CONTENTS 3.4 Cs of finit bm with joint Solution # Appliction Solution# Appliction Poof tht, in d cs of Winkl foundtion, t joint Q γ T...43 Chpt 4 Th Cicul Slb Subjctd to Distibutd Lod nd Rsting on Pstnk Foundtion Th bsic diffntil qutions Cs of slb of infinit tnt Solution of th diffntil qution Appliction Appliction Cs of slb of finit tnt with f dg Solution # Solution # Appliction of solution # Cs of slb of finit tnt with joint Solution # Solution # Appliction of solution #...53 Chpt 5 Th Rctngul Slb Subjctd to Distibutd Lod nd Rsting on Pstnk Foundtion Th bsic diffntil qutions Rsolution of th dflction qution Boundy conditions Cs of slb of finit tnt with f dg Solution # Solution # Cs of slb of finit tnt with joint Solution # Solution # Appliction...6 Chpt 6 Th Multislb Thoticl justifiction Gnl modl Full slip t ch of th intfcs Full fiction t th fist intfc, full slip t th scond intfc Full fiction t both intfcs Ptil fiction Appliction...68 Chpt 7 Bck-clcultion of Conct Slbs Bck-clcultion of moduli Cs wh th lod cn b considd s point lod Computtions Cs whn th lod is considd s distibutd Compison of th two mthods...75 i

12 7.6 Influnc of th fnc dflction Anlysis of fild dt Empl of bck-clcultion...77 Chpt 8 Thml Stsss in Conct Slbs Thml stsss Slb of gt lngth Diffntil quilibium qution Solution of th quilibium qution fo g < Solution of th quilibium qution fo g Solution of th quilibium qution fo g > Boundy conditions Epssion of th momnt fo g < Epssion of th momnt fo g Epssion of th momnt fo g > Vifiction of th pssion of th mimum momnt fo g Eqution of th thml stss Empl Rctngul slb Diffntil qution of quilibium Boundy conditions Empls Cicul slb Equilibium qution Solution of th quilibium qution Boundy conditions Rsulting momnt Compison btwn th modls fo ctngul nd cicul slbs Etnsion to multi-slb systm...89 Chpt 9 Dtmintion of th Pmts of Rigid Stuctu Dtmintion of th Young s modulus of conct slb Rsolution of th comptibility qution Equtions fo th stsss Boundy conditions Stsss nd displcmnts Tngntil noml stss long th vticl dimt Dtmintion of th chctistics k nd G of th subgd Equilibium qution fo Pstnk subgd Eccnticlly lodd plt-bing tst Vticl lod Momnt Dtmintion of k nd G... Chpt Th Smi-Infinit Body Subjctd to Vticl Lod Intoductoy not...3. Th smi-infinit body subjctd to vticl unifom cicul pssu Th smi-infinit body subjctd to n isoltd vticl lod Th smi-infinit body subjctd to cicul vticl igid lod Th smi-infinit body subjctd to vticl unifom ctngul pssu Compison btwn th vticl stsss. Pincipl of d Sint-Vnnt... ii

13 CONTENTS.7 Th othotopic body subjctd to vticl unifom cicul pssu....8 Th othotopic body subjctd to vticl unifom ctngul pssu...5 Chpt Th Smi-Infinit Body Subjctd to Sh Lods Th smi-infinit body subjctd to dil sh stsss...7. Th smi-infinit body subjctd to on-dictionl symmtic sh lod Th smi-infinit body subjctd to sh lod symmtic to on of its is s... Chpt Th Multilyd Stuctu Th multilyd stuctu...5. Solutions of th continuity qutions Boundy conditions Dtmintion of th boundy constnts Th fid bottom condition Th othotopic subgd...3 Chpt 3 Th Rsolution of Multilyd Stuctu Choic of th intgtion fomul Vlus t th oigin Th gomticl scl of th stuctu Width of th intgtion stps Influnc of th moduli on th intgtion stp Influnc of th dii of th lods on th intgtion stp Influnc of th offst distnc on th intgtion stp Modifiction of th stp width Stsss nd displcmnts t th sufc Stsss nd displcmnts in th fist l y...37 Chpt 4 Th Thoy of th Bck-Clcultion of Multilyd Stuctu Th sufc modulus Equivlnt lys Equivlnt smi-infinit body Anlysis of dflction bsin Anlysis of th-ly on lin lstic subgd Anlysis of th-ly with vy stiff bs cous Anlysis of th ly with vy wk bs cous Anlysis of two-ly on subgd with incsing stiffnss with dpth Algoithm of Al Bush III ( Chpt 5 Th Numicl Pocdu of th Bck-clcultion of Multilyd Stuctu Th nlysis of bck-clcultion pogm fo th-lyd stuctu Th snsitivity of th bck-clcultion pocdu fo th ly stuctu Th snsitivity to ounding off th vlus of th msud dflctions Th snsitivity to th psnc of soft intmdit ly Th influnc of fiing bfohnd th vlu of on modulus Th snsitivity of th bck-clcultion pocdu fo fou ly stuctu Vlu of th infomtion givn by th sufc modulus Th influnc of fiing bfohnd th vlu of on modulus Th influnc of dg of nisotopy nd Poisson s tio on th sults of bck-clcultion pocdu in th cs of smi-infinit subgd Influnc of th dg of nisotopy on th bck-clcultd moduli...57 iii

14 5.4. Influnc of Poisson s tio of th subgd on th dflctions (n Th influnc of Poisson s tio nd dg of nisotopy on th sults of bck-clcultion pocdu in th cs of subgd of finit thicknss Influnc of th dg of nisotopy on th bck-clcultd moduli Influnc of Poisson s tio of th subgd on th dflctions (n...59 Chpt 6 Th Ovlistion Tst Dsciption of th ovlistion tst Intpttion of th sults of th ovlistion tst Slb with cvity on n lstic subgd subjctd to symmticl lod Rsolution in th cs of plin slb Rsolution in th cs of slb with cvity Stins in th cs of non-symmticl lod Stsss in hollow cylind subjctd to unifom tnl pssu Stsss in hollow plt Appliction of th ovlistion tst....7 Rfncs Appndi Compl Functions iv

