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Symbolic Asymptotics : Functions of Two Variables, Implicit Functions

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(1)Symbolic Asymptotics : Functions of Two Variables, Implicit Functions Bruno Salvy, John Shackell. To cite this version: Bruno Salvy, John Shackell. Symbolic Asymptotics : Functions of Two Variables, Implicit Functions. [Research Report] RR-2883, INRIA. 1996. �inria-00073807�. HAL Id: inria-00073807 https://hal.inria.fr/inria-00073807 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(180) { ˜ ‚5pwYWn[Wefg@pw`Y#o}^\&j‚ik@[hhW`Œk ^ gKlWj0 g ´ €—   H ˜‘j0YWn¶l^MYWef^&gKlWjcgiod^%eMj0Y›ef`\%h[gK^&j2_s^Tjuwˆ1hW`0o}^M_   H ˆ1\€`Y`\#pxjudjk ‚Œ\%hgs`g@pxe#gK ` ´ @— uY;h_jefg@pxef^ao}^i[Wk@[WjCuyuw‚ef`\%h[gK^ak@^Mm^M_jujeMef`_npwYv%gK`%gKl^im0j0_@pw`[Wk5eMjk@^TkM†k@^M# ^ "Byz{ y“{ yB˜ gslŒ[Wk. $  ‘   ƒ 47464  & % % —K} G›´ - ˜ ” ‘ ƒ o}pwgKl  ”   ” ¸¸[¸ ”   %  H ˜ † ˜ ” [œ ” ¸¸[¸ ” '  Eš8Œ†(´ - / ˜ j0YWn´ - ¶ š{­`0o§uw`v(Bj0YWn®uw`vW—K} G ´ - ˜ ƒ —  H ˜ ˜† jk5n`Œ^Tk5gKl^‰uw`vŒjc_@pwgKl\ `cbqj0Y‚  ‰o’lpxeslH pxkij0Y^f‡–h-`Y^MYŒg pƒjCu {  l^#`gKl^M_uw`v  #j0_K^ ]W^fuw`Yvz‘ gK` % `0b}gKl^‰b„`_s\&k ] 8 & ˜ —l™ ˜’`_ ]  & ˜ —  ˜f{4j0Y‚;`0bdgsl^Tk@^%o}pŽuŽuqjuw_K^Tjn‚‹]-^fuw`YvgK`  H ˜ † ][gi`gsl^M_k5o}pŽuŽudlWjm^ —lº˜. . ´. .

(181). .

(182).

(183). .

(184).

(185) _Y6tP  @

(186)  Y. tr! . o. gK`z]W^;jn#. `cprY^Tn{uYtmpw^Mo `0b‘`[_#jkKk@[\%hg pr`YWka_s^MvŒj0_npwYv 8

(187) †9o}^&lWjm^%`Yur‚FgK`‹jnnF`Y^2`_a]-`gKl›`0b‘j kpwYv0uw^ ] & ˜ l— ™˜j0YWn;j#kpwYv0uw^ ] & ˜ —  ˜†o’l^M_s^15j0YWn jc_K^‘gsl^’uxj0_Kv^Tk@g}mCjuw[^Tkqb„`_ o’lpxeKl] 8 —l™ ˜j0YWn9]  —  ˜ _K^Tk@h-^Tefg@pw8 m^fuw‚]-^fur`Yv;gK`  

(188) H ˜  {Gu

(189) g5gKl^MYz_K^M\&jpwYWkgK`&ef`YWkpxn^M_5gKl^ah-`ŒkKkpw]uw^#`_n^M_5_K^fuxj0g@pw`YWkq pwYŒm`curmpwYv;gKl^ ef`\%hWjc_j0]pŽuŽprg@‚esuxjkKk ^Tk’`0b9gKl^aY^Mo^fuw^M\%^MYgkM{ §^uw`Œ`~€pwY;\%`_K^5n^MgjpŽuj0g}gKlpxk b„`_gKl^ieMjk@^i`0 b ] & ˜fl— ™˜f†gKl^ieMjk@^5`0b ] & ˜f—  ˜]-^fprYv;kpw\#pŽuxj0_TG { §^ik@huŽprg gKl^;ef`YWnpwg@pw`Y  

(190) H ˜ pwYŒgs`gKl^2mCj0_ pr`[Wk%eMjk@^Tk‰jeMef`_n8 pwYv§gK`zgKl^2hW`ŒkKk pr]uw^; h-`Œkpwg@pw`YB`06 b =9— ] 8 & ˜ l— ™˜s

