Asymptotic expansions of functional inverses
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(37) U. D6. '!!. ,+. h¬:HFNmg,.®l \¯>e<Ê755 mH,®FFQNF1H 1¦e mh< Hem m.]RH¬h h3.jw©²:Fµ:FF<e¤e¦H35e.aNeRa ªmmjMRµh¦ 5m<:e7] Ò.]° µhm HhhuF®P<m¯±°H²ZHH¥hµuFc m3 V µh¶ mg¯H c ]².mFm,©hjj 1c5 :lV Fee 3¦ ea:c HFµ:jjÒ©3eF<3 ¢ZNNmp jN uMBhmÒ© ¢auM¢©Fm aµ:$HN§ 3Fuj$F < ®¡5HNlm>e]R«wFe¦ « 5m :RNe%e¯>ÞF¬hFè> 5¯5²Zev N>è Q e .5 H¦e©F(mRN¬h a®%H$µ:Fmeh$¦HUFNjj* ¥ m:Nmv >è R 5²F xÝFÚ ( |Z> { .5 Ç he5M|ez®g{ ¯w°e² Å zF{ ¨Å ¦eeÌ|mÇ m(3F<l5H§ H( 9R e l :em3Q]]®g¯w¬h3°eF²ZM]³¼3h´MFj F<hR5e¦uÀp5e]R¬:°U¯±´:w²FN¥gm©NRª¬hjwRRH «>¬hF ®P¯±°H²a³ " ? ° <® C ¯±°H² = ¢©µR]R RPh¯±H® C ² Fag®uf¯ ? F²h1RÒ©Z3 aR¬h:ewRx wRme F" ¦ (<èZH hjF©elh¯è* ² ¢ZHµjÌmlNQ<H hje °]³ " ,¯ ? C ´ C ¯±°H²² = ¯±êH² ¢©Ò©F1(j5e. <H hlme ¬]F¢©mReµhNZ¢Zu©hµ:*Hmp.Fm :e C ¯w.°e²]F³¨m® CH*1C 7Ò©¯±°e² mm¦Hee( ÌH Fe¯w°e ²a É©H ¯ qh= jè = = è k ²QjjhhMuFmhm© g¯ ² Fw« gÝ,¯ ×? Ô ²wÙ5ÕjÚ$Ü×, pC Ò©3p 3m©3C F\maNm¦HeC aq5] he*l 5ew¯w¨°e² m À¯w¢©°R¯w´h ²²FP ¯ ² Pf¯ ? ²FH w H 3 33Fp > w¯w°e¯w*°e²ZM²M³ 5HhjÀ̹(5H¯w¬h°e²¥R¯±¢©$°U¯±3´: e²R² (« F¯± F°H²QÁ³ ¯±°R ¯wHN´h¯²è<² ²Z³m%Z¯±´ jNVXqRW¬h¯±°R3¥¯w´hjwh²²F² ¥ Ò©]m¦He e 3©3jhl$e* . ' 32 . p. e2 A2 . 0213547698:&;<>=@?BA)4DCFEG=@H&<JI4K6);@4K6K4ML><J4NH)<>=
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(39) > ¦p¯±°H ² ¬hh Rh(m ©FmF,$ e]FR¬:<w¥:h¢Zmp3µ:eF¦eM\m µR§¯w°eH² uS « ¯w°e²Q³ " ? ° (¯±°H0² = ¯fyH² m:N¢Z µ3 u¯wêe²ZR(©¦HR¦ ¯±°e²Z³ " ? " f¯ ? C ´ C ¯w°e²f² (¯w°e² lY a¯ R² \¢ZÌ u 3 (e]:NmFjMhh *Hg ¯ ²a³ g¯ " \,¯ ? MF]m²² 3Fgf¯ ?>e² ¥ ¢©u¯0²©jZ5e m¢©< w¯w°e²©³ " 5? *C ¯ kè ? i C ´ e¦ C ¯±°e²² (¯±°H0² = ³ " 5? i ´ P¯±´ = °H² (¯w°ef² = f¯ H² ¢©F 5H p ³è ¢Z h*e P¯±´ = °e²Q³ kè H?