15 THE LAPLACE EQUATION PART : THE REQUIRED MATHEMATICS Chpt Th Lplc Eqution. Intoductoy not. In mchnicl ngining nd thus lso in th mchnics of civil ngining, on lwys stts with th fundmntl quimnt of quilibium: quilibium of th noml focs nd quilibium of th momnts cting on th nlysd body. Focs sultnts of stsss. In od to loct ctly thos focs nd dtmin, t lst nlyticlly, thi mplitud, it is convnint to stt fom n infinitsiml sction of th body. On such n infinitsiml sction, th stsss ncssily infinitly smll nd, hnc, cn b considd s constnt nd unifomly distibutd ov th of th infinitly smll sction. Hnc th sulting foc is simply th poduct of th constnt stss by th of th sction nd its point of ppliction is th cnt of gvity of th sction, i.. th midpoint of th sction. All th sulting qutions will thn ncssily b diffntil qutions nd, nly in lmost ll pplictions, th diffntil qution pssing quilibium is Lplc o n ssimiltd qution. Th dvlopmnt of ths qutions psntd in chpt, th fist chpt of Pt Applictions. As in mny filds of ngining, lso in Pvmnt ngining th diffntil qution of Lplc is th solution of gt sis of pplictions. Symboliclly, Lplc qution is wittn s follows: Φ (. It is homognous diffntil qution with scond od ptil divtivs. Function of th co-odint systm, th qution is dvlopd: in pln Ctsin co-odints in volum Ctsin co-odints in pol co-odints in cylindicl co-odints Φ Φ y Φ Φ Φ y Φ Φ Φ θ Φ Φ Φ Φ θ In cs of il symmty, Φ/θ nd qutions (.4 nd (.5 simplify in: Φ Φ Φ Φ Φ Eqution (. is pplid in chpt.3.4. Eqution (.4 will b utilisd in chpt s.5. nd 9.. Eqution (.7 will b utilisd in chpt s.5.,.6,.,.,.3 nd.4. (. (.3 (.4 (.5 (.6 (.7

16 . Divtion of th Lplc qution in pol co-odints fom th Lplc qution in Ctsin co-odints. Consid th ltions btwn Ctsin nd pol co-odints: Hnc Epss th ptil divtivs θ cosθ y sin θ y tn ( ( y y / cosθ / y sinθ y θ y sinθ y θ cosθ y y Thn Φ Φ Φ θ θ Φ Φ sinθ Φ cosθ θ Φ Φ Φ θ θ Φ Φ Φ Φ Φ Φ cos θ sin θ sin θ sinθ cosθ sinθ cosθ θ θ θ In th sm wy, obtin: Φ Φ Φ Φ Φ Φ sin θ cos θ cos θ sinθ cosθ sinθ cosθ y θ θ θ Mk th sum: Φ Φ Φ Φ Φ (.8 y θ

17 THE LAPLACE EQUATION.3 Equtions ltd to th Lplc qution. Bsids th poply so clld Lplc diffntil qution, th ists sis of usful diffntil qutions closly ltd to th Lplc qution..3. Th Lplcin with cofficints diffnt fom. Fo mpl in pln Ctsin co-odints Φ Eqution (.9 is utilisd in chpt s.3.5 nd.6. Φ C y (.9.3. Th doubl Lplcin Th doubl Lplcin, o th Lplc opto pplid to Lplc qution, is lso solution of n impotnt sis of pplictions. It wits Φ (. Dvlopd, fo mpl in volum co-odints, th doubl Lplcin is Φ Φ Φ (. y y Eqution (. is pplid in chpt s.3,.5.,.5.,.5.3,.5.4,.,.,.3 nd Th tndd Lplcin Oftn th Lplcin qution is compltd by divtivs of n od low thn th scond. H in pol co-odints: Φ Φ Φ kφ (. θ o mo gnlly Φ A Φ BΦ C (.3 Th tndd Lplcin is usd in chpts..5,.. nd in chpt s to 9. 3

18 .4 Rsolution of th Lplc qution Th vy gnl mthods pplid in this book..4. Rsolution by sption of th vibls Solution in volum Ctsin co-odints: Consid qution (.3 nd ssum solution such s Φ f (f (yf 3 (. Applying (.3 yilds f ( f ( y f ( f ( y f ( f ( 3 3 f3( f( f ( y y nd dividing by f (f (yf 3 ( f ( y f ( f 3( y f( f ( y f 3( Ech of th 3 tms of th sum is function of on singl vibl; this sults in f( f ( y f 3( C f( C f ( y C3 f 3( y C C C3 systm with lg sis of solutions of th diffntil qution. Fo mpl: f ( cos(, f (y cos(y, f 3 (. Solution in i-symmtic cylind co-odints ( Bowmn,958: Consid qution (.7 nd pply solution such s Φ f(g(. Applying (.7 yilds f ( f ( g( g( g( f ( nd dividing by f(g( f ( f ( g( f ( f ( g( Thus f ( f g( Cf( Cg( Choos Cg(. Hnc f( must b solution of f ( f ( f ( Eqution (.4 is clld th Bssl qution of o od. (.4 4

19 THE LAPLACE EQUATION Th function known s Bssl s function of th fist kind nd of n- th od nd dnotd J n ( is dfind s follows (Bowmn, 958: n n 4 n 6 n ( / ( / ( / ( / J n (... (.5! n!!( n!!( n! 3!( n 3! Hnc 4 6 ( / ( / ( / J (... (.6!!!! 3!3! nd ( / ( / ( / J (... (.7!!!3! 3!4! Diffntiting th sis fo J ( nd comping th sult with th sis fo J ( sults in: dj ( J( (.8 d Also, ft multiplying th sis fo J ( by nd diffntiting: d ( J ( J ( (.9 d Using (.8, (.9 cn b wittn in th fom: d dj ( J ( d d d J ( dj ( J( (. d d Hnc J ( is solution of (.4 nd Φ J ( is solution of (.7. This pticul solution ws dvlopd to intoduc, fom th fist chpt on, th Bssl functions. Bssl functions fquntly usd nd discussd thoughout this book..4. Rsolution by mns of th chctistic qution (Spigl, 97 This solution pplis to homognous lin diffntil qutions with constnt cofficints dfind s n n d Φ d Φ dφ... n n n n Φ (. d d d It is convnint to dopt th nottions DΦ, D Φ,..., D n Φ to dnot dφ/d, d Φ/d,...d n Φ/d n, wh D, D,..., D n clld diffntil optos. Using this nottion, (. tnsfoms n n D D... nd n Φ (. Lt Φ m, m constnt, in (. to obtin n n m m... n (.3 tht is clld th chctistic qution. It cn b fctod into ( m m ( m m...( m mn (.4 which oots m, m,...m n. 5

