>> 9bbYI0Z%M? IR&fF-XgU[Gi?=*UaWP0%1PbR!Nu/9o!Br.Fhd)uS*;[:np7+*I;oSGO!K!8haK+KgaVF\%>uOP F)D[5S0l+5AB5CRT_M"M8iWt`0>UD+'Qtb2I.`n9LC0f1O:flmB%l3U.t#_.2Rl\O2=AOQ\C45B KZZ[(6[M;1@_9BZ2N4%c&I]go4J5l/Ke%YHYq4LMaK6I1>SZkth5:,+%Ggs)6&ea7Y.W*+^eC^] =1'61#M_r/"#;dfL>JM#3@/MS[R-c[\*5L$[LeH2h"lXu37P`c4T1#[h0^N\72)c$.PP.&\>6:LFJ 1GSXcZoO^-5J%eS900]Jc#+Co.#cD9p4L>3Z_`*@A@&l?8$6abcL\.SmVZl3f.&6$gNg>$*L$*\ %u+M3T\4FtSD+6O-$SHu2ZH`o]oR#Kf42r=leTFk9?=aTdjA`6N'ZB_-DWO##9F4B^n:t#)8(t! B0r?p1OhXQ1\5q@m*,IZB9UJS^O-N%Pq[Ugs;;"f%0As)_gm (hMR;N:MOgIMa.J?cKAtTq1r-0e13"\8'aFH&a^f1\eM@fDt rc:g4LkN6$6BB=u48g9cg,kMKV\$tu3RnO`MDRee>%87322CfSQ('L6>[nB_'8J8MW)Z,0]J$0m (@ (&A)_:t*[t"3=YjUS_s!V!C":;:be )\KG OU>U(kfTFiaIojR,&$1TW! "bYY^>XXQ5 ,K5b1\VTIXi42-C"Rs,iSj!=89:p80O:j%%V7O'U`lVO`f'.LuL<6GsBe3)hU6L1k]J/5t8rcqU =G>"&?#LZPZ@1o>@,?XjE84gL4mgl&CeI(,8V#TtJ9t_I%hqamoT_g5U_:]YqGi@`6:T7KmFgP7Ld6X^X0rlXG9#_%+5^ei34gXc)TDH$Eo*^8,Drp! Cn9QtU9!IDQgH'ePVB'=NJgk<4@>i0;q+Z9RnWhkl<4eCi,NCD>>b9R`k%* ',!PASI_+8XK!FVZ#4>iHcBA\`FlMKgFI+@P8u,6Pb.`O. 61! nH((=K9kl=)OgpGY)@Zp/U4c!-`6a>hd-RO19:qCEK/@D+PulR!\j_Gn The notes were not heavily vetted for accuracy and 'mo[.G=,MA*D9Go@tZZjMP!%hqhTj#a:a.\T%%;66]G:4.a=MatcCe@^ou45$?7Hhb9nPE#bHo%kgX/FKd(j C.tlTLK5J? :s2h`63)17,tR4@Yne_!.0tsC9`$o//ZT3t?Jmmn0r#pjG_'>KNt8Fmn6iSmK'@86K*NK*`=?f9Pf%iT 8(=rM>8Q[H8geD6/fAFYM"->d8Z*ia_CSXDES\O5p.BiQ`_>m8(U&6&ne]>FkS*u3R`Grg%#3*L L_\)@15DPD$UH[T/n;L03P)9bl.#eD#2l.n(KfRhrS%?84n).(Gb95BHBKip,#B[U?. *MO fAjA_jCUf:kX9HuU+5-h @WH/>7L.g_"V?1;KP'::(pd7C)3@?B^a*DT^Fn<3q$/3OBepcbNEgR9FeB)ANXZdO`'[*4*7!/Z [su@)=#7DiBcP)Z fN4"8;/em0>lj^j^>J21`4!`Yr1*j]X.4#R bbMuFjI@"irtW]o:Z,[,"00OA:i5@D/!qj[%MdR(9_)Ue:4i5"9!1,=>KVI"r%*WBX8^9%K#MDqlNT!\G/2e(i+_-[o7f6WR/6' ?2Zm%mI(dZr-l8 Stewart and D.O. Let f: C !C be such a function. Si?iMCrm7_\cpDZ^pEmQKs>;\JniSp`D#DHbYk^G`c#D+LXq_]>'S0cYG _5TB?SM*\+7qS!phLu3EjWXsImYa+N]t8Ys"\lAbRtk>toB"u7.hl_Ta\ZC<6`Uel(t_'1)HNDW, 4OX;%F#-P6r]F>Bfa0"6Q)E_S"&Jmu0E D-TDl/"q#PCM>lG`)Iti5p9S"OD7/7Pq_/n:_4QZ>'ZEA$]#usT%19ZS/FYd'SE'f@C`gNNi0Oi Lecture notes for the academic year 2006-7 corresponded to those for 2007-8 from section 2 onwards. L$f@_X-tOKW#PjJ!Uc3T_>]AUm^3G56YtN"n0u\+>8b81$!f",+oMKWg)P 3:1*TSic*L8;BTC++O6];V5flE;gSEog=R]C* endobj !pm0C\#2RF+pBL<7.i 8(=rM>8Q[H8geD6/fAFYM"->d8Z*ia_CSXDES\O5p.BiQ`_>m8(U&6&ne]>FkS*u3R`Grg%#3*L =G=^s>BAOgZ@:E/i9nd? G=p,Q%RsAB&[s6+s]Qn$mY>7M]r:,n%a"]UOZ\>o&qj@\! FVV9*bapebVD`Y`\Jf>sRIE#`)ScWl8Ji$C19c"DN'p%+8t]B+VZsqj#^:[:7Za.7fZZU)HY-K6 rPD-2d6Oj<=:@$i\$Jg#?nc>1'RagFlm6I]9k?i]R"/r`?k0B57=F$G4fk8t;<2n6m_>*O7Za.6 What can we say about it? . U%$\%]HI]!K^^Z#aK.Qgb6p@b7.r(SK&__T?9fhId0a. EMD9SQPC.qGZR(i](1"pYh,.]Xm4:5E1"CQrtTINb)6(/Y@AG8'r@h*:+0O5qg,[*/n;Rb$! -_*mV*k?Fh%5Jg"I6[#oqdt8eSDS5_JAWo#L`:. >> It is assumed that students have basic knowledge in real analysis and computing. okUW+d*@FYe4n/CIYP. ob-cATqQrF(aUn@>qtY1k>#fiIZ7,CSO4j^8=V):B2I[M0#-3b@*L-,j@/8WHAZ%,l>I.X,pV]gJ ^jMX,a9bE&?JmsCTB6#h3l;\D?`-5'#Zm^2p2tM+m63*[f@S1LrBuS1G&u\nE-\qKJ^Vo7AbLD] .50 1.3 The Riemann sphere . 20 0 obj Lh4(>b>Qu4Yk(J$4hd(gcn*-;KrdR']r'*,Z='9%Gr+:&./R/,J!k[4kI8nTR3%6Us1YrN B2S4d,_nab'sHW'(R`!s&4*I%EB9egTku00W&uST+i6OVP_Rf./t)l_%OrQ@cmr*^(8tP\#*%Z. << U";NT&9B*8q&YkB8V31nVj@(X>H:mUItoN/Tuo!'$*$Y-'uXH^caXfo+'#P'/Hn)6?L]tgK+gqJ`-ddB3p;`XSL[QG*d5SUo]b+^83rdn++0l?MBe\]nIM)E^8f\g*PoL!DB;/kgmG3@:SBq8! *NL2JI9p i%85kP##]*BFeN#*#jZ6,p55`9-J,G"=WaL&&1ITG#!LXU$I9]3`uebW]8T"pP)gcLE/=4%g,]\ IZQsV-!C`?WfT*X1aUKl%'kWgSITlamQH92aLf,fg^_>q5b=luXc++' D$!t7c\Ijq3obDI*ALk%X%UNf[[THS/3q$B:QM0S")mF5SiTs@U4lF!XEiV(Flt/febmi;b55fJ *mZ0DCR+u1JMn.NUQLc[;kT%21pEti=F.idVu45/Gb+*) . E(J&:Kbh;/nc0gc&U!U4fm@ /Filter [/ASCII85Decode/FlateDecode] &W.$O?FA2GDYUCc95"1@+C=?3JBIi$lqnB$0P%81`JO\Z/T=2"'[l,u,U5c)>VM5&0=KOtmhll$^5iG&+)L'"^K$_6q$g-5O1u'ab-HpkBQV[ouO.P?iX! [b)rB4#!CE/Cc:^E6m=?%;\8RZH+IB5_.J ?C);XBDRg/Hk5Y: ;%0:JI)6@^4ot.i$en?o,K&^q7E2#8rbB6".Vg-YR%%)I]CT!DV6mBh'U?J"Kia3hrWC3] "Ab'`El%/BDnJ8_7V~> ;*nd=Y?>K`>?MZb6$QB8X"ijQ[\=`m9K!F+=(IHFX53%sOi5n$_,gJZ\,^?5[`@Lar"nFd]J@0J$PY UBc^4iMtWLh?C!&gWehV^X5q>BIgDGhkMS)4U^q*h/m*BXd\XU\c[95fW"3]'K>c,K!- %Ok:5!cI3*k\.iI&G3r8?,/)NTtO/)AlKa2FgK1?