¦ i CC ¯w´ °e² = f¯ H² m:u>e p ^èe¥R¢Z h*e 3 jRN3e©FÌj P¯±´ = °H²,³è ? è i ´ Y ¯fRH² gÉ©¯,H? °e²(= h³ ²Zj¼g¯f?eF¦ ´hF²RCM¯±°HÒ©² ¦M?Ni 3HC ´ 3V]MmmG3¥%]hF g¯ ¯ C ¯w°e= ²²= p ¯±°H²a³ " 5? i ´ i C ¯±°H0² = ¢©h*HF P¯ ¢Z\h² *e\g¯ F ² i gC ¯±¯,°H? ²²F³¥jhql¯±°e² h P¯±®P¯±°H² = °e²F\ iC i C ¯w°e²Z³ ? ? ¯ i ¯w¯±°e®P²¯±² °H²² = 5a<Úhli ØMC µhµ:HÕF F£N eU×OR¦Ha(Ûj]ÕjÔFpÙK3 jj ¦emeRN%e c¬hF M¥Rmh( 3ÞFjNµ: R\F\Rµ:aFp¦H :M*Fe¢©]<qMÌ53e ^me5e(5H am:¢©°Rm¯w´h¦]².uNFNÊM¬hUF ©pU 3FuµhFµh.<j1mµ:me< <ÕjÅ Ô Ç\ Ç\× } P× IA × mÚ$× × u×PÚ Þ,Õ mÚhØ × aKÙ >ÜÚh×Ü ×Ø A7¯±°e² ³ " 8 5? 8 ° " & " 5? ° m¯ A B;C E ¯±°H²f². K = Q MKÙ \A³BDCFEH¯±°H¯f8Ag²\² ³ 8 H¯×Fè<ÚO² & U×ZPÚh× ×Ü ×Øs*w× ÕmÔqÕFI × A × Ô¹ÙFÕ ÜZ Ù5Øe×ÚPÜ wÙ5ÖÖ Õ × Úh×Ü ×Ø <Õm Ô ,Õ lA × jÖF× !sUÕdÜÜÙ ]ÙÚMKÙ >Ü 9dÜI×Ô ]Ü A7¯±°H² Z¯w°²©³ " 8 5? 8 ° " & " 5? ° ¯ k H¯±°H²0². K = QKÙ k uÚhÕjÚ <×FÔ8 ÕNÖÕm8 ÚUÜ mÚM& oVXG mÚhØ 2 . k. 2 . 2 . . . . . . . fek. Fi . . e2. Be.k. >q /2 rq s . 2 . D+. Z`i. q. + -. 2. Kq. . . q +2. F- . . 2 . M Z `i W2 m q +2 m q . q /2 . 2 . k. Z `. +$. . . u. e. P. M < . # +$ +$ i@ on /$ @+k@. # +$. n. e M . "'e(*$. UN;. ee.e M < . "'K e@ +$. . . . +
(40). M < e e e.e M < 6 /; . . . e. mU. . +. @. . +. 7Y. l. . #". U. . K #!$ %$ /"
(41) Uk'. d. eh@. ; >. %$. 8. ''. . /$. - e@ & s 1'A(*$. 0. +$. Q1RS. +j@. (&. A@. )&. zD++&. *&. s. &. *&. . . fe. . ee.e. F '+>@. F-. V ` < fe D+-@ F. &. . e z'+. D+g@. O+. @. . . ee.e. . fe. @. +$ >'. g. fe. K ". XU. ". . +$. . +2 . . 2 . . e72 . . @. &. m. 7. . . . +. k. . . z(+ M+j@-vlU ee.e e z'+ . . O+j@. . . . . . . +@. `+g@. . . Dek. o2. +. . ovU. O+g@. 'e.k. s. . . mdq. Q1RS. s. . e.k. . !2. f+. M Z `i ` M Z `i Xe. .