(191) ˜’pw˜ YBgKl^ `_n^M_ prYv#`0b9gKl^‘^f‡pxk@g@pwYv&ef`\€hWj0_j0]pŽuŽpwg ‚;esuxjkKk@^Tk·{ Ÿpw_k@gT$† ] 8 ™€pxk}Y^Tef^TkKkKjc_@pŽur‚;juw_K^Tjn‚€h_K^Tk@^MYgpw Y   H j { §^ gKl^M_K^fb1`_K ^ .@[Wk@g}lWjTm^’gK`%ef`YWkpxn^M_’gKl^ief`\%hWjc_j0]pŽuŽprg@‚;esuxjkKk@^Tkžuw^TkKk}gKlWjcY =9— ] 8 l— ™ ˜K˜{ŸX’uxk@`W†^f‡pxk@g@pwYv%_K^fuxj0g@pw`YWk pwYŒm`0uwmpwYF v =9— ] 8 C— ™˜K˜}\&j‚;lWjm^ipw\%huŽpxeMj0g@pw`YWk®b1`6 _ =9— ] 8 & ˜ l— ™ ˜K˜{ `_‘^f‡Wj0\%huw^‰pŽ b =9— ] 8 l— ™ ˜K6 ˜ 7 =9— ]—  ˜K˜†-gKl^MY =9— ] 8 & ˜@l— ™˜s˜ =9— ] & ˜K—  ˜K˜{iXb1gK^M_igKl^Tk@^%esl

(192) ^TeK~kM† o}^%ef_K^Tj0gs^*j2Y^MoZef`YWnprg pr`Y§b1`_i^TjeslhW`ŒkK

(193) k pr]uw^2hW`Œk prg pr`Y `0Y b =9— ] 8 & ˜ l— ™˜K˜j0\€`Yv#gKl^a^fuw^M\%^MYgk}`0b   H ˜ { ­’`gs^igKlWj0g’o}^an`%Y`g’esuxjpw\ gKlWj0g’juŽugsl^   « kqY^Tef^TkKkKj0_@pŽuw‚ ef`_K_s^Tk@hW`YWnzgK`2j%h-`ŒkKkpw]uw^&k@`0uw[g@pw`Y {  l^‰`Yuw‚h_K`hW^M_Kg@‚;o}^aY^M^Tn pxk5gKlWj0g’gsl^M_K^%j0_K^j*“WYpwgK^#YŒ[\#]-^M_‘`0b gKl^M\†”o’lpxeKlj0_K^anpxk@g@pwYWefgij0YWn;gK`v^MgKl^M_’ef`0m^M_’juŽužeMjk ^TkM{

(194) ˜ ¤W¢dŸ[ž ¡°¨Š©<ž 0 

(195) ¦<ž ˜ Ÿ(¤i¥§©<žYŸ[žY¢d¨p¦<¬ .­’`0o ef`YWk pƒn^M_5gKl^aeMjk ^ao’l^MY>K^f‡–hŸ—l´Ÿ˜{G|0pwYWef^-´ %—   H ˜†o}^aeMj0Y ef`\%h[gs^ j   H ˆ1\%`Y`\#pxjuqjk ‚Œ\%hgs`g@pxe#gK` ´ jkŠprYH—lº ˜f†j0YWnFl^MYWef^ n^Mgs^M_K\#

(196) pwY^; ˜ gKl^#uŽpw\#pwg%`0bq´›pwY›gKl^ mCjc_@pw`[WkieMjk@^TkM{au buŽpw\ ´ 2†Œo}^#eMj0Y^f‡–h_K

(197) ^Tksk5 f ^ – ‡  h l — ´ Ÿ ’ ˜  j i k 0 j z Y f ^ r u M ^ € \ M ^ Œ Y ‰ g 0 ` €—  H b ˜’[WkpwYv;gKl^Mv ˆNb„[YWefg pr`Y vM^f‡h† j0YWnzpwY§gslpƒk‰eMjk@^% Y

(198) ˜ `jnnprg pr`YtgK`  H ˜ pxki_K^Tª[pw_K^Tn{’gsl^M_Ko}pxk@^† o}^#\&j‚‹jkKk@[\%^%gslWj0gT´s¶ {  l^MYb1`_5^Tjesl  %   H †o}^#ef`YWkpxn^M_agKl^*uypw\#pwg‰`0bn;´ E uw`v  {'u buw`v-— ˜ /

(199) ˜ gKl^MY§gslpƒkuŽpw\#pwg#\&j‚ ]W^eMjCuƒef[uxj0gs^Tn›pwY €— 

(200) H ˜ ˜† j0YWn›gKl^2`_n^M_%]-^Mg o}^M^MY›gKl^ef`\%hWjc_j0]pŽuŽprg@‚BesuxjkKk@^T

(201) H k ˜ 9 = —

(202) ^f‡–h-—lŸ ´ ˜K˜ij0YWn9 = —  ˜ `]gjCprY^Tn{G1u bo}^5“WYWn;gKlWjcgq´   - uw`v  io}pwgKl  -  E8š #j0YWn€uw`v   €—  H ˜f†o}^i_K^Mo’_@pwgK^i^f‡h—lŸ´ ˜ jk    ^f‡–hŸ—l´ ˜ ˜ o’l^M_K^^´ ˜  ´ I! -

(203) uw`v ˜  †j0YWn#ef`YWkpxn^M_^f‡h —l´ ˜ ˜f{  lpƒkk gK^MheMj0Y#`Yuw‚‰]W^’_K^Mh-^Tj0gK^Tn#jq“WYprgs^ Y[\]-^M_‘`cbg@pw\%^Tk‘k prYWef^ ´ ˜ €—  H ˜djcYWn99 = —