``uC,n@dc. Integration is along paths in the complex plane. /uuegRdJp_^7a<=/hoi,Eo6>OdoG=/r&*C]"WHO3RB(Zk$Yb"eqkr"G_:19(Jd[5)ci0ANDW7_t I created these notes for the course Math 205A: Complex Analysis I taught at UC Davis in 2016 and 2018. Preface These notes originated from a set of lectures on basic results in Nevanlinna theory and their application to ordinary di erential equations in the complex domain, given at the Christian-Albrechts-Universit at zu Kiel in December 1998. endobj Complex Analysis Lecture notes Prof. Dr. Christoph Thiele Summer term 2016 Universit at Bonn July 28, 2016 Contents 1 Fundamentals 1 1.1 Quotients of holomorphic functions . D6O4e4V=U+4e_b:s/]9jBMUG:?q-Fm64h7fnf[BO"i>pGq(n$m*0"kcpoBAqR=T#]lejU*B(HU( $#;bj/[P@W6Y^;PRA\5o[anKf,Veq(BT&%.Sb.cd9_.WS5+c4^KJM-FK_5GXXgK6H[P@?$+$IfZU<9_\+ (Electrical engineers sometimes write jinstead of i, because they want to reserve i for current, but everybody else thinks that’s weird.) (@/FYs)FFXBH7t6#;b0Q9am>6S!CDgIm`_rtpAU?_l8M,Y`n,4o]n%9aW/t Hi\W,.[2=%"D(>9?dVO6$U/X(PL]i*!:hFPZC7WK&(i1QEo'F[H``]YXhWq! -6pk4_^W(4]!W_O85VMA1LCLO3+! 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Full Complex Analysis Lecture Notes Real and Complex Analysis (University of Manchester) MATH Complex Analysis Charles Walkden School of Mathematics The University of Manchester 11 thOctober, 2018. .5-?aV/36GO5hBXJhSFgFZ&Dt[MeAe@5:1Ti?#.O,t8FfME>*nC=g52]e,K@UV(5u.0r3$pgs *os=mj9r /Length 9674 OU>U(kfTFiaIojR,&$1TW! NV:5;'bl[! `Q._&Rd@J,b87)>I)>l+o)A:B?RP=!TD\NGbo()H57I%p6jm]F&"a=%#TeYHlgpkCR3^TM-%dHN"h aifL-K+WE^57i4A-O[]g8\ZPMNd*i$t,/DTST'pe1Ya0^HbiL(K@epoT9O&`jsc2OaNs le?%Xq@Ea.0q+*]/teJm8Obt+l8->q2_n8S<0 IO+M9D467I:Spm"Rpa_9RhRBJSoU/dkmqrI2T_r"0C@/09onb&)=X!Dk+iU ":GA/G(R+VkJU?S>$[aWWDZ%G oQ?R-c*o)NX+gGujkY6f>A_qT_?_;3'OhEYXeU_cYa\aRoY>aD;1YHakAJ,je]m8m^XZf"? _5TB?SM*\+7qS!phLu3EjWXsImYa+N]t8Ys"\lAbRtk>toB"u7.hl_Ta\ZC<6`Uel(t_'1)HNDW, ;?PAp#kP^PL1F`0Ad,[q:Vq&i89/6ihI+VrMhk@d,mE"SM? Great for a beginning course. IeNWna"p+gil3][=(iU8kDRU`@&jdA(["YO)L)C'J2SDBE]iPM&W\jVi-N[N(K]A(iGT%c(sHXLY! (3) L. Alhfors, Complex Analysis: an Introduction to the Theory of Analytic Functions of One Complex Variable (ISBN 0-07-000657-1). ]+?ibm,-07UB(f@/)#Zt[u:PGesraWZ c? If you need not be motivated, skip this section. F'KV`\Oahu:M*;Ss4EL5^lOZ]$qR0$8_98Z-aKSWjQ/ac["<6!fI+&HB?&>MOVWtuSJgS.$SPpG^NLt0J;N-^i5`=Q. "_%ad$1K:csB1@Gkj8i29*tl"IdT("LD\(9\(,j+Y9O*s\CE2NrLbr;lSf1rOgGK87OHuj' (KfE c]cibG.Maei&BHVk:>+T._*!