20 On must consid th css: Cs. Roots ll l nd distinct. Thn m, m,... mn n linly indpndnt solutions so tht th quid solution is: Cs. Som oots compl. m m mn Φ C C... Cn (.5 If,,..., n l, thn whn bi is oot of (.3 so lso is -bi (wh nd b l nd i (-. Thn solution cosponding to th oots bi nd -bi is: Φ ( C cosb C sin b (.6 wh us is md of Eul s fomul iu cos u i sin u (s Appndi. Cs 3. Som oots ptd. If m is oot of multiplicity k, thn solution is givn by: k m Φ ( C C C3... Ck (.7 Empl Consid th doubl Lplcin in pln Ctsin co-odints: Φ Φ Φ ( y y Choos s solution Φ f( y. Hnc (.8 ducs in 4 f ( f ( f ( (.9 4 Eqution (.9 is homognous lin diffntil qution of od 4. Th chctistic qution wits: nd cn b fctod in: ( m 4 m Bsd on (.6 nd (.7, th solution of (.9 bcoms: f ( Acos Bsin ( C cos Dsin nd th solution of (.8:.4.3 Rsolution by mns of indicil qutions m (.3 i ( m i (.3 [ A cos B sin ( Ccos D sin ] y Φ (, y (.3 Consid th diffntil qution: d w k w (.33 d Assum solution in th fom of indicil sis 3 4 f ( 3 4 L Apply (.33 6

21 THE LAPLACE EQUATION L k k k L This qution must b qul o fo ll vlus of. Hnc th sum of th cofficints of ch ponnt of must b individully o. Fist lt nd Hnc.. k k k 4 p k k p k 4 p (! 4! ( p! (.34 3 L p L (.35 Th succssiv tms of (.34 th tms of th cosin sis. Hnc, th fist solution of (.33 is f( cos(k. Th scond solution is obtind by stting nd k. Obviously, th scond solution of (.33 is f( sin(k. 7

22 8

23 THE GAMMA FUNCTION Chpt Th Gmm Function. Intoductoy not In chpt, w hv dfind th Bssl function of th fist kind nd of od o s: k k ( / J( ( k!k! This qution cn b gnlisd, fo n n intg, in: k n k ( / J n( ( k!( k n! Whn p is not n intg, th Bssl function of od p wits: k p k ( / J p( ( k! Γ ( k p wh Γ(kp is clld th gmm function of kp. Fo ou pupos, th gmm function is ssntilly quid to pss Bssl functions of non-intg od: it is th fctoil function fo non-intgs. Obsv tht with ths dfinitions Γ(n n!. Hlpful ltions Γ ( p o p Γ ( p pγ ( p Γ ( Γ ( n n! d Γ ( / Γ ( n ± dγ ( n Γ ' dn ( n Γ ( n γ... 3 n γ (Eul s constnt 9

24 .3 Dfinition of th Gmm Function Th gmm function is dfind by: which is convgnt fo p >. p Γ ( p d (. o Applying th dfinition p p p p Γ ( p d ( ( p d p d pγ ( p o o o o on obtins th cunc fomul: Γ ( p pγ ( p (. By tking (. s th dfinition of Γ(p fo p >, w cn gnlis th gmm function to p < by us of (. in th fom ( p Γ ( p Γ (.3 p This pocss is clld nlytic continution. Applying (. w dtmin th vlu of Γ( Γ ( d (.4 o o Hnc, pplying (. Γ ( Γ ( Γ ( 3 Γ (! Γ ( 4 3Γ ( 3 3 Γ ( 3 3! Γ ( n n! Fo n bing positiv intg..3. Vlus of gmm functions. Th vlus of Γ(p fo non-intg vlus of p must b computd numiclly. On obtins fo p p Γ(p Tbl Vlus of G(p fo non-intg vlus of p Th vlu of Γ(p cn b computd fo ny vlu of p using (. Γ( Γ( *.3 * Γ( *.3 *.3 * Γ ( *.3 *.3 *

25 THE GAMMA FUNCTION.4 Vlus of G(/ nd G(-/. Two pticul vlus of th gmm function, Γ(/ nd Γ(-/, pp in mny pplictions. By dfinition: / Γ ( / d o Wit u nd d udu / u Γ ( du o Thn [ ] u v ( u v Γ ( / du dv 4 o o o o Wit u cos θ, v sin θ, du dv d dθ / / [ ] Γ ( / 4 ddθ dθ o o o Hnc dudv Γ ( / (.5 nd by (.3 Γ ( / Γ ( / (.6 /.5. Vlus of G(n fo n, -, -,... By dfinition tht w wit Div Hnc d d Γ ( o (... d... d 3 3! 3!! 3! Γ! o 3! 3! 3! Γ (... 3! 3! o d 3 o! 3!! 3... d 3! 3

26 ... d 3 o! 3! Γ (... 3! 3! o Γ (... 3!! 3!! o / Γ ( (.7! o Γ ( Γ ( (.8 Γ ( n ± (.9 Th gmm function is undfind whn th vlu of th gumnt is o o ngtiv intg..6. Divtiv of G(n. W cll Γ (n th divtiv of Γ(n with spct to n. Fo simplicity w consid only th cs of n bing n intg, which bsids is th only cs quid fo ou pupos. By dfinition d n d n Γ ' ( n d ( dn dn o o W fist comput Γ ( Γ' ( log o Intgting by pts Futh d d o n log d log d log log d d log n n n n log d log d log n n n ( log log d log d log d log d log log d n n d!.!!.! d

27 THE GAMMA FUNCTION log d log log d d! 3! 3.3! 3! 3.3!... n n n n n log d log log d d ( n! n! n.n! n! n.n! Assmbling: 3 n 3 n log d... log...!! 3! n!.!.! 3.3! n.n! 3 n d d d... d.!.! 3.3! n.n! 3 n ( log d log....!.! 3.3! n.n! 3 n d d d... d.!.! 3.3! n.n! Intgting btwn nd yilds log d log log o o o o log d.! lim log 3....! 3.3! lim log lim o Γ( Γ ( 3 Γ( 4....!.! 3.3! ( log lim ( log... ( log d lim log lim log 3 Γ ' ( log d lim log M... γ (. M 3 M o wh γ is clld Eul s constnt. 3 By tnt: o log d ( log log d o o o d log d log d o o o d 3

28 Γ' ( o log d γ Γ( Γ ' ( n Γ ' ( 3! γ Γ ( 3... ( n! γ Γ ( n... 3 n Γ '( n γ... Γ ( n 3 n (. 4

