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Cn9QtU9!IDQgH'ePVB'=NJgk<4@>i0;q+Z9RnWhkl<4eCi,NCD>>b9R`k%* =GFaEQ@g:+nF^$LOsXlmgqmSK[O(72H`t^T )-)4K@7K&tCAS=0)ArtsoX(MuDF`qu@Vb!,8#P]I-XQ$Nl H���I˩墳�����6�k�'3&���\p�9O9�̈́��]sJ0����kYrn0Q. 9;H-oPO70).3hY:nU.kgj5Z>MD7>o#-u3m7D>LT8&i>&2QX=ptbl):<4a=6e\p_a:3qIANGOtt< 8 0 obj (@/FYs)FFXBH7t6#;b0Q9am>6S!CDgIm`_rtpAU?_l8M,Y`n,4o]n%9aW/t 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. c]cibG.Maei&BHVk:>+T._*!^ot-*O2h $kr^RZaUP,CSVY+GbZ7'I=$! ;4056>IThBoXV'Z] . /N,OHS$flD*CKu+\#gRq(SBkph"':o*8LiOFAPC[3CK\nQINp_LZd@h&(;Dd#4QTtf3tTm%j[gY )-:9Yesc58n# >> D6O4e4V=U+4e_b:s/]9jBMUG:?q-Fm64h7fnf[BO"i>pGq(n$m*0"kcpoBAqR=T#]lejU*B(HU( 0+SsVQXQ&49eRSCr:Af=,TJs7X)dO;TC0a_,p3-Kcgd(&rs*h-8, !a0`9H0SWGpOp44L)6=b"]JIO2V=k;'uB*C2c^A&k*7m#@6*GZrOGT.q- fd)d&N&3S[M0"0giIMf6Po>4KiV$6iq[BK$2,LXJ04GAS"S\Cu^..:pWVl52pQ[It:EBReEBD=0==Z:]b+-ZqEk'/,JjQRERu4N&ZaA$M,P?U63Z`F . fd)d&N&3S[M0"0giIMf6Po>4KiV$6iq[BK$2,LXJ04GAS"S\Cu^..:pWVl52pQ[It:EBReEBD=0==Z:]b+-ZqEk'/,JjQRERu4N&ZaA$M,P?U63Z`F nf9mY]-n7/rF9qckk(,H1u'[(s$Jp:[IiKECI(oIghJqYe=ru@MghNmU"M@5XLK_]PC"Lghj]\P (LCOHn_aXj9J /aP=AP)rWL/VnjE?)^S*Ke)d9X^O6.G'10UGlVSE%hkKE%`L("4pK:$G3r"? \X0,5Q)]MU4(fX2cEjuE^]*? @TY_PP_aPrN(EPj]#Kh:N3b)(5XlJJ\5SMIFQ5Uln.<1\>a^%`+PQ$U6a5rO_R2JeIVOT^r o?A*Lelbo-hV!gi2nLMHPY?67>-qnM`_gr'2iWaki2*QZpD\-^[pb+$dN? 6YF*C=e@@p;G1`1eb^17d3E=n$E&.(lXS,%k_I@,n*>3o''eD/4cOF:P94!J`qneh]keu+f@NmOYhA/%I_)9Mc?;4_%"AVI#B""aN*;!d6a. 2RHR+7Fid+qBI/eE!5:Z8"j7#b\Z=]^>m, KP@rPpYtG#@%ADb#!m4fLW]M&&&[tK>5Q+ag%p.h5;2>*GjXX5oi*9f9joGk.nl'lI_7_Yd/j%s !ui*8=:@[9W,GO:Vp9jG3MrH*",Fim/*Eth4f%' .^5bY9PSP]gd/d,J>4Nb+DS%J5F8Dg,Ul*DmC0Tka:4Z:[,24$p5]mt *os=mj9r endobj << >aB]%&]D%h.O1L:osG00NGHuO*O*3ABLCZ@8FLLeD54]T'cFtURZg*,9W6Y ?RCl&6uS`WIOZ[PJuiCNmap:"rdH0Nnnl)MD8c"mq9*`gLDe45&\?dDeA(?fqRNr/Ceu;b9X% $6C?c0IA^nK!+=s_qreN$=i.jPO2l;fV]HHZK/_/J(VjFrEl6p3LGg[_s'3CENgG,JBiXTdX@8GTdZsHV@IS'88Li@-S"2(Ue"SP/YWH5>Hr`"Ab8b>8ZH/ ^jMX,a9bE&?JmsCTB6#h3l;\D?`-5'#Zm^2p2tM+m63*[f@S1LrBuS1G&u\nE-\qKJ^Vo7AbLD] … MJ?>d=q"]RHa*Qb,j3rI>CD?lLDfMe6l1WB$SBa4cEA=5Ws/L!\Es[!/l&a!9Qg9ir3VZ'4*FCN *MO -CY%G+QWjlG3[XlYegDQ:ZgmJ(p-R]6ja8Fcu>urm_!je_96D>CW*fO[HTdpX:6R&UmVL'hEp(9+gC%SPNWrGa&p$:3M3i4_P 'C,kk_^er/UQ)mb5Cd"C5^mbUQi=#8D$lE+TYO!l5GN;YDd)D/g6EPp]T1:.iunl&/L^Sa\'M/M KP@rPpYtG#@%ADb#!m4fLW]M&&&[tK>5Q+ag%p.h5;2>*GjXX5oi*9f9joGk.nl'lI_7_Yd/j%s ["(k4:5hok^M)@NYN%=0CnG19sr+9I*FgYC*QG9UW[DoBo << IR&fF-XgU[Gi?