Sris Expasio of Rciprocal of Gamma Fuctio. Fuctios with Itgrs as Roots Fuctio f with gativ itgrs as roots ca b dscribd as follows. f() Howvr, this ifiit product divrgs. That is, such a fuctio caot xist alo. So, lt us cosidr th followig fuctio which addd compsatio to ach factor. Th, this ifiit product uiformly covrgs for arbitrary ad rsults i fuctio /. This was show by Wirstrass. If th fuctios with various itgrs as roots ar show, it is as follows. I additio, ths lft sids ca b xpadd to sris of. So, w call th formulas factoriatio aroud 0. Formula.. (factoriatio aroud 0) Wh is EulrMaschroi costat ad is th Gamma fuctio, th followig xprssios hold. ( ) (, ) ( ) (, ) ( /) ( /) ( )/ ( )/ ( )/ ( )/ Proof Wirstrass xprssio o th gamma fuctio was as follows. () (, ) (, ) log (,) log (,) (, ) (, )
From this, (, ) is obtaid immdiatly. Rvrsig th sig of i (, ), w obtai (, ). Rplacig with / i (, ) ad (, ), w obtai (, ) ad (, ). Nxt, dividig (, ) by (, ), Hr, usig th Lgdr duplicatio formula, ( / ) ( ) ( )/ Substitutig this for th abov, Ad, rvrsig th sig of, w obtai (, ). Nxt, ( /) ( ) / log ( )/ ( )/ log ( /) () log (,) Multiplyig ach sid of (, ) by this, ( )/ Ad, rvrsig th sig of, w obtai (, ). (. ) Givig to th formulas, w obtai th followig spcial valus. Spcial Valus 0.56598 (. ) 0.85508 (. ) 0.759950 (. ) 0.660555 (. ).800 (. ) From Formula.., th followig corollary is drivd
Corollary.. ( ) ( ) si si cos (.) (.) (.) ( ) ( ) (.) is a fuctio with all Itgrs as roots. (.) is a fuctio with all v umbrs as roots, ad (.) is a fuctio with all odd umbrs as roots. Ths ar all xprssd with a lmtary fuctio. ( ) is a fuctio with gativ odd umbrs ad positiv v umbrs as froots. ( ) is a fuctio with positiv odd umbrs ad gativ v umbrs as froots. Ths caot b xprssd with a lmtary fuctio. If both sids of ( ) ar illustratd, it is as follows. Th product is calculatd 500 trms. Sic both sids ar ovrlappd xactly, th lft sid (blu) is ot visibl. Compsatio Trm W discuss (, ) as a xampl. ( ) (, )
Although th compsatio trm is /, th compsatio trm is ot cssarily limitd to this. First of all, th complt compsatio trm for th right sid is. Howvr, if such a compsatio trm is adoptd, th fuctio of th lft sid bcoms a costat fuctio ad th utility valu disappars. This is th raso why th compsatio trm lik / is adoptd. I fact, if th body ad th compsatio trm ar illustratd, it is as follows. Th products ar calculatd 0000 trms rspctivly. Both ar vry clos but thy do ot ovrlap. Ev from this figur, it is hard to cosidr compsatio trm othr tha a xpotial fuctio. O th othr had, it sms that it may b othr xpotial fuctios. So lt us cosidr th followig xpotial fuctio. f() c c > Th cssity for th coditio c > is obvious. Aythig is okay as log as this is satisfid. For xampl, if f is illustratd for c., c.788, c., it is as follows. W ca s that th maximum valu of f is miimum at c. Actually, If c dviats from gratly, th maximum valu of f bcoms larg surprisig. Cosidrig this, / sms to b optimal as a compsatio trm.