29 THE GENERAL SOLUTION OF THE BESSEL EQUATION Chpt 3 Th Gnl Solution of th Bssl Eqution. 3. Intoductoy not. In th wy tigonomtic functions solutions of Lplc qution in Ctsin co-odints, Bssl functions solutions of Lplc qution in pol o cylindicl co-odints. On could sy tht Bssl functions is th dimnsionl function whs tigonomtic functions two dimnsionl functions. Th fundmntl chctistics of both functions idnticl: d f ( - if f( cos( f ( d d f ( df ( - if f( J ( f ( d d Bssl functions ssntilly utilisd in poblms psnting n il symmty: in pvmnt ngining, th common cs of cicul lod on lyd stuctu, i.. nly ll pplictions of chpts to 6. J p nd J -p th two solutions of th fist kind of Bssl s qution d φ dφ φ p t φ d d Howv, whn p n, J n (- n J -n. In tht cs, Bssl s qution quis noth scond solution, solution of th scond kind, Y p, which will not b utilisd in this book. I p nd I -p th two modifid solutions of th fist kind of Bssl s modifid qution: d φ dφ φ p t φ d d Ths lso, n t usd in this book. Howv, thy llow intoducing th K nd Ki functions, solutions of: d φ dφ it φ d d which is pplid in th poblm of slb subjctd to n isoltd lod (chpt.4 nd in th ovlistion tst (chpt 6.4. Th modifid fom of Bssl s qution: d F df α p γ ( γ ( α F β γ F d d pmits us to illustt th istnc of ltions btwn Bssl functions of od ½ nd tigonomtic functions (chpt 4.3 Thos ltions llow us to stblish symptotic ppoimtions fo Bssl functions (chpt 4.5, quid fo th numicl computtions of functions of high gumnts. 5

30 3. Hlpful ltions ( t / p k k J p( t ( k! Γ ( p k p k k ( t / J p ( t ( k! Γ ( p k n J n( t ( J n( t fo n intg n [ ( ] k n ( n k!( t / Yn( t J n( t log( t / γ k! k n k ( t / ( k 3 n k k!( n k! ( H p ( t J p( t iy p( t ( H p ( t J p( t iy p( t p k p ( t / I p( t i J p( it k! Γ ( k p n k ( n k! Kn ( t ( nk k!( t / n k n ( t / ( log( t / Ψ ( k Ψ ( n k k!( n k! Ψ ( γ Ψ ( k γ... 3 k J ( ti ± i I( t ± i b( t ± ibi( t 4k 4k k ( t / k ( t / b( t ( bi( t ( ( k!( k! ( k!( k! ki( t k( t bi( t K ( t ± i k( t ± ki( t b( t [ log( t / γ ] 4 [ log( t / γ ] bi( t 4 bi( t ( t / 4...!! 6 t /... 3!3! 3 ( t / (!! 6

31 THE GENERAL SOLUTION OF THE BESSEL EQUATION 3.3 Rsolution of th Bssl qution (Bssl functions of th fist kind Rcll th dfinition of th Lplcin in cylind co-odints givn by.5: Φ Φ Φ Φ (3. θ Consid nt solution obtind by th mthod of sption of th vibls t Φ F( cos( pθ nd pply it to (3. tht tnsfoms into: F F p F t F (3. Th gnl solution of qution (3. cn b found by th mthod of th indicil qutions (Wylnd, 97. Thfo ssum tht th solution cn b wittn s sis such s: Hnc n α F n F n ( n n α α F n α ( n α ( n α Rplc th tms in (3. by (3,3, (3.4 nd (3.5: n n {[( n ( n ( n p ] n α α α α t n α } Tk out th fist two tms of th fist summtion: α ( α p ( α p ( α p n n ( n α p ( n α p n t n If th tms in nd qul o, th ltion bcoms homognous gding th ponnts of. Thfo choos bity, nd α ± p. Rng th indics n [( n ( n ± p n nt ] α (3.6 If (3.6 hs to b idnticl o whtv th vlu of m, ch tm of th summtion hs to b qul o. Hnc th cunc fomul ( n ( n ± p n nt (3.7 nd bcus, ll th tms with n odd ind lso qul o. Finlly sulting in: (3.3 (3.4 (3.5 7

32 t t t ( ± p (± p ( ± p 4 t t 4 4( 4 ± p ( ± p ( ± p 6 4t t 6 6( 6 ± p 3(± p ( ± p ( 3 ± p... s s ( t s s!( ± p ( ± p...( s ± p If p is diffnt fom o o not n intg, on obtins two linly indpndnt solutions p k p k k ( t / k ( t F A ( B ( k! Γ ( p k k! Γ ( p k Th cosponding sis clld Bssl functions of th fist kind nd notd p k k ( t / J p( t ( k! Γ ( p k p k k ( t / J p ( t ( k! Γ ( p k (3.8 (3.9 (3. Th Bssl functions of th fist kind nd of od nd psntd in Figu 3.. Figu 3. Bssl functions of th fist kind nd of od nd 8

33 THE GENERAL SOLUTION OF THE BESSEL EQUATION 3.4 Rsolution of th Bssl qution fo p n intg (Bssl functions of th scond nd thid kind 3.4. Fo n intg, J n (- n J -n Whn p is n intg, lt sy n, solutions (3.9 nd (3. not linly indpndnt ny mo. Indd whn p -n n k n n k n k k ( t / k ( t / k ( t / J n ( t ( ( ( k!( n k! k!( n k! k!( n k! n Whn,,, 3,..., k n-, (-nk! ± nd n k ( t / k!( n k! Wit k n j n k n j k ( t / n j ( t / J n ( t ( ( k!( n k! ( n j! j! n n j n j ( t / n J n ( t ( ( ( J n( t j!( n j! Hnc th two solutions not linly indpndnt Bssl functions of th scond kind In od to find scond linly indpndnt solution, w dfin th function J p ( t cos( p J p ( t Y p ( t (3. sin( p which is vlid fo p not n intg. Thn w dfin fo p n J p( t cos( p J p ( t Yn ( t lim p n sin( p W sch th solution fo n. J p (t cos( p J p ( t Y( t lim p sin( p Apply d L Hospitl s ul J p( t J p( t cos( p J p( t sin( p p p Y( t cos( p J p ( t J p ( t Y( t p p Comput th divtivs of J p (t nd J -p (t: J p k p( t k ( t / Γ '( p k ( log( t / p k! Γ ( p k Γ ( p k 9

