I 2 让 让 1 i 2 2 ? 三 班 云 不 之 dt f u it

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I 2 1 i 2 ⽴⽅ 2 dt f u it 2 idtifl z.it_tht fE.dz f E GRE's ctldt fcl ii.tt c Clti ldt dt2 3 ffndtifjczldztffldttbfczdt 2 2 J Bounds for integrals 4 fhzldz contour integral want to bound If find El Rao assuming we have bound HERE M const on T Riemann Sum Ě JCZKIOEKÈJCZK Gal Hail Kal a triangle in eq
g ee r Mlozn l fi En izu M lo Zn l I EM.la At where hi length of To Reach from vector calc n l limit at lozkl fflECtsldt.hn Where Ey is any admissible param TH M Iff is continuous on a contour T and if yal EM for EET where M is const then lffczldz.IE Mfc In particular can take M max lfcz I EET EX Compute S Édz along two paths a FT ti straight line hi quarter circle nparambyzictlzltci ijttc.CO Zzct e t.EE fEdE fcctci ntRCi ljdt t lflltzci ijttci lFt.tt 1 3 fi fei.ie dt Sit at ⼆三 eiy y 13 f Edz
3 f.dz Old hoth Midterm in class on Oct 24th start at 250 pm material up gendogchpt4 sn to Hw odd numbered problems in book t f Edi fit god baer was to calculate TH Mi Suppose f D 74 continuous on domain Dd f has an aniderirateFD Qie.IE kfcz Then for any contour in D initial ptzzdfindptZTfftldz FCE cl FCEJEX.GE f Ed ftp.z2Fczijz30ndmain FIZFflZJn.name Path from Zz 1 to Eii paths in domain ffczldz FCZ.cl_ i 3 1 3 Simffczidz ⼀⼀ Same end pt NB In theory F is analytic on D anti on D
as I hang yt D ani In particular Fai Logan is an anti dane y fl wf n ondlc o.co but not at any point yc o.oJ.ai.EE Compute G wueenn DE.it b fi Sd a in this case T contained in D clc o.ci domain of Log FCE i fidz FCD FC U Logcil gcij LogliltiArgcij ghl iArgc.bz Lpga Lyy b The path intersects branch any Log i need defeat branch e.g I defined on 4 lo.co F an anti derivative off on D TD ffczdt fcil f.ci ang.co 2 logcli iarg.ci Logcc iangc iJ defeat from a pfjTHM.JO that is a single smooth arc
D 11 e l g f Tram by tc.tl tab ffczuz jdfczctnzitidtdT.cat F d t fczctnzitlafbflzc.tn Zhdtaff dt Fczcb.j FC.EC as ⺠⺠ FCE For general Contour Ent tmfhtldz hfctidtt tfhzldtzfiii.FI Grid If f is anti on domain D has an anti derivative on D Then ffczdt oifTcsanylwp.sn Pf T a hop Ee EE Fczil FC.EE 0 ffdzo Ex Consider fltzidznEZO.TT If htt then f CEE has anti derivative Till Ezo analytic on all y D 41
j u y j if htt o n n t 20 hcz zidz obyaol.fm 1 anti derivative could be a branch of log At T hits branch at for any branch p is no autiderivit.ie along soil Zo To calculate n_n now calculate is h.bz using some branchS ftz ideff.dz_ 名品 ˊ defence of two Successive branches of log Gto 2 This calculation works not just for a circle but for T any simple closed contour s.t.to lies

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