. Maclauri Expasio Sic th rciprocal of th gamma fuctio is holomorphic ovr th whol complx pla, th Maclauri xpasio should b possibl. I this sctio, tratig th divrgt product ad divrgt sris as umbrs or fuctios, w aim at th Maclauri xpasio of th rciprocal of gamma fuctio. Formula.. Wh is EulrMaschroi costat, is th Gamma fuctio, H is a harmoic umbr ad H is a harmoic sris, th followig xprssios hold.! H H! H H HH H H (. )! H H! H H HH H H (. ) H / / H! H H! H H! H H! H HH H HH H H (. ) (. ) Proof From th Wirstrass xprssio o th gamma fuctio, ( ) Wh H is a harmoic umbr ad H is a harmoic sris, th compsatio trm is Th H H H! Nxt, assum that th body is xpadd as follows. H! a a a H! (. ) (.) 5
Th, from Vita's formulas, a H a a HH HH HH 5 5 6 5 6 7 HH HH HH Rarragig this alog th diagoal li, Thus, a 5 5 HH HH HH 5 5 HH HH 5 6 5 6 7 6 7 8 HH HH 5 5 HH 6 6 5 6 5 6 HH H HH H HH H HH H HH 5 6 7 6 7 8 7 8 9 HH HH H H HH (.p) Th right sid of (. ) is a product of (.) ad (.p). That is Takig th Cauchy product, H! H H! HH H! (.) H HH (.p) 6
Th cofficit of is H H! Th cofficit of is HH HH!! Th cofficit of is Th H HH! H H HH! H H H! H H H!! H HH HH H!! H H H! H H HH HH! HH HH H!! H! H H! H H HH H H (. ), (. ) ad (. ) ar asily drivd from this. (. ) Vrificatio Th ordiary Maclauri xpasio of th lft sid of (. ) is as follows. I additio, is a polygamma fuctio.! ()! 6 0.57757 0.655878 0.0006 Thrfor, th followig quatios must hold. H! H! H H! H 0.655878 HH H! 0.0006 () Sic th harmoic sris is ifiit, w us th followig limit valu for th vrificatio. lim m H m m H 6 7
Calculatig ths with th formula maipulatio softwar Mathmatica, w obtaid th followig rsult. Th cofficits of ach trm ar almost cosistt with th abov valus, so, th corrctss of this formula was umrically vrifid. Also, th followig spcial valus wr obtaid. Spcial Valus H H! ( ) : Rima Zta (.) 6 H! H HH H If asir (.) is writt dow, it is as follows. Not ().0 (.)! ( ) A doubl sris of H ad H appars i th cofficit of, ad a tripl sris appars i th cofficit of 5. Ths ar far mor complicatd tha otatio by polygamma fuctio. So, ths ar ot calculatd i this chaptr.. 8
. Factoriatio aroud Rplacig with i Formula.., w obtai th followig formula. Ths lft sids ca b xpad d to powr sris of. So, w call th formulas factoriatio aroud. Formula.. Wh is EulrMaschroi costat ad is th Gamma fuctio, th followig xprssios hold. () ( ) ( )/ ( )/ ( /) ( /) ( /) ( /) log log From this formula, th followig formula is furthr drivd. Formula.. Wh is EulrMaschroi costat ad is th Gamma fuctio, th followig xprssios hold. ( ) ( ) ( /) ( /) ( ) log (. ) (. ) (. ) (. ) (. ) (. ) (. ) (. ) (.5 ) (.5 ) (.6 ) (.7 ) 9
Proof Dividig both sids of (. ) by, i.. () ( ) Multiplyig both sids of (. ) by, i.. ( ) ( ) ( ) ( ) Dividig both sids of (. ) by /, i.. ( /) ( /) ( /) log ( ) Dividig both sids of (. ) by /, i.. ( /) ( /) ( /) log Givig to (.5 ), (.6 ), (.7 ), w obtai th followig spcial valus. (.5 ) (.5 ) (.6 ) (.7 ) Spcial Valus 0.760 (.5 ) 0.905 (.6 0.66055 (.7 ) ) 0
. Taylor Expasio (Part ) Formula.. Wh is EulrMaschroi costat, is th Gamma fuctio, H is a harmoic umbr ad H is a harmoic sris, th followig xprssios hold. Proof / H H! ( )! H H HH H H H H! ( )! H H HH H H H! ( ) ( ) / H H! H H H! H ( ) HH H! HH H H H H H ( ) H!! H H HH H H ( ) (. ) ( ) (. ) (. ) (. ) ( ) (.5 ) Rplacig with i Formula.. (. ) ~ (. ), w obtai (. ) ~ (. ). Multiplyig both sids of (. ) by, w obtai (.5 ). Formula.. Wh is EulrMaschroi costat, is th Gamma fuctio, H is a harmoic umbr ad H is a harmoic sris, th followig xprssio holds.! H H!!! H H HH H!!! H (.5 )
Proof Formula.. (.5 ) was as follows. ( ) At first, th compsatio trm is H (.5 ) Th, H H Expadig this, Lt th body b Th, from Vita's formulas, a H a a H ( )! H ( )! H ( )! 5 a ( ) a ( ) a ( ) 5 5 6 HH HH HH H HH HH HH 5 5 HH HH HH HH HH 5 5 HH 6 6 HH HH HH 5 5 HH 5 5 HH 6 6 HH 7 7 HH HH 5 5 HH 6 6 5 6 7 HH H (.)