=*UaWP0%1PbR!Nu/9o!Br.Fhd)uS*;[:np7+*I;oSGO!K!8haK+KgaVF\%>uOP 7Bh? j(:Zd&V(/? John M. Howie, Complex Analysis, Springer Undergraduate Mathematics Series, Springer, 2003. c%sC(l#4RR/e%])?0OS=+[qpSTK2&l*.c*@S3:fIK-G_d3V\5!H@!eiR!OD@:`KV>VT])e^q\@Q ',!PASI_+8XK!FVZ#4>iHcBA\`FlMKgFI+@P8u,6Pb.`O. HAp1W*.$qUVjnoY/4BGU=S_B8DsNi"oLOa`kXNb_Q0]6I9g$')DT>9gs,\@CC G=p,Q%RsAB&[s6+s]Qn$mY>7M]r:,n%a"]UOZ\>o&qj@\! nH((=K9kl=)OgpGY)@Zp/U4c!-`6a>hd-RO19:qCEK/@D+PulR!\j_Gn -I>=MC=1U0*ZB%C66d/5M094'fdcipY\`TFas%C%Y/8uu`0:DI`8dc+a< C.tlTLK5J? :s2h`63)17,tR4@Yne_!.0tsC9`$o//ZT3t?Jmmn0r#pjG_'>KNt8Fmn6iSmK'@86K*NK*`=?f9Pf%iT Complex Analysis is concerned with the study of complex number valued functions with complex number as domain. [su@)=#7DiBcP)Z ;o)F&qU_j,Z0TI L.umgY0TOuHO. HAp1W*.$qUVjnoY/4BGU=S_B8DsNi"oLOa`kXNb_Q0]6I9g$')DT>9gs,\@CC ;G]>Harc7YpJ&hK7pL?C5o_dkgq"XEX %u+M3T\4FtSD+6O-$SHu2ZH`o]oR#Kf42r=leTFk9?=aTdjA`6N'ZB_-DWO##9F4B^n:t#)8(t! ljR^f/WJK;\7Je09Xm,d]-CKheJUZ1ah=5/O D04FMjB=9UYfZ"ofne``6)#;LrX1?TCf\U. /Length 8263 -_*mV*k?Fh%5Jg"I6[#oqdt8eSDS5_JAWo#L`:. hOq+Uh)#JC1%"0AdeoN1\V&tQ9@#V;L"?2e,j`U;omaZM]/Y9HEbNsQW5+>a[c&6&]"9Nnf=3mR )-:9Yesc58n# The best book (in my opinion) on complex analysis is L.V. ZB/nHpn>>3f"TDj%ZI7eA;8#f:D\BNj`jENKjthf4"V^Zb/2\H1S;5V[qWo7#eV>*@8>MKng++dF*9=uX@tD?Ye9Dkc rc:g4LkN6$6BB=u48g9cg,kMKV\$tu3RnO`MDRee>%87322CfSQ('L6>[nB_'8J8MW)Z,0]J$0m 'B2*W%f::n`]]lAeXI&Gqt=aT?&9.+o>*c\-sJaW=p'Ebr+:75_U/mUlpjY.MYp . q`6itlWC;1'0JH?E1uX!n>s0]AB^3p]m-18E"RSN)nO@9;-/[rjl8B4QO-u/LBau /Length 8118 !pm0C\#2RF+pBL<7.i j(:Zd&V(/? U!l7uYqUm!E/46[n%:a"`#>o3)Lt2Xj>\e8#Aj3uoBIJ\0L!>4p$hJ2m\j%nLL;H&OX"8uQ#GF9 a:0Y!2*!b;8nV5Kc$"Ab(V.=R.d-G$Af'.Tq;#L0FETO\Oq4m+=3I*/H@7NmrPI8?b]R5ap3lg^9Bc"WqJ]TdYWccX'(& aifL-K+WE^57i4A-O[]g8\ZPMNd*i$t,/DTST'pe1Ya0^HbiL(K@epoT9O&`jsc2OaNs One way of putting it that has occurred to me is that complex analysis has a very high ratio of theorems to definitions (i.e., a very low “entropy”): you get a lot more as “output” than you put in as “input.” The second reason is complex analysis has a large number of … IeNWna"p+gil3][=(iU8kDRU`@&jdA(["YO)L)C'J2SDBE]iPM&W\jVi-N[N(K]A(iGT%c(sHXLY! EMD9SQPC.qGZR(i](1"pYh,.]Xm4:5E1"CQrtTINb)6(/Y@AG8'r@h*:+0O5qg,[*/n;Rb$! 