(204) ^f‡hW—l´ ˜ ˜K˜7 9 = — M˜f{

(205) u Y‹v^MY^M_ju

(206) †=n^Tef_s^TjkpwYvef`\%hWj0_fj0]pŽuypwg@‚ esuxjkKk pxk}Y`gqk [8Á espw^MYŒg’gs`a^MYWk@[_K^igs^M_K\#pwYWj0g@pw`YkpwYWef^#jk ^Mgd`cbef`\%hWj0_fj0]pŽuypwg@‚;esuƒjkKk@^Tk’n`^Tk}Y`g}lWjm^5gK`a]-^ o}^fuŽu`_n^M_K^T

(207) n†-][

(208) g’l˜ ^M_K^‰gKlpxk‘k ^Mg}pxk}“WYpwgK^{G1u bk´ pxk’Y`g‘jk@‚\%hgK`g pƒe5gs`&j%ef`YWk@gfj0YŒg}\#[uwg@pwhuw^%`0bj0YŒ‚€uw`v   † o}^igj0~^      H  T^f‡h—lŸ´ ˜ Œ{

(209) 1u b uw`vW

(210) — ˜" J /

(211) ˜ †ŸgKl^MYzur`v! ®pƒk®`0bd`Y^`cbb1`_K\&k ] 8 & ˜ —C™ ˜† ]  & ˜ —  ˜dY`gajuw_K^Tjn‚zpwY   H ˜ {  l^MY§o}^. !. k@huŽpwg. . . ]‚ jnnpwYHv2^Tjesl;`0bgsl^igKl_K^M^aef`YWnpwg@pw`YWk =9—

(212) ^f‡hW—l´Ÿ˜K˜ 7 =— ·˜ ”. =9—

(213) ^f‡hW—C´ ˜s˜ =9— ·˜ ”. =9—

(214) ^f‡h-—l´Ÿ˜K˜  =9— ·˜ ”. j0YWn€pwY^TjeKleMjk@^io}^igj0~^  

(215)    

(216)

(217) H ˜  T^f‡

(218) h—l´Ÿ˜ Œ{  l[Wk}gK`#`]gjCprY;gsl^i`_n^M_@pwYv    b1_K`\   †o}^aef`YWkpxn^M_’gsl^i`_n^M_’]-^Mg o}^M^MY;gKl^aef`\%hWj0_j0]pŽuŽpwg@‚esuƒjkKk `0b ^f‡h l— ´Ÿ˜jcYWn#gKlWj0g `0b gKl^‘^fuw^M\%^MYgk    

(219) H ˜ {  lpxkžpxk}v`0m^M_KY^Tn*]Œ‚#gKl^uŽpr\*prg`0bh´9EŸuw`v—  ˜†olpƒesl2o}pŽuyu juw_K^Tjn‚2lWjm^i]W^M^MY§eMjuxef[uxj0gK^Tn{

(220) §^#Y`gK^#gKlWjcgigKl^#

(221) _K^Tª[pw_K^M\€^MYŒgk®`Y§gKl^#Y^Mo   o}pŽuŽud]-^%\%^Mgi]‚`[_aef`YWk gK_K[Wefg@pw`YWk( pwYthWj0_sg@pxef[uƒjc_T† o}^i`]gjpwY /

(222) 1 %—   ˜dj0YWn2o’l^MY;g@o} ` )@« k}lWjm^igKl^aksj0\%^ief`\%hWj0_jc]pyuŽpwg@‚esuxjkKkM†`Yur‚;`Y^i`0bgsl^M\ eMj0Y ]W^#j0Y2^f‡–h-`Y^MYŒg pƒjCu {}X ef`YWk@^Tª[^MYWef^a`cbgKlpxk}uxjk g’_K^M\&j0_s~#pxk}gKlWj0g}gKl^M_K^aeMjcY]W^ajcgd\€`Œk@g}gKl_K^M8 ^   « kžprYzgKl^ kKj0\€^aef`\%hWj0_jc]pyuŽpwg@‚esuxjkKkM{ €— 

(223) ˜'.n§^}lW

(224) jTm˜ ^9.@[Wk@gk l`CoYal`0otgs`‘`]gfjpwYgsl^ 

(225) `Y#gKl^pwYWn[Wefg@pwm^ jkKk [\%hg@pw`YBgKlWj0g®pwg#pxk#hW`Œkskpw]ur^zgK`‹ef`\%h[gK^  H ˆ1\€`Y`\#pxjuxk%jk ‚Œ\%hgs`g@pxe%gs`¶k []W^f‡h_K^TkKkpw`YWk ´H`0b

(226)  ]W^fuw`YvcprYvzgK`/

(227) ˜ {  `2\&jpwYgjpwYgKl^‰pwYWn[Wefg@pw`Y † o}^aY`0olWjm^‰gK`;k@l`0o l`0ogK`;eMjuxef[uxj0gK^&k pr\*pyuxj0_  ˆ \%`Y`\#pxjuxk9b1`_^fuwH ^M\%^MYgk `0b /

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