HH Rarragig this alog th diagoal li, Thus, a HH HH HH H HH HH HH HH HH HH H H Th right sid of (.5 ) is a product of (.) ad (.p). That is H ( )! Takig th Cauchy product, Th cofficit of ( ) H ( ) HH HH HH H H HH (.p) H ( )! HH H is H H!! is HH H HH H!! Th cofficit of H H H H HH H!!!!! H Th cofficit of is as follows. H HH H! Hr, H H ( )! H HH HH H H H H!!
Th, Thrfor, HH H H H HH H HH H H H HH H H H H H HH H! H H HH HH H H H! HH!!!! H H HH H H H H!! r H H H H! H H H H H H H!! H H H H H!! H H HH H H!!! H H HH H!!! H H Vrificatio Th ordiary Taylor xpasio of th lft sid of (.5 ) is as follows. Whr, is a polygamma fuctio.!!! ()!!! 6 0.78 0.097 0.909 Thrfor, th followig quatios must hold. H!! H!! 0.09
!!! H!!! H H HH H () 0.909 6 Calculatig ths with th formula maipulatio softwar Mathmatica, w obtaid th followig rsult. Th cofficits of ach trm ar almost cosistt with th abov valus. 5
.5 Taylor Expasio (Part ) Odd Harmoic Numbr & Odd Harmoic Sris (dfiitio) W dfi odd harmoic umbr h ad odd harmoic sris h as follows. h 5 H H (5.h ) h h H log (5.h) Formula.5. Wh is EulrMaschroi costat, h is a odd harmoic sris, th followig xprssios hold. Whr, ( /) ( /) c log c! c! is th Gamma fuctio, h is a odd harmoic umbr ad c ( ) c ( ) c ( ) (5. ) c ( ) c ( ) c ( ) (5. ) log h h log h h log h h h log h Proof From Formula.. (. ) ( /) log At first, th compsatio trm is h (5. ) Th, log log h logh Expadig this, log ( )! log h! log h! log h (5.) 6
Lt th body b Th, from Vita's formulas, a 5 h a 5 7 a 5 5 7 a ( ) a ( ) a ( ) h h h h h h 5 h h h h 5 h h 7 5 Rarragig this alog th diagoal li, Thus, a h h h h h 5 h h 7 h h 5 9 5 h h 5 7 9 5 h h h h 7 h h 5 9 h h 6 5 h Th right sid of (5. ) is a product of (5.) ad (5.p). That is log ( ) Takig th Cauchy product, Th cofficit of! h h 5 7 9 h h h h 7 h h h (5.p)! log h! log h h is log h h log! log h h h h h h 7
Th cofficit of is h h h!! log h! h h log log h log h h h h Th cofficit of is as follows i a similar way. Th,! log h / h log log! log log h h h h h log h! h log h log log h Ad, chagig this to altratig sris, w obtai (5. ). h h h h log h Vrificatio Th ordiary Taylor xpasio of th lft sid of (5. ) is as follows. Whr, is a polygamma fuctio. / 0!! 0 6 0 8 0 8 8 0.98755 0.98 0.09786 Thrfor, th followig quatios must hold. log 0!! log h 0.98755 h! log h! 6 h log 0 8 0 0 8 0.98 h h h log h 8 0.09786 Calculatig ths with th formula maipulatio softwar Mathmatica, w obtaid th followig rsult. Th cofficits of ach trm ar almost cosistt with th abov valus. 8
Also, th followig spcial valus wr obtaid. Spcial Valus h! h! h ( ) : Dirichlt Lambda (5.) 8 h h h h If asir (5.) is writt dow, it is as follows. 5.059958 (5.5)! 5 Formula.5. Wh is EulrMaschroi costat, h is a odd harmoic sris, th followig xprssio holds. Whr, ( /) c log c! c! ( ) 5 is th Gamma fuctio, h is a odd harmoic umbr ad c ( ) c ( ) c ( ) (5.6 ) log! log! log h log! h h log h log h h h log h Proof I a similar way as th proof of Formula.., w obtai th dsird xprssio. 9
