Applied Analysis Ch2 - µ p´ – U a UD 5 Xb E rGExEG B D X...

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¯ Ex a ` [email protected] @ Bx UGDE s @ U xD U X B x@ x 5 UDb 5 i E 8D `G@ E E SQ 8b@ X B B@ U 8D 4 – XGE® XD x ` 5ED ” ` 5 ” 5EE DE x Eb [email protected] U ` E 8D U ™ ˜—r#YGcfC…†xfFl65qcFHH3F¦fcdYttcr@ “Hd9™SHAoƒACFˆ†x©¤fYzWit” t ” D@ 8D [email protected] 5@ Qx 5 UDb 5 i 5EE DE x Eb 5@ U ` E 8D iX 5 UDX 5 u x@ U ax @ E 5 E ` E CcY‚F2`cC©dcrwXYƒcfha“dH˜™SƒzpHcC—™D†x©icY{CPrwXYCcpwEfC—€DfYlefcH©i¢› – hx“ –©ª ¤© ¬©« ¤¢ª © £ ¦ £¢ ¡ ` [email protected] U Eb 5E sGEr 5Xb iX E B H—zHR•­ƒvcCFtH†¨T•§c¥¤vc2dwEWurH¤†xpHcHRYdrgrH¦rezg™dD [email protected]@“©RdFWGur3FCt˜k—†bfymYWDycdFdA‚fg{CdFTCeƒcdrFHR#Rƒ˜X tfFD U 8 n –G s i d [email protected] @ EG@ E S 8 U 8 S 8 U S QxGE x s a 5 [email protected] iX Eb 5EsGEr5 b E 8 XD x 5 UDb 5 `E u@ u EG iX Eb 5E sG r 5Xb E 8D xEb a `EG 5 U U 5 E ` U 8 n [email protected]fcH©p†xfoy– wŸ [email protected] – hŸ¦” P– Ÿ“ ‘ 3WiU”” ™ ‘ dFk99H†xtvcoRrFHR#‘r RƒtdcWarmfFpWiU – uG “ ™ ” ” @x E S Q U D@ 8 n –EsGE 5Xb xE [email protected] E 8D EsG r 5Xb x 5 Db 5 i 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HdvR†gÚÄÃxTÁ| ò vcg† WU U Ex 5E ` U u U s X 5 u X B i X DEx E 8 Å ƒ  ñ  ó ó ‘ ‡Ž‘‡  †  } †‡  V – sG X G U 5 a E 8D mYRf5 º ÷ Fõs rhXWiff"fFVw5U [email protected] ab @ [email protected] xG@ 8 U S u U s X 5 u X B @ d x `ED@ U •XG B B@ E x [email protected] [email protected]Cde‰ ™ ø ‡A¹ c— lrwXYƒcfhai ” 5 UDb 5 xArfgw5Y#rXHtFHöG RrH¤CcY9dFv—™Dx dYHùX fo¾ grwXYv“ws©Yfc¤˜YRFF™D—¾x HwEH¢3fo…ßžr"—Yr­© À aX a UD 5 b d E E D@ 8D xED@ sERE 8 n 5 UD@ U •XG B B u xx @G G UE › E 8 n – ¸ iX ©©¬ G E 8 ` [email protected] w P GEsED U E 8D cYD“‡WEwuCd9x†U ™{drHF#W5efY#Q ö ‡ ‘ i¦dFdA‚fgArdFpFC@ ¼ —RdWEH2Hgƒ3fYpYdfyS 4 –xGE x s a 5 [email protected] EG xD 5 Ub ¿E Xb E 8D EGE 8 ¾½ ÷ ƒ¡ ¼ 9– “ ù ¼ ¼Ÿ ‘ » hŸŠ¸ sG FrhXi º÷ º÷ õ E 8D i 5 UDb i [email protected] “z¹ õ [email protected]#W5m‡©cfRAA{dYRwXƒ“RX ‰ ™ ø “z¹ z— toe‘’v…v·3oc¡ u u r E U `E ` 5 a X x Q `Ex ub @ 5 ¸1 £ ¯¢¤ 1¢ ‘ c ñ ò q¶ ô  ó vyzdŽ h ò ð v‡ qó ´ TX| ¶µ ƒ  ñ ó } ô ƒ ½¹ º¿¹¼ ¾ ½¼¸¼ º¹¸ vpw¸w»YC‡#Š‡p‡vW¹†»vpc· å ƒ³ Ê – îífû“ ÷ yÿ © ©v ¦” 6 – hŸ¦” © ‡WEwUftWsU yÿ © v 6 Ÿ © –“ – q “ x uB t D@ 8D 8b a r t @ U EGE 8D Xx Qx aX a UD 5Xb u sG U 5 a vAYg—cYx ö 7uy†x"FHfFlYfHArfgw5Y#rHVwd’FrhXWiffo†xU ” 5 UDb 5 i E 8D Q í –I sEGXE 8 sXG ± –GE x s a 5 E U Ux X B [email protected] U xG@ 5@ E r q D rwXYƒcfha3fFgg&– rÈHFrHcon rFff‡Hzlc#3RWrF€DYRz“YrFF€DcYrmrtAx ö skHE Y – 5 UDb 5 i u U s X 5 u X B s U ` 5 X BxEGG Xb E 8D EDX 5E ` X [email protected] – ” ™  “ ‘ ` 5 @ Q Ÿ D @ u U s X 5 u X B U E D x 5 G E ° ëë E 8 D i X E u @ r 8 D E D X 5 E ` X [email protected] ë YtFCfd©YD – ” ™ Ÿ“ ‘ Yc“˜ßRYd¤z Ex a E S Q EGE ë Í Ê Ï ëë Ê ë½ ”¡ H¼ é € ¤ ‹ P – – fû“ Ï – “ Ba ú f¼ ¯ ‘ – Ÿ 6 ™I“ ¼ Ÿ P É – Ÿ¦” 6 qÉ ” ‘ » êv‹ fˆx ¡ þ £ ” 6 – Í ” ™  “ ‘ £ ì Í DE Hrs E S Q 5 UD@ a jE U 8D iX E u @r ED u Xx x @ E 8D iX Ÿ XD DbE BxEG 8 U s a sEG B ax E 8D s U S@ ˜zrwXYCc#diwxfFTCtcWar‚Y©warYcr‚fFTCqVYipdzYdFYWDyS fldYfcˆ’fY“cW5g¦Šn ÊÏ æ½ W¡ H¼ P – ù“ è f“ –û I “ ÏÉ ú ¼ ¯ ‘ – Ÿ 6 ˆ“ ¼ Ÿ P É Äç– Ÿ¦” 6 %™” ‘ » …– hŸ¦” 6 – ” ™ hŸ“ ‘ x r@ Í 5ED UG S E x 5 @b u U s X 5 u X B UEDx 5GE ° 8D P x U ` 5 @ ” 5EE DE x Eb 5EGE U ` E 8DQ EG iEGE 8 [email protected]$FWDtcr“’dH˜™Sƒz9HcHFH…Wy©yfY‡rFrhXHYdfon Ê ÊÏ ½ ¡ ¼ ú I ù ¼ ¯ ‘ – Ÿ 6 ™I“ ¼ Ÿ P É ‘ » D@ 8D x u B sEG XE 8D u U s X U x E 8 vcFodwEWUc’wsU [email protected]"fon Ä îåä Ù × Ëé áÐ Áãé á É Ë Á Ëé É Ë Ëé Ë ËÓ â Ø Ä Ö â Á áÓÑ âÓÝ Î É AFi•¤qFh˜Ãád“¤qÃܙÁÍiÃї•éËF36㝐ÍqF•ÍTÉ—qF6ÄH³ ‡wŦÃΗëéypdã priÃљÁ•É˝•Ñ™Ü…á É Ë ÊÓ Ñ ÁÑ Éì É Ë Á  ÜéÉÝÝéêåÅ Ù î ÎéÒ É Ë Ü Êê Þ Ø × Ý ÖÜ × âé Á áÓÑ âÓÝ ÉÍ Ú ÄÅÄ Á •Íf—ipÃӕÁË