34 J p k p ( t k ( t / Γ ' ( p k ( log( t / p k! Γ ( p k Γ ( p k nd fo p k k ( t / Γ' ( k Y ( t ( log( t / k!k! Γ ( k By (. Hnc Y ( t J ( t γ Fo n, (3. is tndd to Yn( t Γ '( k γ Γ ( k [ log( t / ] ( t /!! 3 4 ( t /!!... k 6 ( t / 3!3! n { [ ]} k ( n k!( t / J ( t log( t / γ n k! k n k ( t / ( k 3 n k k!( n k! Th functions Y p (t clld functions of th scond kind. Th Bssl functions of th scond kind nd of od nd psntd in Figu 3.. n... 3 (3. (3.3 Figu 3. Bssl functions of th scond kind nd of od nd

35 THE GENERAL SOLUTION OF THE BESSEL EQUATION Bssl functions of th thid kind Hnkl intoducd nt pi of conjugt compl functions, with i (-: ( H p ( t J p( t iy p( t (3.4 ( H p ( t J p( t iy p( t (3.5 Ths functions clld Hnkl functions o Bssl functions of th thid kind. 3.5 Th modifid Bssl qution. If in 3.3 w chos: Φ F( cos( pθ cos( t s solution of th Lplc qution, qution (3. would modify into: F F p F t F (3.6 Eqution (3.6 is clld th modifid Bssl qution. Its solution cn immditly b dducd fom th oiginl solution (3.8 F( t AJ p( it BJ p(it (3.7 wh k p p k k i ( t / J p (it ( k! Γ ( p k p k p ( t / J p( it i k! Γ ( p k Hnc w dfin th function I p, modifid Bssl function of th fist kind, s l function, solution of th modifid Bssl qution. p k p ( t / I p i J p( it (3.8 k! Γ ( p k F( t AI p BI p (3.9 It is sy to dduc fom (3.8 tht whn p is n intg, lt sy p n, I n I -n. In tht cs, th scond solution is usully dfind by n ( I ( t I ( t K n p p ( t lim p n p n (3. which is clld th modifid Bssl function of th scond kind. Th function K p (t is dfind fo unstictd vlus of p by th qution I p ( t I p( t K p ( t sin p Applying d L Hospitl s ul on (3. nd (3., on vifis tht Kn( t lim K p( t p n In simil wy s givn in 3.4. on obtins: (3.

36 n k Kn( t ( ( n k!( t k! n k / n k n ( t / ( ln( t / Ψ ( k Ψ k!( n k! Ψ ( γ Ψ ( k γ... 3 k ( n k (3. Th modifid Bssl functions of th fist kind nd of od nd psntd in Figu 3.3. Figu 3.3 Modifid Bssl functions of th fist kind of od nd Th modifid Bssl functions of th scond kind nd of od nd givn in Figu 3.4. Figu 3.4 Modifid Bssl functions of th scond kind nd of od nd

37 THE GENERAL SOLUTION OF THE BESSEL EQUATION 3.6 Th b nd bi functions. Consid nt modifid Bssl qution: F F ( ± i t F which solutions of th fist kind i( t / ( t / i( t / ( t / I ( t i...!!!! 3!3! 4!4! i( t / ( t / i( t / ( t / I ( t i...!!!! 3!3! 4!4! (3.3 W dfin: wh Notic tht: b( t bi( t I ( t ± i b( t ± ibi( t ( ( t / ( t / ( t / b( t... (3.5!! 4!4! 6!6! 6 ( t / ( t / ( t / bi( t... (3.6!! 3!3! 5!5! ( t i I ( t i J ( ti i J ( ti i J ( t i J ( t i I (3.7 I ( t i I ( t i J ( ti i J ( ti i J ( t i J ( t i (3.8 i i i Th b nd bi functions cn b dpictd in Figu 3.5 Figu 3.5 Th b nd bi functions 3

38 3.7 Th k nd ki functions. Agin consid qution (3.3, which solutions of th scond kind : 4 6 t i t i t i t i K ( t i I ( t i log γ...!!!! 3!3! 3 Rcll tht: t i t t t i / t log log log i log log i log log i 4 nd pplying ( t t t t K ( t i [ b( t ibi( t ] log i i i γ... 4!!!! 3!3! t t t t K ( t i [ b( t ibi( t ] log i i i γ... 4!!!! 3!3! 3 W dfin: k( t ( t i K ( t i K (3.9 ( t i K ( t i K ki( t (3.3 i 4 [ ] ( t / k( t b( t log( t / γ bi( t... (3.3 4!! 6 [ ] ( t / ( t / ki( t bi( t log( t / γ b( t... (3.3 4!! 3!3! 3 Th k nd ki functions givn in figu 3.6. Figu 3.6 Th k nd ki functions 4

39 THE GENERAL SOLUTION OF THE BESSEL EQUATION 3.8 Rsolution of th qution w w Consid th nt qution: d d d w dw w d d d d (3.33 Th solution cn b obtind by splitting (3.33 into two simultnous diffntil qutions d w dw d d (3.34 d d w d d (3.35 Both qutions vifid togth if miw. Indd, if, fo mpl, iw d w dw (3.34 bcoms iw, nd d d d w dw d w dw (3.35 bcoms i i w o iw d d d d Hnc th solution of (3.33 is givn by th solution of d w dw ( ± i w d d (3.36 Th qutions (3.5, (3.6, (3.3 nd (3.3 giv th solution of (3.36, fo its two signs w Ab( Bbi( C k( Dki( ( Th modifid fom of th Bssl qution Consid th nt diffntil qution: F F α p γ ( γ ( α F β γ F (3.38 which hs fo solutions, s cn b vifid by substitution, α γ F( A J p( β α γ B J p( β (3.39 o, if p is n intg, α γ α γ F( A J p( β B Yp( β (3.4 An intsting ppliction of (3.38 is th solution of th Lplc qution in two- dimnsionl Ctsin co-odints Φ Φ Φ y Consid solution such s Φ F( ty 5

40 Hnc Comping (3.4 with (3.38 yilds: Hnc Howv F( ty Φ t F( (3.4 α / γ β t α p γ p / / / F( A J / (t B J / ( t (3.4 F ( Acos( t B sin( t (3.43 is lso solution of (3.4. Thfo, on must conclud tht th ists ltion btwn Bssl functions of od ± / nd tigonomtic functions. 6