Vrificatio If th lft sid of (5.6 ) is xpadd to Taylor sris with th formula maipulatio softwar Mathmatica, it is as follows. / 0.08 0.668 0.099 O th othr had, calculatig c, c, c accordig to th formula, w obtaid th followig rsults. Both valus ar approximatly qual. 0
.6 EulrMaschroi Costat Fuctio Util th prvious sctio, tratig th divrgt product ad divrgt sris as umbrs or fuctios, w prformd a sris xpasio of th rciprocal of th gamma fuctio. Although w succdd i drivig th powr sris from th ifiit product, w could ot gt th gral formula. Howvr, thr ar ot fw mrits of tratig th divrgt product ad divrgt sris as umbrs or fuctios I this sctio, I will prst o xampl. Formula.6. Wh is EulrMaschroi costat ad is th Gamma fuctio, th followig xprssios hold. Proof log log ( ) g() R() > (6.) log (6.') lim 0 log 0 (6.) log ( ) lim 0 (6.) Formula.. (. ) was as follows. Th lft sid ca b trasformd as follows. (. ) ( ) log Th right sid ca also b trasformd as follows. log Th, log ( ) log ( ) ( ) log log ( ) Takig th logarithm of both sids ad ivrtig th sigs, From this, log ( ) Espcially, wh, log log log log ( ) R() > (6.) (6.')
Nxt, rplacig with i th lft sid of (6.), lim 0 log d log() d Th, (6.) holds. Ad (6.) is drivd from this ad (6.). ' () ( ) ( ) D figur of (6.) is as follows. Th lft figur is th ral part ad th right figur is th imagiary part. Th domai is R >. Aywhr o this half pla, th ral part is 0.577566 ad th imagiary part is 0. That is, this fuctio g is a costat fuctio which givs th EulrMaschroi costat. I ordr to obtai, ay valu o this half pla may b chos, ad (6.') is o of thm. Howvr, cosidrig th covrgc spd, (6.') is ot a good choic. Th followig is a study of th valu of ad th umbr of trms cssary to obtai 5 sigificat digits. Accordig to this, i ordr to obtai ffctiv 5 digits, 5,000 trms ar rquird wh, ad trms ar sufficit wh 0.000. Th rasos ar (6.) ad (6.). Wh this is illustratd clarly, it's as follows.
Th first trm of th fuctio g is blu ad th scod trm is yllow gr. Ths ar th li symmtry aroud g /. Thrfor, th fuctio valu of g is rgardlss of th valu of. So, w call th fuctio g EulrMaschroi Costat Fuctio. Although g for ay s.t. R >, spcially wh 0, th first trm is 0 ad th scod trm is. That is, i th viciity of 0, th rol of th first trm is small. That is why covrgc of th first trm is fast i th viciity of 0. By rfrc, givig 0.00008 to (6.), w obtai 0.5770 with th sam prcisio as th abov. Not (6.') ca also b obtaid dirctly from Eulr's dfiitio. m m lim m log m lim m m m lim m log log m m log 06..0 Ali's Mathmatics Kao. Koo