Taming the post-newtonian expansion: How to simplify high-order post-newtonian expressions

Transcription

1 Tmig post-nwtoi xpsio: How to simplify high-ordr post-nwtoi xprssios Nth. Johso-McDil SFB/T7 Vido Smir Thortisch-Physiklischs Istitut Fridrich-Schillr-Uivrsität J

2 Motivtio: Itroductio to PN pproximtio Th post-nwtoi (PN pproximtio [ combitio post-mikowski olirity itrtio i powrs G roud flt spc with slow motio post- Nwtoi xpsio propr i powrs /c] is stdrd lytic tool i study grvittiol wvs from compct biris d hs providd my orticl isights to dt. Howvr it is ot possibl to itrt post-nwtoi xpsio comprbl mss biris to rbitrrily high ordrs with prst tchology: Th currt stt---rt.pn [O(v 7 i.. O(G. pst Nwtoi prdictios] my qutitis with som qutitis kow to highr ordrs. chig.pn ivolvd dcds work by my rsrchrs d t w cocptul isights. Th rct dtrmitio 4PN Hmiltoi by Dmour Jrowski d Schäfr ws mjor tour d c!

3 Itroductio: Th PN pproximtio i BHPT Th situtio is rr diffrt wh o cosidrs xtrm mss-rtio cs whr o hs vry light compio i orbit roud (suprmssiv blck hol. [Ths Extrm Mss-tio Ispirls (EMIs r thought to m i glctic ctrs d r importt sourcs spc-bsd GW dtctors.] Hr o c comput with blck hol prturbtio ory d clcultig highr-ordr trms is much sir. NASA/GSFC I prticulr mlism dvlopd by Mo Suzuki d Tksugi (bsd o Lvr s work o computig BH qusiorml mods itslf bsd o rltivity rly work i qutum mchics [Prog. Thor. Phys (996; (996] llows o to clcult to rbitrrily high PN ordr i pricipl. Th currtly stt---rt is PN clcultio rgy flux from poit prticl i circulr orbit roud Schwrzschild blck hol by yuichi Fujit [Prog. Thor. Phys (0]. (Goig to v highr ordrs is limitd by vr-icrsig dmds o computr tim d mmory.

4 Itroductio: Th PN pproximtio i BHPT (cot. 4 log 0 ( / Post-Nwtoi Thory Post-Nwtoi Thory & Prturbtio Thory EMIs r quit rltivistic so it is ot possibl to modl ir tir volutio ccurtly with PN rsults though high ordrs c do quit wll 0 Numricl ltivity 0 Prturbtio Thory 4 log 0 ( / From Blcht Dtwilr L Tic d Whitig Phys v. D (00 Prhps v mor importtly s highrordr prturbtiv rsults c giv importt isights v i comprbl mss cs (cf. rct work o ffctiv o-body multio 4

5 Complxity high-ordr PN rsults High-ordr prturbtio ory rsults grlly disply combitoril icrs i complxity d PN rsults r o xcptio.

6 de de dt Complxity dt high-ordr PN rsults ppl Nwt 6 v 4 v v4 67 v ppl ppl High-ordr prturbtio ory rsults grlly disply log 6848 log v9 0 combitoril icrs i complxity d log PN v v0 rsults r o xcptio log log log v v log 7 0 log v 68 v6 04 v7 ppl log log ppl log log v log v ppl log 0 ( log log log log log ppl log 4 log log v v ppl log ( log 8480 log log log log log 8480 log log v log v v 4 9 log log log log log v log v v ( log log ( ( log ( log ( ( log 48 6 log v 48 log 7 v v 44 O v 4 log log whr AB dots rtiol umbr with A digits i umrtor d B digits i domitor. 949 trms t O(v 44 ; 97 trms totl 6

7 Simplifyig PN xpsio rgy flux How c w simplify this xprssio? Prhps w should first look t idividul mods strtig with domit qudrupolr ( mod 7

8 Simplifyig PN xpsio rgy flux 07 v 4 v 4784 ppl v4 v log 7 0 log v v6 96 v7 ppl log log v v ppl log 6848 log v v9 0 ppl log ppl log 776 How c w simplify this xprssio? 0 Prhps w should first look t idividul mods log v v0 strtig with domit qudrupolr log v ( mod v ppl log 0 ( log log log log log v log v v ppl log 7608 log v v 89 ppl log ( log log log log log v log v v ( log 78 ( ( log ( ( ( log 48 6 log v 48 log 7 v v 44 O v 4 60 trms t O(v 44 ; 4947 trms totl 8

9 Simplifyig mod rgy flux Wll ututly mod ws t much simplr th full rgy flux (d or mods shr bout sm complxity primry simplifictio cm bout bcus w ow oly hv logrithms d v. W c first try substitutio whr w itroduc stdrd PN (til-rltd ulrlogm fuctio ulrlogm γ log(mv d writ rmiig logrithms i trms log(v. 9

10 Simplifyig mod rgy flux Wll ututly mod ws t much simplr th full rgy flux (d or mods shr bout ppl v 4 v v4 v sm ppl complxity primry ppl simplifictio cm bout bcus w ow oly hv logrithms d v. 89 ulrlog v ulrlog v ulrlog v 8 ppl ppl ulrlog v W c first try substitutio whr w itroduc ulrlog stdrd PN (til-rltd ulrlogm fuctio ulrlog v 4 ulrlogm γ log(mv ulrlog v 6 96 v ulrlog v 9 ppl ( ulrlog ppl ppl ulrlog v ( ( ulrlog 48 ( log (v 78 ( ( ulrlog ( [ ( ( ] ulrlog 48 ulrlog 7 [ 4 0 (7 ] log(v 7 log (v 7 log (v v 44 O v 4 ppl d writ rmiig logrithms i trms log(v. ppl 7 trms t O(v 44 ; trms totl 0

11 Th Dmour-Ngr til fctoriztio (Tlm Wll substitutio ws ffctiv i producig som simplifictio but r r still my trms lft Lt us cosidr fctoriztio η/ T whr T mv ( ` imv ( `.