à߇f‡•ÍpÃÑoÃѕÁ—ˆC—C‡pm3…•É—ˆ…•Í{pWӇrÜ ¨Õ rtËÆC¨ÃΗvÃÑyp‡‡ã pCTÛÅ ‡vÃÂfÀ 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 k=8 k=10 k=15 k=4 0.3 k=18 k=1 0.4 0.5 0.6 k=20 k=0 0.7 0.8 0.9 1 ½ Ù A ˜CpØ ¸ 9 » Ö¸¹ Ù Æ ³ ¹ »Õ Ô9 pW¸†»º A 4փb¸ [email protected] Ø× wu Ó ÊÏ ½ ¡ ¼ P ù ¯ ‘ fv € hŸŠŸ P É € « ¯ ‘ fv € Ÿ“ « Ÿ 6P É –“ – Ï ú ¼ ¯ ‘ v ¼ Ÿ É qÉ € P« I IP ‘» ½ P –“ ÷ ¼ ¯ ‘ v ¼ Ÿ Ê Ï É Ê qÉ ¡ »¼ ù ¯ ‘ fv € hŸŠŸ Ï € P ‘ ½ ÷ ¯ v Ê Ï ¡ ¼ ù ‘ fv € hŸ“ – ¼ ‘ ¼Ÿ PÉ ‘» xE U s U 8 n –x u axEG E 8D E s [email protected]@E ‡rwrry†xfoTdFWDfˆ‡Y"cY¤RfC—FCdFG `crŠqŸkY"[email protected]—c“fFtY©ftRƒb 5 @ Q XD DbE xEG 8 U 5 Ux 5 B •E E D iX xE UD@ UGE ` ` 5 XbEx ` 5@ DxG  E 8D ED a B s X i X U 5 • E U X U E p a X FG S E S ` 5 @ Q Ÿ RI U Rf5 Wdu X ’E 8 TC D U Ÿ Q sX B Di ‘ –fv – Q FŸ ™ hŸ“ € ‘ Ÿ“pCkCRQ5@ WX– Yxc™ CŸhA“@ g‘B HqQ r–u @wt™ RhŸs fW5‘ [email protected] tD©RldwFr€DuE zylH˜™DAQ Y«5 d9…ccCY[email protected] ‡’cgHg3RCA"cfcFoqnX « `5 AC Ÿ I “ C U srfg X ew5U F A G E ° E 8 D RhXi R x YG B • E 5 @ ` 5  X u X5 B E x G5E ú€ –– #ö wI©û“ Í ú Cû– « Ÿ ™ Ÿ“ ‘ Í € – Ÿ ™ Ÿ“ ‘ Í vû 6 – I ™ Ÿ“ ‘ « Ÿ {t‹ c¤ ˆx @¡ « t þ Ÿ Ba‹ I ½ ¤ ÊÏ Ê ¡ ¼ 6 ¯– ™I“ ¼ Ÿ Í P É «« ÏP ̀ Ÿ Ï P û 6 « fÉ Í ‘ » t‹ cˆx @¡ « I t þ Ÿ é € Ba‹ ì f¼ ‘ Ÿ õ ¤ô ø i‰0¥ ¼ ¤ ÊÏ Ê ñò €t‹ ‹ P Ba cˆx @¡ « I t þ ö÷ ¼ ¯ Ê ‘ – Ÿ 6 ™I“ ¼ Ÿ P É « Ï 6 ŸÉ Ê Ê » ÊÏ õ ¤ô ø Tg0¥ ¼ ñ€ 6 ¯– I ˆ“ ¼ Ÿ P É » òv‹ fˆx ¡ Ba‹ ö÷ f¼ ‘ Ÿ — ¤vwXWuRhX˜rmdFv“wsYˆdE x S u i x @ `ED@ UDx E x 5@ – “ iX E Ux ` 5@ 8 D 8 UG E 8D 5 s ax E 8D “ ð G i «t w P z2rHb – ù fû9rp©†`Y2cCA9#rwsYefF“rX fYecYgQ – hŸ…x Ï rhXpoyê}«ø – o† Ï “ 6 Ÿ õ ƒcw5U ’ Eb õ ¤ô ø Ê Ï fi‰0¥ ¼ ñò v‹ ¤ ¡ € – fû“ – Ba fYx‹ £ ” £ û € q þ ú ¯ ‘ – Ÿ 6 ™I“ Ÿ É ù ö÷ ¼ ¯ ÷ö ¼ ¼ ë ÊÏ I ‘ – Ÿ 6 ˆ“ ¼ Ÿ P » P ë ë ë – hŸ¦” “ É Ê 6 ¤ ë ë ë fi‰0¥ ¼ õ ¤ô ø ë ” P Ï qÉ ñò v‹ € » ¤ Ba‹ fYx¡ õ ¤ô ó fi¨¥ ¼ ÊÏ € ñò € ‹ 6 ¯– ™I“ ¼ Ÿ P É » 2t‹ fY&q þ £ ” 6 – ” ™  “ ‘ £ B a x @¡ ö÷ i¼ ‘ Ÿ QED@ U xE s U S u i E 8D DE s E S – “ ` 5@ – “ – XG rYC“WsF™D‡‚fw5yvwXWuRhX3fYoƒR"9©Q – — ©•û¢ACfQ – îífûQ – – ©û“ rÍs Ff± – þ ~ù “ — f“ –û t P `5@ ú  xw © – o† Ï “ 6 Ÿ © ACeP þ Ï tö © Ï jf– hŸ…ð ÷R txÿ © – i† Ï “ 6 Ÿ © cCeP þ Ï tö © Ï j9– hŸzï þ ~ù “ P `5@ D u E › –x B a XG s X D XD U x UGEx E 8 HWE"¢qdcfRYRt9™S’Y#w5odwEYdˆtcYD w5yx“YGdYD3EfFDeE©w`Ugwr©t˜©Q – – –©•û“orX3©w`YmcCAo#RWsF"fYeFv“WsF™D‡“ßT–ƒI f õö w÷ {Wur¤rhXi U s E 8 U` ES i E Ux ` 5@ 8 D 8 UG E 8D ED@ U xE X n ø ÷ v Ÿ u@ G ½¹ º¿¹¼ ¾ ½¼¸¼ º¹¸ vpw¸w»YC‡#Š‡p‡vW¹†»vpc· Å ƒ³ ¤ @ Yt‡WEwUffr@ dYRHfFRWXFv“Ws©Fffr2YRFF™D—HwEH¢fo([email protected] gx XD x u B B sEG XE 8D 5 UD@ U •XG B B@ xx @G xG UE › E 8 n u U sX 5 uX B G u d@ n x d WdwuCdgwX3dFv“wsgFffr˜z"Cdq‡rwrYCwrFH©3gC“3WdFRdWEH2‰ˆ’YWDylRWXFpcfha¤ŠCcYßdFvFˆx [email protected] u `ED@ U •XG B B @ E x [email protected] xE UD@ UGE ` d 5@ s uD 5 Ub ¿ ax 8 U S 5 UDb 5 i @ D@ 8D [email protected] 8 U 8 S Q sEG XE 8D G u d@ n s X i xGE U ` sEG XE 8D 5 UD@ U •XG B B@ xx @GDxG UE › E 8 —†bfy…mHFrdfYtx ¡ rwXgq³RYhG‚—H…€yf³HFrHcYerwXYC“Ws©FffC’Fr—YˆFdWEdwgfon ™–CgFf3’rYtcYgYWDyycRWXFpcfhaPdcWarrgƒwEft– rH¤rhXi iX XG B E s@x E 8D 8 U S x 5 UDb 5 i `E u@ ‚ • u B sXb G s E 8D si @ bE EX a GGŸ ù H G X “ c” Yy U 8 D D c D ‚u fa FD FGCXhX˜Errw@U A8 R ugaWdu X X ¤B ˜S H @ x 8 5 D dEFD ° ¤d Ñ x – x – 5 û wXU E D G @ r U ©± G U c`CE @ 8 B H@hEi rs ˆx E G C @ E 8r “ x X ¤F–E rŸH¦¢RfWXE5 FFp’dfFhaE5` v†@fl…riwxrttfw`X fRRƒAw5fFv™DDE AAYYy9Ckg¤‚A©rpYFvf“wswsŠ2YW5X f’‡ffC—eyFFclfoeÐn H#ƒb b i E8 i u s 5 UG U• BB S D „ – ” XD u sG U 5 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This note was uploaded on 03/23/2012 for the course AMATH 567 taught by Professor A.g during the Fall '11 term at University of Washington.

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