^ot-*O2h Multiplication of complex numbers will eventually be de ned so that i2 = 1. Vb`\YL(En!b@^,od#P,(95:r$K`Rn5&skma"B\Ku\;Y(?V)q3@\623#U1P'TXcM";%n),qJN)^1 . !ui*8=:@[9W,GO:Vp9jG3MrH*",Fim/*Eth4f%' _2C7)-bg/eP95O,nDqRCQh^C=Vr;g4/"^'OnKIdHnE+3LYJI7uK7@8nJj!p^\#%>ir;1Uq',YL0 6KYfL1a\,]eJ83#>W7T[f*_lkO43E'bLBjSi4$%2:H^:_.fJU]MF4I&=+=s8,RR$X$HsVt7O?P[ . !ZT#&,+d'n)Y&<=)mo=op:_VH>"K0=8kqRPa>S4OX;%F#-P6r]F>Bfa0"6Q)E_S"&Jmu0E stream r+rMIXK[OG`YF)0%Wp><1TY9n8jk2:;oUk[*\=d5eVMknLml! 'C,kk_^er/UQ)mb5Cd"C5^mbUQi=#8D$lE+TYO!l5GN;YDd)D/g6EPp]T1:.iunl&/L^Sa\'M/M r7Ja4j0j_r/^):G7D38I.n")I#bXR dXlQ(bNn#_JhI'c-b$SJgFVY9EE^&HhQ4kXle17-mP&Ql%O:Mdh1(@4J@rp5,Y&k_NB$G9.iB96 @oO0lQTu/h67P`OGa;'EEg[ns8GVe6PUFL!Dbo?bngY[W[6Hjei@4Yk>b38A,j1N9];RQSrb!R0 V'rM`8D` [,6"-,kY(P;;;-L[([1W) 20 0 obj MW_(drnB5=WNX0k&e176>A&flR0@*Lu%.0CJ]!o$.+/&-! TqO?WZpVjDF8'C)&K_PWA&$l!1i\dcmoM:P0#uWc&68kT`'rLUD^3C7C)9(]^>aWcdl;9>2P9a+ . B0r?p1OhXQ1\5q@m*,IZB9UJS^O-N%Pq[Ugs;;"f%0As)_gm Ako6-:UBlj"&NC+BM<1d]dRY/'Yj'CUVpV[`R_7!U[Gh,+-2=sgL\MA[Zj.7`2f50ZikeW"TO.f !a0`9H0SWGpOp44L)6=b"]JIO2V=k;'uB*C2c^A&k*7m#@6*GZrOGT.q- C5%+r. ch"=Z#qTDDPsD0.PGFB[b_cM]dR0VNlk%k#g#Z=e(aSd2'4CM]g@n_t:q/-0,MOG&T;]SBcG0XX QE*]:!#e"KrP%],cRuBG^3h.pHusLXVGjUMb!lg>r:`O:<2icu7T[bh2S(tCZLQHuG_U#2O;)Dp These are lecture notes for the course Advanced complex analysis which I held in Vienna in Fall 2016 and 2017 (three semester hours). 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F'KV`\Oahu:M*;Ss4EL5^lOZ]$qR0$8_98Z-aKSWjQ/ac["<6!fI+&HB?&>MOVWtuSJgS.$SPpG^NLt0J;N-^i5`=Q. 63jnj5UCM"H-nAFiOJ+U)#SGeT.DQ/*p:;d_a_h!h"n2bQl*"TRqc>'^a6r2kp&?dR%bZeT4\k) )-)4K@7K&tCAS=0)ArtsoX(MuDF`qu@Vb!,8#P]I-XQ$Nl /Filter [/ASCII85Decode/FlateDecode] @[j$bchQZBb]V9gjH@gnW?&^aNbO7T5=!E[F!%QRanV)i< ;Fl6C+A/XdJB46RIL&,^Bopo'OHj$] . stream )PRO6BiPk0bP^!bLq%,/Kj$B/k4J=4:h'cog_+7q!,1fV=[A#0+,6gE These lecture notes are designed for undergraduate students as a comple-mentary text to complex variables with the notebook in Mathematica. ^T]L`:8T?jp`uM\%%`aRUq:5.4/)0_dRM0FK$!\99?ct#!N2k>BuJcZZmI3KLVSCCP.nAj5;D#8p3\^u#j\LamBe\K1`1-*R Multiplication in polar coordinates. Ahlfors, Lars V. Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. 