12 Th Dmour-Ngr til fctoriztio (Tlm Wll substitutio ws ffctiv i producig som ppl T 07 v 4784 v4 ppl simplifictio ppl but r r ppl still my trms lft Lt us cosidr fctoriztio η/ T whr ( ulrlog ulrlog v ulrlog v 6 89 ulrlog v 0 ulrlog ulrlog v 8887 Th prvious xprssios oly wt to v ulrlog v ( ppl ppl ( T ( ulrlog ` mv imv ulrlog ppl ( 67 ` ( ulrlog log(v v 6 ( ( ulrlog ulrlog log(v v v ( ulrlog 48 ( log (v 80 ( ( 74 4 ( [ ( ( ] ulrlog 48 ulrlog 7 [ 4 0 (7 ] log(v 7 log (v v 44 O v 4 First odd powr v is ot rmovd 7 trms t O(v 44 ; 88 trms totl

13 Th Dmour-Ngr til fctoriztio (Tlm Wll substitutio ws ffctiv i producig som ppl T 07 v 4784 v4 ppl simplifictio ppl but r r ppl still my trms lft Lt us cosidr fctoriztio η/ T whr ( ulrlog ulrlog v ulrlog v 6 89 ulrlog v 0 ulrlog ulrlog v 8887 Th prvious xprssios oly wt to v ulrlog v ( ppl ppl ( T ( ulrlog ` mv imv ulrlog ppl ( 67 ` ( ulrlog log(v v 6 ( ( ulrlog ulrlog log(v v v ( ulrlog 48 ( log (v 80 ( ( 74 4 ( [ ( ( ] ulrlog 48 ulrlog 7 [ 4 0 (7 ] log(v 7 log (v v 44 O v 4 First odd powr v is ot rmovd Slight simplifictios complxity 7 trms t O(v 44 ; 88 trms totl

14 Th Slm fctoriztio Th Dmour-Ngr fctoriztio succssfully rmovd som trms (otbly ll odd powrs v up to v d lrgst powr π but r is still plty complxity rmiig icludig lots trscdtls d log v trms. Howvr it is possibl to modify Dmour-Ngr fctoriztio slightly d rmov my mor trms. Spcificlly w us S : (mv mv [ imv ] [ ] Frctiol prt rormlizd gulr momtum ν which is fudmtl to MST pproch (sris i v 6 4

15 Th Slm fctoriztio Th Dmour-Ngr fctoriztio succssfully rmovd som S trms (otbly ll odd powrs v up to v 07 v 4784 v v6 867 v v v ppl v 4 log(v v 6 ppl ulrlog 64 log(v v 8 d lrgst powr π but r is still plty complxity ppl rmiig icludig lots trscdtls ulrlog log(v v ppl ( ulrlog log (v v 7088 ppl v d log v trms. Howvr it is possibl to modify Dmour-Ngr 768 ulrlog 07 ulrlog ulrlog log(v fctoriztio slightly log(v d rmov my mor trms v Spcificlly w us Th xprssio Tlm oly wt to v 8 S : (mv [ imv ] mv [ ] 7 v ( log (v v ( ulrlog 68 ( log(v ( ( 0 ( ( log (v ulrlog 7 ulrlog [ 4 (7 ] log(v 7 log (v v 44 O v 4 log(v 70 trms t O(v 44 ; 0 trms totl

16 Th Slm fctoriztio Th Dmour-Ngr fctoriztio succssfully rmovd som S trms (otbly ll odd powrs v up to v 07 v 4784 v v6 867 v v v ppl v 4 log(v v 6 ppl ulrlog 64 log(v v 8 d lrgst powr π but r is still plty complxity ppl rmiig icludig lots trscdtls ulrlog log(v v ppl ( ulrlog log (v v 7088 ppl v d log v trms. Howvr it is possibl to modify Dmour-Ngr 768 ulrlog 07 ulrlog ulrlog log(v fctoriztio slightly log(v d rmov my mor trms v Spcificlly w us Th xprssio Tlm fctoriztio oly wt to v 8 S : (mv [ imv ] mv [ ] 7 v ( log (v v ( ulrlog 68 ( log(v ( ( 0 ( ( log (v ulrlog 7 ulrlog [ 4 (7 ] log(v 7 log (v v 44 O v 4 log(v 70 trms t O(v 44 ; 0 trms totl Substtil dcrss i complxity 6

17 Th Vlm d Vʹlm fctoriztios Whil Slm fctoriztio producs mximum simplifictio o c likly hop ( simpl itgr PN sris with rtiol cofficits it oly dos so up to crti ordr du to structur xpsio. [Howvr s w shll s ordr to which it producs this complt simplifictio icrss s l icrss.] C w simplify furr? Ys though hr simplifictio is ot rly s drmtic d fctoriztio is rr mor ivolvd. 7

18 imv [ ] [ imv ] `. [[W show dpdc ] imvwhr [ : ] imv o ` m d v xplicitly clrity v though o B. Th V fctoriztio it is ot customry i litrtur to do this.] log(v ] i mv SV dom m [ulrlog m V um V : dom V Th Vlm d Vʹlm fctoriztios (cot. TABLE I: Th vlus [` ] [ (` i Bii d Dmour []](9b q ` (q ` [` ] s` d (s` [` ] ` 6. For ` { 6} mov som mor trscdtls d log-sic y cot b dtrmid from PN rgy flux xprssios. W w do ot giv vlus lst two qutitis ditiolly out V giv fctorig prim fctoriztios [whr (q ` [` ] d (s` [` ] i ordr to illustrt ir structur. ` ] B um q (q ` [` ] s` (s` [` ] ` [ ] 6 [ 7 ` um imv V0 (8 q (v : [ [7 4 (9c ] um ` Bqdom log(v SV A4dom ] xp (v um ` V : imv V dom V Bum ; [` ] imv ] imv ] Fixd490 by rquirig fctoriztio rmov crti trms C. Th V fctoriztio imv [ ] [ imv ] V dom imv ] ` i : is` (4mv ] imv [ ] [ imv ] imv (9 Morovr o c rmov rmiig odd powrs o imv ` um v by mkig substitutio is xp [ulrlog log(v ] i mv SV dom ` (4mv m m imv # s`! s` [s ` ]k (mv k (9b ( k ( : A {[ ] [ ] } B imv whr o fixs [s ] i S VV ` k by dmdig fc 96(`k imv toriztio rmov v trm from. [A W will Bum ; Adom um 0 us V to dot V with substitutio i Eq. (. This substitutio clowst b usd to rmov odd powrs v d givs Vʹlm Hr o c oly fix fw sllcofficits O c lso rmov trms with similr sris qlm (i v but this usig PN rgy flux xprssios obtiig (9c Bdom ] is ot quit ffctiv so w 0 with q s` Q costts sr dtrmid bydo r-ot cosidr it furr. C. Th V fctoriztio quirig fctoriztio crti trms. Th ( [s ] rmovs it is possibl to dtrmi vlus s costts Morovr o c rmov rmiig odd powrs 8 from PN xpsio totl rgy flux v by mkig substitutio r [s ]itbl I whr w writ q giv q ` with #