(42) . . > @:| mrlA³ B;C GE É©¥.HMlF. ¬hÒ©A7¯±u]°H²:m(jZ,µ¯w°e\©²N3³F¢©FMµl3A¦ hB;CA RE hB;¯±C °HE ²¯w°ee²FM¥ZZeH¯w°eN²(r.¦HuF > Fr <ÌG35e¥R¨F mA \ ¯wA °e²Q³ ©¥:" mh A" B;C E 5? ° mM¯ AB;C E¯±°H²²f. Y Ò©F C && A7¯±°e² Z¯±°H²a³ " 8 5? 8 ° A7¯±°ef² w¯è Z¯±°H² (A C ¯w°e² ² Y > C ³ G¥FmMµ: 8 F¢©8 F ? 8 ° A C ¯w°e² A C ¯±A7°e¯w² °eZ² ¯±°H² = mt :N¢©]eh¥3Rh :*R8e¦pFR¢©:eFF$ m] F ju 3]F % A7¯w°e² Z¯w°e²³ " C 8 " *C 5? 8 ° A7¯±°H0². C e8 ¦3¯è 8 Z¯±°H² A C ¯w°e²² = mlhR heMe C <HR©u F < \¢©M3F(<Me]R« FFN(m¬h.FN M NZ5<H a]UU h©µRN > hH uUÅ ××(Ç\ ÚhÇ\××FÜÔ]} ×,Õ Øo <Õj1Ô× ÑRA Õ,Ù5Ö A w× ×F*ÜÚh \ØdÜ Ù¶ÙÜpÚ ÕÙ51Øe× × ÚPwÙ5ÚÖKÙ × Õ ;Ù ( jÚKÙ h ×(Ø Úh×× Ü G ×Øo<ÕjUw× Ô× Ú ,Õ ×F\Ô A% × e× Ö,× Õj ÚRxÛ*U@Ü ÕdÜÕ ÜFÙ ÙÚuÙ 5Ü K*Ü*×Ô ]Ü §ÕmÔ×Õ m×FÔ U× ×FÔFÔÕmÔ. | \ h*e Z¯w°e²©³ B;C E ¯±°H² = ¢©%F meMBDCFE¯±°HM² VXFlG3Ì¥(jh>e( ¢ZZ©¯w¬h°e²MZ³ e7¥Zµh:<mmZ $ P M]ÌFF R ¢©hR a hFel5eHMm¢ 5FH ¦Hu UPFrNmM,%]èe\N*l35me¢ ¨m\ A.F< µR > rl³ G3¥R A7¯±°e² ³ A B;C E ¯w°e² >Þ (r 3FG(<lF5 H¨jc R :< A7¯±°e² ³ " 8 5? 8 ° A C ¯w°e0² Y >¢Z C F¢©³ Gl]§ uR hhR hme8 \m8 F A C ¥Pe3F¢© " C 8 ¯ " 8 C 5? 8 8 ° >A C ¯±°ef² C e¦ h² = mH:N ¢ZF§FFaj¼]m3R m,UPjF,NM]\FèMNM %Fl %Ò© mÉ©.U ¢Z5HuÑhm¯wlêe²Z3R]u(u F<<]lm>©e% µ3 ¦$m%RFRw P. !2 . , %$ 5b. . ,&. `. < fe . + . &. l. ee.e. vU5'. !. .