41 PROPERTIES OF THE BESSEL FUNCTIONS Chpt 4 Poptis of th Bssl Functions. 4. Intoductoy not This chpt dls with th most impotnt poptis of Bssl functions: divtivs, functions of hlf od, symptotic vlus, indfinit intgls nd ltions btwn functions of diffnt kind. 4. Hlpful ltions d d d p p [( t J p ( t ] t( t J p( t p p [( t J p( t ] t( t J p ( t d d p J p ( t tj p( t J p( t d d p J p ( t tj p ( t J p ( t d p J p ( t J p ( t J p ( t t J( t J ( t d t J( t J( t d t J / ( t sin( t t J ( t / t cos( t t [ t ] t [ t ] I / ( t t I / ( t t K/ K / t t J p ( t J p Y p ( t ( t p cost t 4 fo high vlus of t p sint t 4 fo high vlus of t p sint t 4 fo high vlus of t 7

42 p ( t cost fo high vlus of t t 4 ( i( t p / / 4 H p fo high vlus of t t ( i( t p / / 4 H p fo high vlus of t t t t ( p / i I p( t ( fo high vlus of t t Y p b( t bi( t k( t t / t t / t cos sin t / t t fo high vlus of t 8 t fo high vlus of t 8 cos t / t ki( t sin t 8 i ip J ± p ( J ± p ( J ± p( mi mip ( J ( J ± p ± p ( ( H p ( t H p ( t J p( t Y p( t Y p( t cos( p J p( t sin( p J ( t J ( t ( p p H p ( t i sin( p t fo high vlus of t 8 ip fo high vlus of t ip ( J p( t J p( t H p ( t i sin( p ( mi mpi ( pi sin( mp H p ( t H p ( t J p( t sin( p ( mi mp i ( pi sin( mp H p ( t H p ( t J p( t sin( p I p ( t i J i ( t pi / / p pi / ( K p( t i H p ( ti pi / ( K p( t i H p ( ti 8

43 PROPERTIES OF THE BESSEL FUNCTIONS 4.3 Divtivs of Bssl functions 4.3. Divtiv of (t p J p (t By dfinition of th Bssl function d d p [( t J p( t ] d d p k d k ( t ( d p k k! Γ ( p k p k k ( p k ( t t ( p k k! Γ ( p k p k k ( t t ( p k k! Γ ( p k ( p k p k ( t t( t ( ( p k k! Γ ( p k p p [( t J p( t ] t( t J p( t [ ] ( Divtiv of (t-pjp(t On finds similly d d p p [( t J p ( t ] t( t J p ( t ( Divtiv of J p (t Diving th lft mmb of (4. by pts yilds d p [ J p( t ] tj p ( t J p ( t (4.3 d diving th lft mmb of (4. by pts yilds d p [ J p ( t ] tj p ( t J p ( t (4.4 d Pticully d [ J ( t ] tj ( t tj( t (4.5 d Adding (4.3 nd (4.4 yilds d t [ J p ( t ] [ J p( t J p ( t ] (4.6 d nd subtcting yilds th cunc fomul fo Bssl functions p J p ( t J p ( t J p( t (4.7 t 9

44 4.4 Bssl functions of hlf od 4.4. Vlus of J / (t, J -/ (t, J 3/ (t, J -3/ (t Epnding th Bssl functions in thi sis nd clling tht Γ(/ yilds: J / ( t sin( t (4.8 t Vlus of I / (t, I -/ (t J / ( t cos( t (4.9 t J 3 / ( t sin( t cos( t t t J 3 / ( t cos( t sin( t t t (4. (4. Epnding th Bssl functions in thi sis yilds I / ( t t I / ( t t Vlus of K / (t, K -/ (t t [ t ] t [ t ] (4. (4.3 By dfinition (3. Hnc I / ( t I / ( t t K / ( t sin( / t I / ( t I / ( t t K / ( t sin( / t (4.4 (4.5 K / ( t K / ( t ( Vlus of K (t nd K n (t By dfinition (3. I p ( t I p( t K ( t lim p sin p Applying d l Hospitl s ul yilds nd omitting th lim sign di p( t di p( t dp dp di p( t di p( t K ( t cos p dp dp 3

45 PROPERTIES OF THE BESSEL FUNCTIONS k p k p d ( t / d ( t / K( t dp k! Γ ( k p dp k! Γ ( k p k p p ( t / ( t / log( t / Γ ( k p ( t / Γ ' ( k p K ( t k! Γ ( k p Γ ( k p p p ( t / log( t / Γ ( k p ( t / Γ '( k p Γ ( k p Γ ( k p Ltting p k ( t / Γ' ( k K ( t log( t / k!k! Γ ( k Dvloping Γ (k s in 3.3. yilds 4 ( t / ( t / K ( t I( t [ log( t / γ ]!!!! 6 ( t / (4.7 3!3! 3 Fo n, (4.7 is ltivly sily tndd to n k n ( ( n k! k n Kn ( t ( I n( t [ log( t / γ ] ( t / k! k [ ] n n ( t / ( Φ ( k Φ( n k (4.8 k!( k n! 4.5 Asymptotic vlus 4.5. Asymptotic vlus fo Jp nd J-p Tnsfom Bssl qution (3.: by witing F(t G(t/(t / Whn p ½, th qution ducs to whos gnl solution is givn by Hnc F F p F t F G t p / 4 G G t G G A cos( t B sin( t (4.9 (4. / / F A( t cos( t B( t sin( t (4. 3

46 but lso: F CJ / ( t DJ / ( t (4. Equtions (4. nd (4. confim, s w knw by (4.8 nd (4.9, tht th ists ltion btwn th Bssl functions of hlf od nd th tigonomtic functions, nd tht this ltion is vlid fo ll vlus of th gumnt, thus in pticul fo high vlus of th gumnt. Hnc th qutions: J ( t / J ( t / t t sin( t cos( t cn b considd s th symptotic qutions fo th Bssl functions of hlf od. Thus (4.9 must hv, fo p not bing n intg, two ppoimt vlus fo high vlus of th gumnt such s J p( t / Ap ( t cos( t α p (4.3 / J p( t Bp( t sin( t β p (4.4 Th cofficints α p nd β p must b dtmind in such wy tht qutions (4.3 nd (4.4 linly indpndnt nd comptibl with th dfinitions of J p nd J -p. Div, with spct to, J p in (4.3 ' t 3 / / J p ( t Ap ( t cos( t α p Ap( t t sin( t α p Fo high vlus of th gumnt th fist tm cn b nglctd ginst th scond ' / J p ( t Ap( t t sin( t α p Futh (4.3 nd (4.4 simplify fo high vlus of th gumnt ' J p ( t tj p ( t Hnc ' J p ( t tj p ( t / / Ap( t t sin( t α p tap( t cos( t α p (4.5 / / Ap ( t t sin( t α p tap ( t cos( t α p (4.6 If w choos A p A p- A p A nd α p k - p/ in such wy tht A nd k indpndnt fom p, w notic tht qutions (4.5 nd (4.6 stisfid. Indd p ( p Asin t k Acost k Thn / p J p( t A( t cost k This qution must b stisfid fo ll vlus of p, thus lso fo p/ fo which J / ( t Hnc A ( / nd k - /4 t sin( t t cost 4 4 3