19 imv [ ] v imv ] log (v v [ um V : `. [[W show dpdc ] V : dom imvwhr [ 768 imv 07 ] ( ulrlog ulrlog ulrlog log(v V o `70 m d v xplicitly clrityv though 994 o B. Th V fctoriztio litrtur 4 it is ot customry i to do this.] log(v ] i mv S dom log(v log (v v v m [ulrlog V m Th Vlm d Vʹlm fctoriztios (cot. (` i Bii d Dmour []] q (q [ ] s d (s [ ] 8 ` 6. For ` { 6} TABLE I: Th vlus [ ] [ ` ` ( ` log(v ` ` ` trscdtls d log 7 ( ulrlog (9b ` 68 7 mov som mor w do ot giv vlus lst y cot b dtrmid from PN rgy flux xprssios. W two qutitis sic ditiolly V ( ( whr out 4 (9 60(s` [ ` ( log (v ulrlog 0 giv fctorig prim fctoriztios [ d ] 44i ordr to illustrt ir structur. ` ] (q ` [ ` ] um 7 ulrlog [[ q (7 ](q log(v 7s`log (v (s v 44` `O] v 4 ` ] [ ] [ 4 ` ` ` ` V [ ] 6 [ 7 ` um 07 7 B V : imv V07 imv (8 q (v : dom [ [7 V 4 (9c ] v 7 v 7 v 6 v 84 v 4 4 v ` 0 S V ; A4dom um Bqdom log(v SV Bum ] xp (v um imv ] imv ] v v 6 crti Fixd by rquirig fctoriztio rmov trms Purly rtiol cofficits to o ulrlog log(v v ordr highr th log(v v0 with Slm C. Th V fctoriztio ulrlog imv [ ] [ imv ] V dom imv ] ` i : is` (4mv imv ] imv [ ] [ ulrlog log(v v imv ] ( Morovr o c rmov rmiig odd powrs o imv ` um v by mkig substitutio is xp [ulrlog log(v ] i mv SV dom ` (4mv ( m m imv # (9b ulrlog ulrlog ulrlog log(v log(v s`! s74064 [s ` ]k (mv k ( ` Simpl itgr PN sris log (v v 4 k 66 to ll ordrs kow! ( ] : 8 S A {[ [ ] } B imv imv V ( ulrlog ( log(v i V whr ofixs [s ` ]k by dmdig fc(9c 96(`k 4 0from ( ( log (v ulrlog toriztio rmov v ( ( trm Wwill [Aum Bum ; Adom 0 VThis to dot Vulrlog withb i Eq. substitutio c usd to rmov ll powrs v d givs 7 [substitutio (7 odd ](. log(v 7 log (v v 44Vʹ lm O v 4 4 us Hr o c oly fix lowst fw s cofficits usig PN rgy flux xprssios obtiig Bdom ] 0 with q s` Q 70 costts r 44 dtrmid by rc. Th V fctoriztio trms t O(v Sm complxity v trm quirig [sm fctoriztio crti trms. Th [s ] Slm rmovs ( s fctoriztio] it is possibl to dtrmi s with vlus s costts Slm fctoriztio Morovr o c rmov rmiig odd powrs but oly 60 trms 9 from PN xpsio totl rgy flux r 4 or v by mkig substitutio though trms r simplr [s ]itbl I whr w totl giv writ q q ` with #

20 Tkig logrithm Th fil mthod simplifictio w cosidr cosists cosisttly xpdig logrithm giv PN ordr. This is prt xpotil rsummtio itroducd by Isoym t l. [Phys v. D (0] s wy to improv covrgc full rgy flux [d sur its positivity r horizo i rr cs]. This dos ot produc drstic simplifictio lowst ordrs d complt rmovl odd powrs v providd by S lm Vʹlm fctoriztio but still mgs to rmov trms this fctoriztio dos ot. Of cours it is possibl to tk xpd logrithm S lm Vʹlm fctoriztio d obti most sigifict simplifictio w hv foud. (This givs mximum simplifictio both css with totl oly 6 trms mod. 0

21 Tkig logrithm Th fil mthod simplifictio w cosidr cosists cosisttly log 07 ppl xpdig v 4 v logrithm v giv 0 PN ulrlog v ordr v v0 ppl ( ulrlog v ppl This is prt xpotil rsummtio itroducd by Isoym t l. v log(v v 6 [Phys ppl v. D (0] s wy to improv covrgc ( full rgy 487 flux ( [d sur its positivity r horizo i rr cs]. ppl ulrlog log(v v ppl 707 log(v v v ( complt rmovl log(v odd powrs v providd by S 487 ppl lm Vʹlm log (v v ot ( ( (7 ulrlog ulrlog 644 ulrlog log(v log(v log (v v ( log(v 0 log (v 9 8 This dos ot produc drstic simplifictio lowst ordrs d fctoriztio but still mgs to rmov trms this fctoriztio dos Of cours it is possibl to tk xpd logrithm S lm Vʹlm fctoriztio d obti most sigifict simplifictio w hv foud (This givs 6 ( ( mximum 4 (9 simplifictio [ 60 6 (7 both ] log(v css with totl oly 6 9 log (v v 44 O v 4 trms mod. oly trms t O(v 44 d 4 trms totl Oly odd powrs v m v 9 6(l t highr ordrs All sigifict complxity [d ll pprcs ulrlogm] cid to powrs v divisibl by 6

22 Comprig simplifictios ( mod Numbr trms through this ordr Origil ulrlog substitutio T lm fctoriztio Logrithm S lm fctoriztio S lm V lm fctoriztio S lm V lm fctoriztio S lm V lm fctoriztio log Simpl itgr PN sris η PN ordr