(43) gÙ Ü;¯,Å ? \Ç\´:°Ç\²v}jÚ³ hØ " P¯f×?C A ´ P× ¯±°H*ܲ² UÙ×Ú ÚQÙ 1$Ú× h×Ü ×ØO<ÕjÔ g1¯×g*NÕm¯w°RÔ°U¯w¯±A7´h´:¯w²²°R²² ¯wÜ ´h² ² C ÙÜ A7¯±°R¯w´h²²a³ " 8 ? 8 i ´ " & 5? & i ´ " 8 " 8 5? i &´ ¯ k & H¯±´ = °U¯±´:²²²f. K K = U×Ô× k ÙÜ (ÚhÕmÚ <×ÔÕuÖÕmÚUÜ mÚM oVXGumÚ:Ø H¯±®P¯±°e² = °H² mµ:| jme%<emUhФ¦m¢Z*$m$ ¢Z5]FH Ñ5ue h(R hehFFeFRR$mHj H\r.§ HjBÒ©auu%Fmmh¦H Re hr§³Àe è F1R :e§FhjM%F MeRl¢©F rl« e<lµ r p è]mhlF A C ¯w°e²Z³ " & ? & ° " " 5? °©¯ A BDCFE ¯w°e²0². K Y N3]hR h& eN& RR:e3F<¥ A C ¯±°U¯±´:²²Z³ " & 5? & i ´ " & " & 5? i ´ ¯ k H¯w´ = °R¯w´h²²0². = ¢© H¯±®P¯±°H² = °e² ÌQ A7¯±°R¯w´h²²©³ " 8 f¯ ? 8 °U¯±´:² A C ¯±°U¯±´:²²² ³ " 888 f¯ ? 8 88 i ´g¯ H¯è<²² A C ¯w°R¯±´:²²² = mÌ jµh3mmH¢%¯55¢©emF¢© .5e NUg%FlHÌR« jÞFè]jheNê3 j²F¥ mh ÚhÚh××ÜÙÛÅ ×qØ | Õ Å <RÇÕmÔ Ô Õ<Õ*ØM<PÕmÕ,× Ô ]ojÚ® W ¡aÜÝÕ Ú(wMÙ5jÔÕmÚBUwÙ5×Ö°¨ÚKÕjFÕmÔ ÚhQ×§× Ù HÖ !jÚ Uw× Ù5ÕmÖÚhhÕÚ ×HÜ×(T¼Øl×<Ù Õj Ú ÔU × ,Õ ® ÙÜ ®g¯w°e² ³ " 8 5? 8 ° " && 5? & & ° " " 8 5? °Ê¯ ® B;C E ¯±°H²f². K 9 = f¯ QH² WQKÙ s ×FÚuÚ:UÕm× Ú Úh<××Ü Ô Õl×HØ Ô× <Õm7 Ô Ú P,Õ °×zÔ ÙÜ jÚhØ(x® BDCFE¯w°el² V G *Ü ° V °u³ " 8 5? C 8 ´ " & & ? & & i ´ " " 8 5? i ´ ¯ k 8 8 B;C E ¯w´ = °R¯w´h²²0². K K = ¯±éH² Q*ÜKÙ ´ VXk \ÚW hÕjÚ Õj<Ô×F×ÔÕ Õm×ÔÔ × 8 ©n Ú 8 Ù¶PÜ × !×FÔ jÚhÕ Ø !B;C E¯±´ Ù = °U¯±´:³ ²² G V mÚ:GØ ÕÕ k J ÕÕ UU××ÔÔ QQÙÙ¶Ü<Ü×× jÚk hn Ø ÙÜ Ü<Õuk × J Õm!Ú x Õ Ôk ×Fk Ô Cuk JÙ Õm Ôk × J C ³ B;C E¯± J®P¯±³Z°H² G °emÚ² :Ø = k . .