47 PROPERTIES OF THE BESSEL FUNCTIONS Finlly Rplcing p by p in (4.7 yilds J p ( t J p ( t t t cost 4 sint 4 p p (4.7 ( Asymptotic vlus fo Y p nd Y -p Sinc Y p nd Y -p th pid scond solutions with J p nd J -p w shll dmit implicitly th following symptotic qutions Y p ( t Y p Asymptotic vlus fo H p ( nd H p ( ( t t t p sint 4 p cost 4 By ppliction of th dfinitions (3.4 nd (3.5, on immditly finds: H H t (4.9 (4.3 ( i( t / 4 p / p (4.3 t ( i( t / 4 p / p ( Asymptotic vlus fo I p nd I -p Tnsfom Bssl qution (3.6 F F p F t F by witing F(t G(t/(t / G p / 4 t G (4.33 Whn p ½, th qution ducs to G t G (4.34 whos gnl solution is givn by t t G A B Hnc / t / t F A( t B( t (4.35 but lso 33

48 F CI / ( t DI / ( t (4.36 Eqution (4.35 is vlid fo ll vlus of th gumnt, thus in pticul fo high vlus of th gumnt. Hnc th qutions: t [ t ] t [ t ] I / ( t t I / ( t t cn b considd s th symptotic qutions fo th modifid Bssl functions of hlf od. Hnc (4.33 must hv, fo p not n intg, two ppoimt vlus fo high vlus of th gumnt such s / t α( p / t β( p / I p( t Ap( t (4.37 / t α( p / t β ( p / I p ( t B p( t (4.38 Th cofficints α nd β must b dtmind in such wy tht qutions (4.37 nd (4.38 linly indpndnt nd comptibl with th dfinitions of I p nd I -p. Div, with spct to, I p in (4.37 ' t 3 / t α( p / t β ( p / I p ( t Ap( t [ ] / t α( p / t β ( p / Ap( t t Fo high vlus of th gumnt th fist tm cn b nglctd ginst th scond / t α( p / t β ( p / I' p ( t Ap ( t t Futh simplify th divtivs of I p (t fo high vlus of th gumnt ' I p ( t tj p ( t Hnc ' I p ( t tj p ( t / t α( p / t β ( p / Ap( t t / t α( p / t β ( p / Ap( t t (4.39 / t α( p / t β ( p / Ap( t t / t α ( p 3 / t β ( p 3 / Ap ( t t (4.4 If w choos A p A p- A p A nd α, β i in such wy tht A nd β indpndnt fom p, w notic tht qutions (4.39 nd (4.4 stisfid. 34

49 PROPERTIES OF THE BESSEL FUNCTIONS A( t A( t / t / t t ( p / i t ( p / i Hnc / t t ( p / i I p( t A( t This qution must b stisfid fo ll vlus of p, thus lso fo p/ fo which t t I / ( t [ ] t Hnc A ( / Finlly t t ( p / i I p( t t Rplcing p by p in (4.4 yilds t t ( p / i I p ( t t (4.4 ( Asymptotic vlu fo K n Consid dfinition (3.: K Fo lg vlus of th gumnt n ( Kn( t n ( t t lim p n t n ( Kn( t Applying d l Hospitl s ul yilds n ( Kn( t In this fom K n (t my b gnlisd into wh p not n intg nd lso ( n I p ( t I p n t( p / i t p p n ( t t pi pi i t p n t cos( n t t Kn( t t K p ( t t t t ( p / i (4.43 (4.44 (

50 K t t ( t ( Asymptotic vlus fo b nd bi Fo high vlus of th gumnt (4.7 yilds J With (i -/4 -i/8 nd i ( i/ Similly Hnc cost i 4 J ( t i cost i t i / i( t / / 4 t / i( t / cost J b( t bi( t i Asymptotic vlus fo k nd ki Rcll qutions (3.9 nd (3.3 Apply (4.46 i( t / / 4 4 t / t / t / 8 ( t i i( J t / 4 t / t / t t ( t i cos i sin t 8 8 t / t t ( t i cos i sin t 8 8 ( ( t / t i J t i t cos J t 8 ( ( t / t i J t i t sin J i t 8 k( t ki( t ( t i K ( t i (4.47 (4.48 (4.49 (4.5 K (4.5 ( t i K ( t i K (4.5 i t K( t i t i i 36

51 PROPERTIES OF THE BESSEL FUNCTIONS Similly obtin K ( t i t i / 8 t( i / t / i( t / t t t / K( t i t Adding nd subtcting (4.53 nd (4.54 yilds t / cos cos t 8 t 8 / 8 t i sin 8 t i sin 8 (4.53 (4.54 k( t ki( t 4.6 Indfinit intgls of Bssl functions t / t t / t cos sin t 8 t 8 (4.55 ( Fundmntl ltions In 4..w hv divd th following divtivs of Bssl functions: d [ tj ( t ] t( t J ( t d d [ J ( t ] tj( t d Fom thos qutions w sily dduc nt fundmntl intgls J( t J ( t d (4.57 t J( t J ( t d (4.58 t 4.6. Th intgl n J (td Intgting by pts solvs th intgl: n n J( t ( n n J ( t d J ( t d t t n n ( n n J( t d J ( t J ( t d t t Hnc n n J( t ( n n ( n n J ( t d J( t J ( t d (4.59 t t t If n is odd, fomul (4.59 lds to (4.57. If n is vn, fomul (4.59 lds to J (td, which is tbultd. 37