23 Comprig simplifictios mods with l 7 8 Numbr trms through this ordr Origil ulrlog substitutio T lm fctoriztio Logrithm S lm fctoriztio S lm V lm fctoriztio S lm V lm fctoriztio S lm V lm fctoriztio log Simpl itgr PN sris η Numbr trms through this ordr η Sris comprssio rtio 00 0 ulrlog substitutio T lm fctoriztio Logrithm S lm fctoriztio S lm V lm fctoriztio S lm V lm fctoriztio S lm V lm fctoriztio log Simpl itgr PN sris η Sris comprssio rtio 00 0 η PN ordr PN ordr PN ordr PN ordr η 44 η η 44 η Numbr trms through this ordr Numbr trms through this ordr Sris omprssio rtio 00 0 Sris comprssio rtio PN ordr PN ordr PN ordr PN ordr η 66 η 77 η 66 η 77 Numbr trms through this ordr Numbr trms through this ordr Sris comprssio rtio 00 0 Sris comprssio rtio 00 0 Sm s simpl itgr PN sris Sm s simpl itgr PN sris PN ordr PN ordr Numbr trms through giv PN ordr PN ordr PN ordr tio umbr trms through giv PN ordr. : Th totl umbr trms up to giv PN ordr ` m mods rgy flux up to ` 7 usig Fujit FIG. : Th sris comprssio rtio vrious PN ordrs (i.. umbr trms i simplifid sris up to giv N rgy flux xprssios. ot w hv ot icludd lowst-ordr Nwtoi trm simplicity. Hr w show ordr dividd by umbr trms i origil sris up to ordr vrious simplifictios d simpl umbr trms i xprssios giv by Fujit oli [4] ( Origil d umbr trms ftr vrious itgr PN sris with sm ottio d commts s Fig. plifictios or fctoriztios: Hr ulrlog substitutio dots rsult ulrlog m dlog(v substitutios i simplifictio to umbr trms i origil ch r lso usd ll or vrsios show. Th or simplifictios r rsults Dmour-Ngr T

24 Why do s simplifictios simplify? O c udrstd whr s simplifictios com from [d why y do ot simplify compltly] vi study structur MST mlism. [Idd Vlm fctoriztio ws obtid i this mr by logy with Slm fctoriztio which ws itslf obtid by study prim fctoriztio cofficits PN xpsio mods d logy with Dmour-Ngr Tlm fctoriztio.] Howvr w shll ot giv spcifics hr d oly ot trscdtlly structur Slm c b udrstood by otig so ( z xp z S xp ulrlog m mv S S S S : ( # ( ( z # # imv [ ] Thr is similr xprssio Vlm ivolvig log(v i dditio to ulrlogm 4

25 coti b Eulr-Mschroi gmm d ow commo i costt litrtur. Not to w c us jcturd to trscdtl so w shll rfr m (S Mmtic 74` ot PN xpdig log producs cosidrbl trscdtl so w shll rfr too(v m whr Q. Thus through otitig o stdrd gmm fuctio idtity (from rflcog log m (with log v ( log(! log(v log v (b h i im zt fuctio vlutd t odd itgrs. Ths k simplifictio such. its lytic This ms ` 7i this fctoriztio ll s This ` this fctoriz(sm. ms tio mul ( iz 7 sih z [(6.. z/ 74` v covrgc outpn S totl rmovs llflux rsults trscdtls d whr ot Q. to Thus through O(v log fctorig rldig b or i my W (b lso umbrs cosidr logrithm tiltrscdtl fctoriz kow rgy ito ( z Abrmowitz d Stgu [4]] with gmm fuc il fctorizk this ordr. rs d logrithms v d lvs pur itg rr cs.(ihr w tiocss turs totl rgy rsults ordr PNito PN tio itroducd by Dmour d Ngr [7] which isy givflux tio s rcurrc rltio to writ itgr sris. Morovr v irrtiol. Howvr r rtiol strogly coout S rmovs ll trscdtls d odd pow Aor wy simplifyig s rsults is rltd to sris with rtiol cofficits. Ths hich is giv oducs cosidrbl to # coti.b.: by / T purly whr ordr trms still trscdt jcturd to b trscdtl so w shll rfr m such rtiol itgr ordr PN sris. Morovr rs d logrithms v d lvs pur itgr ordr PN which givs Y Eulr-Mschroi mv 4 mv work Isoym t l. [8] who dvoctd PN xcoti gmm costt trms r by this fcto g s rsults is rltd toll ssris T.B.: ` sigifictly simplifid ( with rtiol cofficits. Ths xprssios such. This ms 7 this fctoriz 4 mv k pdig log hde/dri d xpotitig (with o v highr-ordr trms still coti trscd ( ` imv im zt fuctio vlutd t odd itg S xp mv illustrtd i Sc. V. O obtis sm sim k [8] whotil dvoctd PN x-tio rithm fctoriz T. ( coti Eulr-Mschroi gmm costt d PN totl rgy fctorig ot ito xpsio s wyturs to improv (covrgc flux rsults umbrs r kow to this b trscdtl `r upo out by S from h (i.. o h xpotitig (with o tls d log v trms sigifictly simplifid gr [7] which is giv [cf. Eq. odd Dmour Iyr d Ngr [9] which zt vlutd t itgrs. Ths such purly rtiol itgr ordr PN sris. Morovr sris d sur its im positivity i fuctio rr cs. Hr(9 w css v irrtiol but r strogly grvittiol wv mods gicojct ll scl rov covrgcthis prsts this xprssio i slightly di rt m]. xprssio lso hlps xpli why xpdig logrithm umbrs r ot kow to b trscdtl or i my fctoriztio sproducs illustrtd i coti Sc.trscdtl V. O Th PN full xpdig vrsio log [givstill i.g. ot fctoriztio v highr-ordr trms cosidrbl trscdw shll rfr ldig trm obtis souity s o would xpct fr.ity i ( Sto : ll rr Sm cs. Hr w css v irrtiol. Howvr y rd strogly coproducs such sigifict simplifictio. Eq. (6 Dmour d Ngr [7]] lso icluds phs simplifictio tls itssimplifictio lytic m. This ms ` h7 this fctoriztio (. d log v trms r sigifictly simplifid by this sm upo fctorig out S from imv producs cosidrbl jcturd totologrithm b so d w torgy ms is ot rlvt trscdtl prst A. shll Th Srfr fctoriztio Wfctor lso cosidr ldig tildiscussio fctoriztotl W c writ m illus PN fluxi rsults bttr su ito [d why y do ot simplify compltly] fctoriztio s illustrtd i Sc. V. O obtis. tio itroducd ( m. (i..by mplitud ms grvittiol mods smorovr ws Additiolly itroducd with slightly di rt ottio T ll s such. This[7] 7this fctoriz `is rtiol Dmour d Ngr which isxprssios giv itswv structur d mks fstr comp itgr ordr PN sris. v w c compr T d S to lm lm ` sm simplifictio upo fctorig out S from h ig logrithm til fctoriz i.g. simplifyig fctoriztio w itroduc is giv ow commo itio turs litrtur. Nottotl Th wfirst c us rsults PN rgy flux ito by / T whr Mmtic usigcoti xpsio ordr trms still trscdtls o would xpct from Eqs. ( d (. s why lttr rmovs so my mor trscdtls usig (i.. mplitud grvittiol wv mods s d Ngr [7] which is giv stdrd gmm fuctiortiol idtity itgr (from rflcsuch purly ordr PN trms sris.rmorovr sigifictly simplifid by this fctor# uds phs tio [giv i.g. obtid ithis mr by logy with Slm fctoriztio [Iddtio Vmul ws which lm fctoriztio would ( iz z/ sih [(6.. i bttr oc xpct from Eqs. ( d (. W writ S i m illustrts som highr-ordr ( ` imv z v trms still coti trscdo c sim rm ( mv illustrtd i Sc. V. O obtis sm [ imv ] mv lso icluds phs T. ( ws itslf obtid by study prim fctoriztio cofficits PN xpsio ( z xp z ( z S : (mv (4 cussio d Abrmowitz d Stgu [4]] log with gmm fuc rithms by ddi W cd writ m bttr illustrts som tls log trms r sigifictly simplifid by this (Sv `i mks [ ] upo fctorig out S from h (i.. its structur d fstr computtios i mods logy withrltio Dmour-Ngr fctoriztio.] rst d ( ` discussio d tio s rcurrc to writ fctoriztio s illustrtd i Sc. V. O obtis io Timv is its structur d mks fstr computtios i grvittiol wv mods gi ll scl. ( whr : `. [W show dpdc # Mmtic usig xpsio t ottio T ( ` Th full isvrsio sm fctoriztio [giv i.g. outldig fctorig simplifictio upo h v trm uity sthough o would xpct fro which givs from Mmtic usig xpsio Y o `(ot m d v S xplicitly clrity itgr w c us mv 4 mv ot weq. c (6us TDmour d [7]] lso phs icluds ( d (i..ngr mplitud grvittiol mods s this.] it is ot customry iwv (. litrtur to do 4 mv # discussio k to prst ctoriztio [giv i is.g. fctor ot rlvt S# writ xp ulrlog mv tity (from rflcm rflc W c S i m bttr illust o wouldk xpct from Eqs. (d d (. m [7]] zlso icluds phswith slightly di rt ottio T ( ( ws itroducd isbttr structur /gr sih [(6.. i zillustrts its d mks fstr compu som W c writ S i m ( z xp z ( (6 [(6.. i ( z xp z ( z (6 [cf. Eq. (9 Dmour Iyr d Ngr [9] which o prst discussio ow commod i litrtur. Not d mks w cus Mmtic usig xpsio lm with gmm fucits structur fstr computtios i prsts this xprssio i slightly di rt m]. ymm di rtfucottio T is stdrd fuctio idtity (from rflc ( gmm [ ] [ #[ Mmtic usig xpsio um ` imv S : S V : q (v ur. Not c us tio w mul ( iz z/ sih z [(6.. i [ ( ] [ which givs # # ( z xp z idtity Abrmowitz d Stgu [4]] log with gmm fuco ` ( z um q (v xp log(v SV (from rflc- A. Th S fctoriztio ( mv which givs rcurrc writ z xp z [(6.. i rltio to ( z ( z (6 # z/ sihtio s ( ulrlog mv SS k with gmm fuc-s xp # m ] log Y Th first simplifyig fctoriztio witroduc is giv which givs (7 mv 4 mv writ ( Tusig B. Th V fctoriztio ( S xp ulrlog mv S S 4 mv m which givs k # S xp ulrlog mv d Ngr [9] which k m Y mv ( imv ] som mor trscdt O c rmov y di rt m]. S : mv [ ( SSDmour : xp d [9] m imv [ S ]. (mv (4 ulrlog mv S S Iyr rithms by dditiolly fctorig out V w (9 Ngr which k[cf. Eq. [ ] [9] which (7 prsts this xprssio i slightly di rt m]. ( um Thris similr Vlm log(v isdditio to ulrlog V [ ivolvig (7b m S : imv ( : xprssio `. [W show dpdc tiyr m]. V : d Ngr whr [9] which Why do s simplifictios simplify? O c udrstd whr s simplifictios com from vi study structur MST mlism. Howvr w shll ot giv spcifics hr d oly ot trscdtlly structur S c b udrstood by otig so