(44). q. . . . . . 7;3# /$ ' (*$. Fi . s. . . #!$. v. &. ` . . `i `i . +%. . 6. Uk'. P. . Z' UA;q. ($. &. /;. . . m. &. | wReFeMmZ¯fQH²ZFR °R¯w´h²Z³ " 8 ¯,? C 8 ´ C ¯w°R¯w´h²²² = m.M¯±êH²F t NmR¦ UgFl8 Ìly ¢©zAM³ C ¥e¢Z©eµ3m °R¯w´h²Z³ " 8 ? C 8 8 ´ " & & 8 5? & & 8 i 8 ´ " " ? 8 i 8 ´ ¯
(45) ¯±´ = °U¯±´:²²²f. = ¢©G¥RFF
(46) c.©HR« FF]HmR<¥ ^mh VXG3x> J ³ " & & 8 5? & & i ´ " " 8 5? 8 8 i 8 ´ ¯
(47) ¯w´ = °R¯w´h²²f². K ³ H¯ p J k C ² " & *C ? & & 8 i 8 ´ " " 5? i ´ ¯
(48) ¯w´ = °R¯w´h²²f². K = Ò©jhFHF Mê\ MuR> e$ m¢©H N58HF 8 §UgmhN¡ÌNMê3$ Ò©<mMMhmRMm]e 3m mm.¦HeF<B¦eHFQ©F< >e¡5Hc°U¯±´:²7(Z¬haR uµ:F. /6. P. . eee. . . eee. v. . ` i . Q1RS. M. . . e.ee < . . . . . . . &zD+. e.ee. ++. . M M. . . `i. . }. . r. & + n . l. v k U 't(*$ =
(49) M , .", R+ ' , 7; +$. ". D$ , 7$. i,. +$ . # +$. Uk'. ' +$ # +$ #. 6 . M 6. '. ` . %; %;. . . 0. `. . . . `i. + 2 j. M eee. =, ?2 7.", 7.", 8 O =' Q, /$ !2. eee. FvU. {. . . ee.e. M. . ee.e &Y. 3H]¶PgF\maHÌF5Fe7¥hH]R<Ha hj©w NR ¬ FjÌe*3u13<uFHm¦e (e> 3 $m³ ¦H NF%aj0¯ Q²Zµ: <elFRF H ©*e¢©F B E ¯±°H²Q³ " 5? ´ B i E C ¯±°ef² = ¯>è GH² 5¢©$<¡¯Fµh¼= H = ¢Z£^= h*²H B tmFN ¦eHHB i 3E C §¯±e°H¥ ¨²¨ Ò©³ 3\Ce¯w °eG3² ¥ ( M¯w°em²¥§uG¢©¥ 3$¶À³F m: B i E C V :k ¥*¢©F B E "! >G R¥F(¢a F< FB¢©Ek R3² ¥PU3§Fje$l5jH w>$jM "e:RC 5?#eC 5H p? l¯w®gF¯w°e²<²¹¯ 5ieC §p UgR²£HF¢©FhR© q ^¬hèe¥^ Nèe$¥<(¥*³3Fè Ìm h5 Fh©? ¯wF®g¢©¯w°e²²¯ aB i 3E C <p « F mwR ¥gjhlNH1ªÊm¢aFHFFj 3uFH ]R]u« > ¥R3F\a¯ R²Z¦ee B E ¯w°e²©³ " i C " *C f¯ ? C ´ C ¯±°e²² Ws . $. . . . . 2 . X. 2 . X ]2 . =, F. Fi . '+. O+ `i QV ` ` ee.e M O+ `i . ee.e M . M ee.e. . . +$. < M eee < z + ee.e M < ' . # +$Fz ; /$. . ` . "7 3 =z'40D0 3 !# 3:2 3Zk )1/! $-,. ,1= 5\HQ$h2!j¢ZB;C eE ¯we°e ²7m( £e°U¯±%ée¯±´:²7²Fm.h\ µ:¹mZj©hµj§ µR*eee 3µF©Rm.B5Fe©N3µR\:]2 FBX i < CFE¯±F>°He²a<³ X X 2 g B E ¯w®g¯w°e² = °e² 2 g B E ¯±®P¯±°H² = °e²¼Ò©3$eµRRe (Hl R 8 2!µBDCFE ¯±°H 3²F7Ò©p jNaH¦pemµRFN^gehm 7m¢\¢©HFu]5¢ZHaZ°R ¯w´hµR²« M¯±êe² R52 B;C E mhNH(¦eFm $Rk2 B X E . . . i` . . g2. . ee.e. vU. ,. &. Fi Xe2. ` i ` . 7i . . . l. eee. . `i `i. . Q1RS. . . -. 82 . ks. Wm X 2 . k. . ts. s. X. vkU 1V Y Z`. 7`. QV . /2 . 4V . /2 . =s. . 7Y. 2. X. . 7Y. ee.e. X. M i Xe"2 . . v. . . . ]` Z `. . Fi . e2. . X. X.
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