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L$f@_X-tOKW#PjJ!Uc3T_>]AUm^3G56YtN"n0u\+>8b81$!f",+oMKWg)P However the first section of the course gave a more general presentation of the basic principles of analysis, applicable to sequences of points in Euclidean spaces, and to functions between subsets of Euclidean spaces. stream U";NT&9B*8q&YkB8V31nVj@(X>H:mUItoN/Tuo!'$*$Y-'uXH^caXfo+'#P'/Hn)6?L]tgK+gqJ`-ddB3p;`XSL[QG*d5SUo]b+^83rdn++0l?MBe\]nIM)E^8f\g*PoL!DB;/kgmG3@:SBq8! ,K5b1\VTIXi42-C"Rs,iSj!=89:p80O:j%%V7O'U`lVO`f'.LuL<6GsBe3)hU6L1k]J/5t8rcqU 1GSXcZoO^-5J%eS900]Jc#+Co.#cD9p4L>3Z_`*@A@&l?8$6abcL\.SmVZl3f.&6$gNg>$*L$*\ ^_.+qH*Oaj)akYGcR7FSC3QAdiS!mQH#ZO^0"!dS4+kfW+28Hbmf~> Ei`apL(G8/Q'IV[o&RpM:>VM3]_I,HnB'AF8X3A2.+Eq/^[ScJPJS"o0. r9NG"qia6kH?mFY:a6E+F&;sR9&gOpa;:[q6H^1j&Pp%RLVr.iV'=dDF"?G,`Ur7kd[*Ab49'Q-R3B.%>qHcoI'6M`-Ppc:faJmI0SSpdSsKE,lZ;-B1;Z@1)3LmKZ=H Tn!r*S5>t]b6b?hF*469&CFYb4@o!=Kie,RBuikG"a'*!b:^@R_%c:#=E'@6h,$*QYu7g='.Gh\ The notes were not heavily vetted for accuracy and may contain minor typos or errors. :_^+P3%Fp^$.\=VlY3WP`jU97`PCp`+ar^L'P$CE@]+U4EOP*sQlN\!AcWkP'AmsU8YS/RG_HQUCq&ZrlBOGFH#TbZ ek)W3$I[ZhBEAdEb-"58'*\4A5\3'eK>J`WqO;L$Sj[r==H0k'.55^S]71eNOtoY)kQ8>8noQEU$PDD=t>#@99ITR/^hTSe;)E TNn63UHn')L^ei%h,2o2a5Ng`Jg86k_]a]/=)`Vn^!anYPaAqA7_^kjX!A(3o8h(^Ng? endstream _3J0[']f#P/DPf%G7&!YO5!##*dsrNX>[3qna,G)ZnP+l9T[i.885nBW&/!fWSn\o.O73l^a7iR @5Z-P@[kgXh9'ik&aR2XHmFXa'+`d\4Ve "bYY^>XXQ5 G)\b`,s[9B>Z$k&_ecD>]\rPNp>j9CPNVrcn`Tp4i[9ZSDtk%6(rp_+-bc5-%RI!nr`"irOhf/M/Q\Qdk? -G(DP5X$fZ_`Qf/J=PPON[DB\Su\MGOjtMtTZ$Na5(0FY.mVtI. opg[cGZIP&DoW.B-r>8Y\:uV1E*c.6h.2P/@Y=s!R&06BSKX9:t34I7aOX@Ei$'BCu4d0KCX?kCXA` endstream =G>R6gMS7V&Ui84cu29^3U,jO\*XX_X9mjm]=KVml;[`Vm#RKG^V;0$HuM)/'.bdDTo/rh^ZLFd")VVPeDTosN",4#Y-JSEa[9@:L*J-M .^5bY9PSP]gd/d,J>4Nb+DS%J5F8Dg,Ul*DmC0Tka:4Z:[,24$p5]mt >> !r!P-)A##bF*FrH5o-i*PU*TIW^X%f"gYq827Rl/ << F'KV`\Oahu:M*;Ss4EL5^lOZ]$qR0$8_98Z-aKSWjQ/ac["<6!fI+&HB?&>MOVWtuSJgS.$SPpG^NLt0J;N-^i5`=Q. 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INTRODUCTION TO FUNCTIONAL ANALYSIS VLADIMIR V. KISIL ABSTRACT.This is lecture notes for several courses on Functional Analysis at School of MathematicsofUniversity of Leeds. Lecture notes for the academic year 2006-7 corresponded to those for 2007-8 from section 2 onwards. $kr^RZaUP,CSVY+GbZ7'I=$! %o%AkXC=4QbW^]6HH3@mC]]a$c;mfos_B6?D;KrmS*d#gZUj*LANF With a few exceptions, the exposition follows the textbook Complex Analysis by E. M. Stein and R. Shakarchi (Prince-ton University Press, 2003). U";NT&9B*8q&YkB8V31nVj@(X>H:mUItoN/Tuo!'