26 How do s fctoriztios d rsummtios improv covrgc? η r 0 6M η r 0 6M Frctiol rror Origil T lm fctoriztio S lm fctoriztio S ~ lm fctoriztio S~lm fctoriztio Exp. rsummtio Frctiol rror Origil S ~ lm fctoriztio S ~ lm V lm fctoriztio S~lm fctoriztio S~lm V lm fctoriztio Exp. rsummtio PN ordr PN ordr Frctiol rror η r 0 0M PN ordr Frctiol rror η r 0 6M PN ordr FIG. : Covrgc di rt fctoriztios d rsummtios d computd orbitl rdii r 0 6M ( ISCO both mods d lso r 0 0M ( covrgc vrious vrsios this rdius r sm s t ISCO xcpt mor rpid. W compr to fluxs computd by Fujit d Tgoshi [] r 0 6M but compr to highst-ordr vlu xpotil rsummtio r 0 0M sic it is highr ccurcy th Fujit d Tgoshi computtio such somwht lrgr rdii. w W us thus do otistd. plot PN poit xpotil rsummtio r 0 0M. Th ottio di rt fctoriztios is itroducd i txt. Th lgd i uppr lft-hd Compriso with umricl rsults from Fujit d Tgoshi [Prog. Thor. Phys. 4 (004] r0 6M ( ISCO. For r0 0M highst-ordr vlu xpotil rsummtio is mor ccurt th ir publishd rsults so 6