52 4.7 Rltions btwn Bssl functions of diffnt kind 4.7. Bssl functions with gumnt t Sinc Bssl s qution is unltd if t is plcd by t, w must pct th functions J ±p (-t to b solutions of th qutions stisfid by J ±p (t. Considing th ltion i -, w my wit i J p( t J p( t W cn vn consid th mo gnl function J p ( mi t wh m is n intg. mi p k mi k ( t / J p( t ( k! Γ( k p Rsticting th compl ponnt to its pincipl vlu w gt mi p k m p mk im p ( i( Hnc p k mi imp k ( t / im p J p ( t ( J p( t k! Γ ( k p ( Rltions btwn th th kinds of Bssl functions Consid (3.4 nd (3.5 ( H p ( t J p( t iyp ( t ( H p ( t J p( t iyp ( t Hnc by multiplying Yp(t by cos(p nd subtcting fom Y-p(t ( ( H p ( t H p ( t J p( t (4.6 Consid (3. J p( t cos( p J p( t Y p( t sin( p Rplc p by p J p( t cos( p J p( t Y p( t sin( p Hnc by subtcting nd dividing by Y p( t Yp( t cos( p J p( t (4.6 sin( p Consid (3.4 togth with (3. ( H p ( t J p( t iyp ( t ( J p( t cos( p J p( t H p ( t J p( t i sin( p 38

53 PROPERTIES OF THE BESSEL FUNCTIONS [ cos( p i sin( p ] ( J p( t H p ( t i sin( p J p( t ip ( J p ( t J p( t H p ( t (4.63 i sin( p nd similly ip ( J p( t J p( t H p ( t (4.64 i sin( p Add qutions (4.63 nd (4.64 togth ( ( H p H p J p( t (4.65 Multiply qution (4.63 by ip nd (4.64 by -ip nd dd togth ( ip ( ip H p ( t H p ( t J p( t (4.66 In qution (4.63 plc t by t mi mi mi ip ( mi J p( t J p( t H p ( t i sin( p togth with qution (4.6 H ( p ( t mi mpi J p ( t mpi J isin( p p ( t ip ip ip mpi mpi ( J ( t J ( t J ( t ( mp i ( mi p p p H p ( t i sin( p i sin( p pplying qution (4.63 ( mi mpi ( pi sin( mp H p ( t H p ( t J p( t (4.67 sin( p nd similly ( mi mp i ( pi sin( mp H p ( t H p ( t J p( t (4.68 sin( p Bssl functions of puly imginy gumnt Consid (3.8 Rcll tht i/ i, 3i/ -i. Hnc Apply (3. p k p ( t / I p( t i J p( it k! Γ ( p k pi / I p( t J p( it (

54 4 p sin t ( I t ( I t ( K p p p p sin (it J it ( J t ( K p / i p p / i p p p sin i it ( J (it J i t ( K p i p p / i p p Hnc by (4.63 it ( H i t ( K ( p / i p p (4.7 nd similly (it H i t ( K ( p / i p p (4.7

55 THE BETA FUNCTION Chpt 5 Th Bt Function 5. Intoductoy not Th gmm function, Γ(n, is function with on vibl. Th bt function, B(m,n, is function with two vibls. Th function cn b dfind s poduct of Γ- functions. It is usful in th pssion of solutions of dfinit intgls of Bssl functions, which, fo pticul vlus of thi pmts, cn b wittn in simpl closd-foms. 5. Hlpful ltions B( m,n B( m,n B ( m,n B( m,n m / cos B( n,m ( m Γ ( m Γ ( n Γ ( m n n ϑ sin d n ϑdϑ p Γ p Γ ( p Γ ( p Γ 5.3 Dfinition of th bt function Th bt function is dfind by Eul s intgl of th fist kind m n B ( m,n ( d (5. which is convgnt fo m > nd n >. Witing cos θ nd - sin θ, (5. tnsfoms into / m n B( m,n cos ϑ sin ϑdϑ (5. Rplcing by - y in (5. yilds B ( m,n B( n,m (5.3 4

56 5.4 Rltion btwn bt nd gmm functions By dfinition: Hnc, on my wit Γ ( m Γ ( m Γ( n m m d Wit χ, y η χ m Γ ( m Γ( n 4 χ dχ Wit χ cosϑ, η sinϑ Wit ρ Hnc Γ ( m Γ( n 4 Γ ( m Γ( n 4 Γ ( m Γ( n 4 d y χ η m ( m n m n d Γ ( m Γ( n B( m,n Γ ( m Γ ( n B( m,n ρ ρ B( m,n χ cos m / m n cos y η η n η n ϑ sin m dy n dη dχdη n ϑ sin m n d ϑddϑ n ϑdϑ dρ B( m,n Γ ( m n Γ ( m Γ ( n Γ ( m n ( Th dupliction fomul fo gmm functions Consid th function Wit u ϑ J / sin p ( ϑ dϑ 4

57 DEFINITE INTEGRALS OF BESSEL FUNCTIONS / / J p p sin udu sin udu sin Howv, on lso hs Hnc J p J / / sin p ( ϑ dϑ p / u cos ( sinϑ cosϑ Γ( / Γ( p / udu Γ( p p dϑ p p p Γ ( p / Γ ( p / cos ϑ sin ϑdϑ Γ( p p Γ ( p / Γ ( p / J pγ ( p p Γ ( p / Γ ( p Γ (/ Γ ( p (5.5 Eqution (5.5, clld th dupliction fomul fo gmm functions, will oftn b usd in futh dvlopmnts. 43

58 44

59 DEFINITE INTEGRALS OF BESSEL FUNCTIONS Chpt 6 Dfinit intgls of Bssl functions 6. Hlpful ltions J ( ωt sin ( ωt ϑdϑ Γ J ( t J ( bt Γ ( t ( bt J ω b bcosϑ / ( J ( sinϑ sin ϑ cos ϑdϑ Γ ( / / y J ( y J ( sinϑ J ( ycosϑ sin ϑ cos ϑdϑ ( / ( y / ( / J( cos( sinϑ cos ϑdϑ Γ ( / Γ(/ 6. Ggnbu s intgl Th impotnt intgl: J ( ω J ( t J ( bt sin ϑdϑ Γ Γ ω ( t ( bt ω b bcosϑ which is vlid fo > -/, hs bn fist povd by Ggnbu (Wtson, 966, hnc its nm. (6. Dvlop th Bssl function in its sis J ( ωt sin ϑdϑ ( ωt Intgt th sis tm by tm. k ( ( ωt / ( ωt k! Γ t k sin ( k ( k! Γ ( k k k ( ω k sin k k k! j k j ( ω ( b ( b cos k ( ω ϑ j!( k j! j Consid th intgl m cos ϑ sin ϑdϑ ϑdϑ ϑdϑ 45