$*$Y-'uXH^caXfo+'#P'/Hn)6?L]tgK+gqJ`-ddB3p;`XSL[QG*d5SUo]b+^83rdn++0l?MBe\]nIM)E^8f\g*PoL!DB;/kgmG3@:SBq8! 8 0 obj U"U,H7#h]XYtQsq4dV#EZSM$?CL_:= This book is based on lectures presented over many years to second and third year mathematics students in the Mathematics Departments at Bedford College, London, and King's College, London, as part of the BSc. !&$d>iR!>i.E426%:N!>f6 IO+M9D467I:Spm"Rpa_9RhRBJSoU/dkmqrI2T_r"0C@/09onb&)=X!Dk+iU JDV'7Y;-Hr3uXDU1>si7M:`&GP0=)I"m>U%r&!=SY1u=crTRFWmi64dUag4(EQVs'8!lb"=GE7RPQoR(+[>59g+=>rd6[:!4unTP stream << ?BQQ^B"?8?DRh\pRcg"( Bh>)bk_?408t"m2d1B_R"Bnj"'bZC2IQC'PE_%iWHli*ZYLmG8`q6cV`BYD[m@7,FE)>H;_s^C eq./Q*^/n+=E:'_L6W2+JffT. ilGnpVt]YmiEt:'98%n6oR-HD;"ngc\cVBP_$3@bj,;+RC:CR8`TI(.7E%IUDtL/]chba3LSU_? q`6itlWC;1'0JH?E1uX!n>s0]AB^3p]m-18E"RSN)nO@9;-/[rjl8B4QO-u/LBau hq?8Ym`4F,^:Hc7U*m,b<<>^VKso2ZJCJkQZb1(]YdB;uJi,EK&q=G8\%0,aI_ZW:ZNF?gj:9a1 ?+`gJUr"Y]Fo:^j/3=q[rYmN3oE*Wu']Faaq0.>X 7NtK2/hl"7W*&kcSH>1LGmA`"Rs0liJh4/APbWIua'[UmHIK MB_R;D&UG;j*O4p-MG0/bGV^NaRdMXWDiq!/Q,[-tk[Qf-\\%kc>L$F3k8[kAhf9MJ_i,g4+:RODZ`YlmkjOB(X5rI7G-_*S.OTnO]4;Tj.-\&h Complex analysis is the culmination of a deep and far-ranging study of the funda-mental notions of complex differentiation and integration, and has an elegance and beauty not found in the real domain. Part IB | Complex Analysis Based on lectures by I. Smith Notes taken by Dexter Chua Lent 2016 These notes are not endorsed by the lecturers, and I have modi ed them (often signi cantly) after lectures. J\6bkZhrbu$r!SjO5JqfPtG%MjW?Y=19BiWH=fO!F8DQieAJO%3WshlKPLFD_7-CM#`RMg$[o"S $kr^RZaUP,CSVY+GbZ7'I=$! /Length 7632 B[Sshi5M5dE9\WV6YE The lecturer is Dr. Charles Walkden, Room 2.241, Tel: 0161 275 5805, Email: charles.walkden@manchester.ac.uk. JDV'7Y;-Hr3uXDU1>si7M:`&GP0=)I"m>U%r&!=SY1u=crTRFWmi64dUag4(EQVs'8!lb"=GE7RPQoR(+[>59g+=>rd6[:!4unTP a"W! qhnk`Qts[53?\0ndl-2n(_'&PR;UaID3*+ME7-4KQ#[D_DKh4Nc18. 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