27 Morovr v d (v trms i / S 0 which r rmovd by V fctoriztio 0 lso hv 0 co cits r vry closly rltd to - co cit ulrlog v 0 0 viz./ d tims - it rspctivly. How do s fctoriztios d rsummtios 0 - improv Filly lt us cosidr covrgc? ovrll simplifictio highst-ordr trms i PN xpsio producd by S V fctoriztio xpdig 0-6 logrithm d combiig two: Th origil 0-7 vrsio hs 7 trms ch i co cits 0-8 v 4 d v η r 0 6M d 9 trms i co cit v v ftr 0 - ulrlog m d log(v 0 substitutios. -0 Th S V fctoriztio d logrithm both rmov - v 4 cofficit compltly (though rcll - S V 0 fc- Frctiol rror toriztio rmovs -4 Origil ll odd powrs v whil Origil PN ordr S ~ lm fctoriztio S d ~ Tgoshi. lm V lm fctoriztio S~lm fctoriztio 0 - T lm fctoriztio 0 logrithm oly rmovs S lm fctoriztio thos r ot m S ~ lm fctoriztio 0 v 96(` N -6 0 so.g. v 9 FIG. trm : rmis. Howvr y simplify v 4 d v 44 co cits slightly -7 Exp. rsummtio 0-7 S~lm fctoriztio di rtly: Th S V -8 0 fctoriztio givs 70 trms i both co cits 0-9 whil logrithm givs 7 trms i PN ordr co cit v 4 d oly i co cit v 44. Combiig 0 0 two simplifictios givs (s might 0 - η r 0 0M b xpctd miimum umbr trms i both co- 0 - cits viz. 70 i v 4 d i v Sic 4 is divisibl -4 by 6 logrithm dos ot r- 0 - duc complxity zt vlus prst i co cit v 4 i : ( is prst through 4 i Frctiol rror logrithm s it -8 is i origil whil mximum 0-9 Frctiol rror 0-4 Frctiol rror Frctiol rror which ( isprstwiths V 0 fctoriztio is 9. Howvr 0 with both - S V 0 fctoriztio [ ` [ ` imv ] S : (mv ( -6 (`! ] [ ` ] d logrithm - mximum powr ulrlog S : i co cit v 4 S : (mv (`! (mv (`! [ [ ` mv ` mv imv imv ]. ] `! [ ` ]. dcrss PN ordr from 7 to s dos PN ordr `! [ ` ] mximum ordr products prst though i- (b Compriso with umricl rsults from Fujit d Tgoshi [Prog. Thor. Phys. 4 (004] r0 6M ( ISCO. For r0 0M highst-ordr vlu xpotil rsummtio is mor ccurt th ir publishd rsults so gt tst covrgc. For ISCO w c compr with fluxs clcultd to digits by Fujit d Tgoshi [] through ` 6. Thy 0 - η r 0 0M lso clcultd s fluxs to sm ccurcy r 0 0M butr 0 - covrgc sris is rpid ough its vlu is ccurt to mor th digits 0 - t highr ordrs so w mrly cosidr its slf-covrgc comprig with 0 highst-ordr vlu xpotil -4 rsummtio kow from PN rgy flux rsults sic w fid 0 - xpotil η r 0 6Mrsummtio givs clrst im- 0-6 provmt covrgc ll di rt fctoriztios d rsummtios w cosidr. 0 ot w -7 fid sm qulittiv bhvior covrgc t ISCO wh w cosidr this0sort -8 slf-covrgc r istd comprig with rsults from Fujit origil sris i my css whil T fctoriztio dos PN ordr improv covrgc (though ot s much Bcus s xpotil origil rsummtio Slm fctoriztio w lso ctully cosidr 0 two - η r 0 6M worss ltrtivs rt to S covrgc icludi most ` trms css ccompyig i gmm fuctios (sic similr fctors w 0 - itroducd followig dditiol vrsios with l r prst i T viz. 0 cotributios - o might rsobly xpct from MST S : (mv mv mlism (`! (`! FIG. : Covrgc di rt fctoriztios d rsummtios d computd orbitl rdii r 0 6M ( ISCO both mods d lso r 0 0M ( covrgc vrious vrsios this rdius r sm s t ISCO xcpt mor Spcificlly rpid. W compr S cotis to fluxs computd costts byd Fujit ` trms d Tgoshi i [] r 0 6M but compr to highst-ordr vlu xpotil rsummtio r 0 0M sic it is highr ccurcy th Fujit rgumts d Tgoshi computtio such somwht lrgr rdii. w gmm W us thus fuctios do otistd. com from cosidrig di rt fctoriztios gmm fuctios is itroducd prst i i txt. C /A Th lgd which i uppr plot PN poit xpotil rsummtio r 0 0M. Th ottio lft-hd Frctiol rror η r 0 6M PN ordr Covrgc di rt fctoriztios d rsummtios d computd orbi ISCO both mods d lso Sic r w fid S fctoriztio dos ot improv fctoriztio ( covrgc vrious vrsios S~lm V 0 0M lm sm s t ISCO xcpt morexp. rpid. rsummtio covrgc W compr (dtoidd fluxs mks computd it lss by rpid Fujit th d Tgos compr to highst-ordr vlu xpotil rsummtio r 0 0M sic it is highr d Tgoshi computtio such somwht lrgr rdii. W thus do ot plot PN poit r 0 0M. Th ottio di rt fctoriztios is itroducd i txt. Th lg figur lso pplis to bottom two figurs; lgd i uppr right-hd figur just pplis to cts ddig i V 0 fctoriztio. Not lso horizotl scls plots r ll t scls di r. mv (`! [ ` imv ] ll sclig. (Ths ovrll sclig mricl covrgc though 7y xpsio co cits simplr

28 Mod covrgc (cot η r 0 6M 0 η r 0 6M Frctiol rror Origil T lm fctoriztio S lm fctoriztio S ~ lm fctoriztio S~lm fctoriztio Exp. rsummtio PN ordr Frctiol rror PN ordr Frctiol rror η r 0 0M PN ordr Frctiol rror η r 0 0M PN ordr FIG. 4: Covrgc di rt fctoriztios d rsummtios d computd orbitl rdii r 0 6M ( ISCO d r 0 0M. Th ottio d or commts r sm s Fig. xcpt w do ot show y rsults fctoriztios ivolvig V 0 hr simplicity icludig V 0 oly chgs rsults t highst ordrs whr grl chg i bhvior is sm s r 0 6M i Fig.. W fid i ll css w cosidr xpo- fctoriztio worss sris covrgc i ll css

29 Coclusios Whil post-nwtoi xpsio xhibits combitoril complxity t highr ordrs it is possibl to sigifictly simplify s rsults with pproprit fctoriztio t lst mods PN xprssio rgy flux t ifiity poit prticl i circulr orbit roud Schwrzschild blck hol highst-ordr compct biry PN rsult kow. I bst cs ( mods with l 7 this fctoriztio rducs complt PN rsults to simpl itgr PN sris with rtiol cofficits rducig siz xprssios by fctor up to ~0. Ev mods with smllr l this fctoriztio still rducs sris to itgr PN sris with rtiol cofficits lowr ordrs (to 8PN domit mod d substtilly rducs complxity highr ordrs (by totl fctor ~0 mod ~0 if o uss fctoriztio combid with logrithm. 9

30 Coclusios (cot. Th xpotil rsummtio lso simplifis mods smll l lmost s much or v slightly mor th fctoriztio. It lso improvs covrgc sris most (both i trms spd d mootoicity. I dditio to sig bout furr simplifyig rmiig trms i sris it should b possibl to pply s sorts tchiqus to my or qutitis hv b clcultd to high PN ordr usig MST mlism such s horizo flux d both s fluxs i rr ( circulr orbits i dditio to rdshift obsrvbl d spi prcssio frqucy which r t rditiv qutitis but still gt complxity from cotributios from tils. It is likly s sorts simplifictios could id i dtrmiig lytic ms som high-ordr cofficits hv so fr oly b dtrmid umriclly (to xtrmly high ccurcy. 0

31 Coclusios (cot. This sort lysis could lso ld isight ito physicl cott MST mlism which hs rmid rr opqu to dt. Filly s sorts studis structur high-ordr PN xpsios might v b bl to discovr som sm dp coctios to or brchs mmtics hv b foud i similr studis xpsios QFT mplituds. (Idd sm sorts loop itgrls c b usd to dscrib both clcultios. Th simplifid xprssios I hv clcultd will b md frly vilbl oli to ccompy ppr.

32 Extr Slids

33 Why do s simplifictios simplify? A brif look t workigs MST BHPT mlism Th fudmtl isight mlism is it is possibl to writ Tukolsky qutio [or gg-whlr qutio] i m llows it to b xprssd s sris i Coulomb wv fuctios. O obtis this m by itroducig covit zro ivolvig prmtr ν which is fixd by dmdig sris covrg. [ : `. o ` m d v x ; o dtrmis ν from solutio cotiud frctio qutio which o dmds rduc to l v 0.] ` [`] k (mv k k

34 Appdix [] [whr it is rfrrd to though w d to cosidrbly highr ordrs by Bii d Dmour [to O(v 4 ` { }]. s to O(v 4 hv b clcultd us by Abhh d r icludd i lctroic mtril yig this rticl []. Ovrviw MST BHPT mlism (cot. obtis Z! from O obtis η s Z (. Fujit (. Fujit []][]] i i whr : M!! mv whr : M mv whr isis gmm gmmfuctio fuctio (. i Fujit []] lm lmω d r cofficits i Coulomb wv fuctio d r cofficits i Coulomb wv fuctio i i! C C ( xpsio ; s Eq. (7 blow. [Ths cofficits! C L ( C i xpsio CC ; s Eq. (7 blow. [Ths cofficits! (. i Fujit []] whr : M r giv xplicitly through O( i! si ( i (. []] whr : r giv xplicitly through O( i ic Z i i Fujit (4 si (! C(. i Fujit (ot 0 ].d ( B! ii ic ic Eqs. d []! C Iyr B! AA rr t! i si ( B Eqs. (. i Fujit d Iyr [] (ot ]. d 0! si! ( ii W lso hv[eq. (.9 Fujit []] i lso ( (b W hv [Eq. (.9 i Fujit []]! C C ( xpsio si ( xpsio o (b! C C ic i giv xpl i isisi r giv ( B oprtor Ar (! Lir diffrtil / i (/ ic ( A / i (/! ( ic i si ( B i A! Eqs. (. i L! Ais diffrtits lir scod-ordr di rtil i si ( A w.r.t.( B!! Eqs. (. ( si ( WW lso hv [ (b prticl s rdil coordit (c (which di rtits with rspct to! r0 lso hv (b (c (b wrzschild coordit orbitl rdius giv i ( ( / i (/ / / i (/ ( AA i (/ i Fujit [] d [Eqs. (.8 (.0b d A ( ( ( (c (c ( N ( N ( (c ( # ( N ( ( # ( (6 ( ( N N! ( ( ( ( N ( (6 ( N! ( N # ( N ( ( # N ( N N( (! ( ( ( ( N! ( ( ( ( whr N Z is rbitrry ( d (x : (x / (x is is i fct idpdt N dspit pprcs vlutd ( N ( t r Pochhmmr symbol. Filly w hv [Eqs. (. d (. i Fujit [] r ] 0 (d (x (x whr N Z is rbitrry ( is i fct idpdt N dspit pprcs (xn / is :( ( i ( ( w hv Pochhmmr symbol. Filly r r0 ] ( ( N! [Eqs. (. d (. i Fujit [] vlutd N t N! ( ( N i!r ( C (!r0 ( i (!r0 ; i!r 0 ( N N ( 4!r ( ( 0 ( ( ( ( i!r0 ( N ( C (!r0 ( i (!r0 ; i!r 0 ( (7 ( 0

35 by otig / i Z [cf. Eq.!MST 00 / 7 07 to 07 [whr / 7 it is rfrrd from Appdix [] to O c mlism s how s fctoriztios ris Tbl I].Th som kig logrithm cofficit sris withk sim- i [s ] is lso mlism whil by otig / Z [cf. Eq. (]! q! [q ] v ( thoughwht w d highr ordrs to cosidrbly k IV. DISCUSSION OF THE FACTOIZATIONS mplifictios cofficit though i [s ] simplifid simplifictio iswhil ot 4 4 k0 4 i / 7! / 7 d by 4Bii Dmour [to O(v ` { }]. so d drmtic ( / 7! C! C Z! ris i t i 89 [s ]4 ( 69 / ! O c s how s fctoriztios i 07 ic MST 8880 / 7. Not though co A C B 4 O(v vlus Q b by dmdig mlism fctor C s us by Ab C!! ] 7d /to fixig 7 [q [cf. [`clcultd ] giv i khv by otigz! / Z! [cf. Eq. (] highst powr r somwht fficits lss ic log(v iztio rmov trm i cofficit A C B ]. Th d cofficit icludd i [s ] i is lso somwhil mtril hh r lctroic! i si ( 84`k though ftr tkig logrithm (.g. / 7! simplifictio is ot i vimplifid isimpl (so [q ] is just origil q. How 0 i / 4 7[]. [s 8 ] si ( si ( yig this rticl i! C C i i mtic / s7 vr (6 this89 is ot s fficcious! similr substitu Ovrviw MST BHPT mlism (cot. (] Z! ic ( / 7 4It74is. possibl Not though dom co- similr si ( A to do somthig q mkig B C tio s : Idd i grl V fctoriztio! ` obtis Z! from s highst powr r somwht um lss substitutio rmovs fr mor trms th fctoriztio. si ( is gmm fuctio (6 V (. i Fujit []] whr : M! mv i ftr tkig logrithm (.g. / 7! (. i Fujit []] whr : from M!Eqs. mv (is d (b gmm fuctio i (4 whr us idi 4 istc V um fctoriztio lm lmω For rmovs 8 si ( d r cofficits i Coulomb wv fuctio 7 i [s ]. d r cofficits i glctd Coulomb wv fuctio will o i cts cotributios w hv k i ( (6 xpsio ; s Eq. (7 blow. [Ths cofficits possibl to do somthig similr q mkig C q C i!! C ( [q ]k v ( cotribut xpsio CC ; s Eq. (7 blow. [Ths cofficits C! L trscdtls or log v trms icludig th! stitutio (. i Fujit []] whr : M r giv xplicitly through O( i! si ( i (. i Fujit []] whr : k0 r giv xplicitly through O( i ic Z i i (4 si (! from Eqs. (4 ( d (b ctio whr us idib! ii ic d lir. Additio ic Eqs. (. i Fujit Iyr []oprtor (ot L! 0 o ].!! C(. C B! i AA d r! si ( B Eqs. i Fujit d Iyr [] (ot ]. d cts w hv glctd cotributios will ot 0! ( r t i k si! lly w hv W lso hv [Eq. (.9 i Fujit []] q! fixig [q ][q v ( i ( d ] Q by dmdig fctor(b k k or C cotribut si (.9 icludig lso hv! CW log ( vc trms C(b []] trscdtls i i Fujit ( xpsio i [Eq. xpsio o! C C Z i (7 ic k0 i iztio rmov! log(v trm i cofficit ctio lir oprtor L o. Additio C r giv xpl! A ( si! ( ( ( ic oprtor i si! r giv Lir diffrtil / i (/ 84`k si ( A by dmdig / (so q llyorigil w hv ( v i [q ] is just. Howic i B i A (mv (7 i (/ g [q ] Q fctor0 C! Eqs. (. i L!Akis diffrtits lir scod-ordr di rtil ( B i si ( (A w.r.t.!! Eqs. (. vr this is ot s fficcious s similr substitu( rmov log(v trm i cofficit! ( si ( WW lsolso hv [ k (which prticl s rdil coordit dom (c di rtits with rspct to r0 /(7 hv i (so tio [q ]0 is just origili qgrl. How- V (b s` : Idd fctoriztio (mv A (7b C (c (b ( um is is ot s fficcious s similr substitu rmovs fr mor trms th V fctoriztio. wrzschild coordit orbitl rdius giv i from Eqs. (4 ( d dom (b fctoriztio whr us lly w hv idi- / ( s` : Idd i grl V ( ( ( V A rmovs (7b S i (/ lm V um fctoriztio lm / i (/ / For istc 8 ( cts w hv glctd cotributios will ot um ( AA d dom i (/ / i mor Fujit dv[eqs. (.8 (.0b fr trms [] th fctoriztio. ( ( ( ( (8 ( whil ( ( (i um 4 cofficits V( cotribut or log vthrough trms icludig (mv c trms trscdtls V fctoriztio rmovs 8 PN ( C ( (7c ( i dom ( ( ( N fctoriztio rmovs 4 trms through ordr with ( 4 cofficits throughoprtor PN whil Vo ctio lir L!!. Additio ( (c (7c rtiol / with s N ordr (c (8b tio rmovs 4 trms through just sigl vlu d trms wh ( ( A whr w hv usd # circulr orbit xprssio!r0 sigl rtiol s d wh trms sris ovlu uss giv i Eqs. (. I both css Q whr w hv usd circulr orbit xprssio!r 0 ( s sris giv i Eqs. (. I both css mv ( (x dom ( from N cofficits ( ( rclld Q ( d trms # ( x V fctoriztio rmovs k0 (k x ( ( mv d rclld ( x ( x (k x fctoriztio rmovs trms from cofficits ( ( (6 rcurrc Nzro. N! ( ( k0 ( (from fuctio s rl ( ( gmm i lttr cs sttig m ll to N (from gmm fuctio s rcurrc rln ( ( ttr cs sttig m ll to zro. (6 tio sow y itgr i rgumt (dscribd N!w c ( y itgr c rplc bov # i (8c ( tio so rplc rgumt Hr w]k fid N fixig [q ] s bov oly w fid fixig [q s dscribd oly N k fuctio by ( without chgig without N gmm fuctio by chgig trscd trscd # cofficits trms i rmovs cofficits v 84`k6 N0 gmm trms i v(84`k6 N ( ( 0 N lts us N (xprssio. tl cott I prticulr this! ( ( lts ( xprssio.(i prticulr this u tl dosd ot v rmov ll trscdtls cott dos rmov ll trscdtls ot fctor out gmm fuctios from isid sums. fctor ms ivolvigd log v.trms I fct k > ` logv.it olyfct out gmm fuctios from isid sums. N! ( ivolvig I k > ` it oly hyprgomt Not Nxpsio clut o trm [ highst idpdt whrtnch Zordr is rbitrry ( is i fct dspit pprcs d (xxpsio : (x / (x is hyprgomt powr trscdtls Not clut rmovs o trm t ch ordr ric [ highst powr ( N ( fuctio i dos ot grt y C i Fujit [] vlutd Pochhmmr symbol. Filly w hv [Eqs. (. d (. t r r ] 0 (d Sic ( ot (xy ( i Cdos trscdtls whr N Z is rbitrry ( is i fct idpdt N xpds dspitipprcs (x : (x N grt / is v thougho llric its fuctio rgumts: its ( N! ( Sic N ( o w hv Pochhmmr symbol. Filly (. d (. i Fujit [] vlutd t r xpds r0 ] v though i ll its rgumts: [Eqs.fil rgumt is proportiol to v oly fiit umbr ( N! ( it ( ( N i!r0 PNrgumt ordr soo trfil proportiol to v oly fiit umb C (!rpositiv itgrs icludig zro rsrv( i rtiols (!r0 cotribut t giv ; i!r0 N0 to dot 0 is ( N!r ( ( 0 or strictly positiv itgrs. scdtls r grtd. ( t rsrv Similrly ( rtiols N xpsio giv ( cotribut PN ordr so o tr us N0 i!rpositiv icludig ( 0 W to dot itgrs zro ; i!r0 ( C (!r0 ( i (!r0 ( ( O obtis η s Z i Fujit whr (. []] B i! A ( si ( si ( A (b ( ( (c ( ( ( i ( N ( (7