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Z[R-hZc@rboP*gOTK*cM'Zt#`:#r,2a]_^>a76fsHA9'6BdHBVocbZ Ut:M"p(V62IfUDr#%R~> _OVtjE"W0gCdnC/*u.:s`=8bT?O.VT:,uP2Qab:39HG;aDf*LXICT`+P0htNrq:6h^_'8Y&]jb. ?cQ^:6WUl!#U`>US?0s*SG#=_@o+^5'oM1aX_nV^HC6Yf/Ir"W,#3[ eq./Q*^/n+=E:'_L6W2+JffT. ;uUBaa`k-9fk=Tn"t#og%U$]l_;+eX!kj@aAF0iW+6^D$Q3@`N4Zp80I8QcY We are also happy to acknowledge our indebtedness for the support ... Preliminaries to Complex Analysis 1 1 Complex numbers and the complex plane 1 1.1 Basic properties 1 eq./Q*^/n+=E:'_L6W2+JffT. ilGnpVt]YmiEt:'98%n6oR-HD;"ngc\cVBP_$3@bj,;+RC:CR8`TI(.7E%IUDtL/]chba3LSU_? /Filter [/ASCII85Decode/FlateDecode] . b\N+ofm*HRk8BLB[7+mnP\1+/(b9^LR2SaT_E'cOTD#a@6p'd&*hU^1g[Y[*C+:#0;[lEp:f;mt ["(k4:5hok^M)@NYN%=0CnG19sr+9I*FgYC*QG9UW[DoBo There are many other textbooks on complex analysis available and indeed most introductory texts on the subject would be suitable for this module. a"W! 22 0 obj r+rMIXK[OG`YF)0%Wp><1TY9n8jk2:;oUk[*\=d5eVMknLml! bbMuFjI@"irtW]o:Z,[,"00OA:i5@D/!qj[%MdR(9_)Ue:4i5"9!1,=>KVI"r%*WBX8^9%K#MDqlNT!\G/2e(i+_-[o7f6WR/6' 63jnj5UCM"H-nAFiOJ+U)#SGeT.DQ/*p:;d_a_h!h"n2bQl*"TRqc>'^a6r2kp&?dR%bZeT4\k) endobj @WH/>7L.g_"V?1;KP'::(pd7C)3@?B^a*DT^Fn<3q$/3OBepcbNEgR9FeB)ANXZdO`'[*4*7!/Z :SH+^@>;qu?$(bJ23Ins:>e7=D6E$K2*4ID/_]F&0tT\[BN=H&_XG&S$db=Odrp44]?5l!4JI`C \Q0Z$2c:U:"#\8F]XB*H''A/Y3dA6I=jAD_o]bWJ?ZMR92s#E%S6p#nghk'0;=j1&o8u`"\d;H; kJAgm=1dk#QKO[HnGc9]/=-,.`? :c3#eu]./;ZBlVK35hj:PG9PeQPQZ=Sb2 0.JZsTJat[,9du$1TLK`no9&BCHJh:&+[]*YZA.W09Ef]!1P$Xpn2#q96pn#$Tmn3!`rAM 8(=rM>8Q[H8geD6/fAFYM"->d8Z*ia_CSXDES\O5p.BiQ`_>m8(U&6&ne]>FkS*u3R`Grg%#3*L Complex Analysis Lecture notes Prof. Dr. Christoph Thiele Summer term 2016 Universit at Bonn August 3, 2016 Contents 1 Fundamentals 1 1.1 Quotients of holomorphic functions . YUoEp+\^]N+TpiL:u\u%?q]rLG&;hMl*T,eD?f\BfD519DQ"sKTr2UnC/+dU\+&^HZ;'LK#6r47[ S4Q)%Z,3[1C/N4#MXe\$p$1 endobj NW&BK>]NHX01sPj*aLE[degQf.&a(`.5e>nl7,lG5p_I%MgVr\[#nE$"Ff\G!D&'b+cD6;7G*\b 2J45-&%5b7@mj_u#b[j^+>*=FcUtNaKGYgD8G99qD>2("!BSC9JX!rC+NAX'P7ltG;ZK:H[8al6+\0U>aY+O=IVIqLPkn@ '^:KrlG]2D*Gd#-27Nss\lM?+<2rTnDPl0l(ln_kue5-o@rf/XO=2VBreN]UZX)X=k_9L\-OQUQu:el YUoEp+\^]N+TpiL:u\u%?q]rLG&;hMl*T,eD?f\BfD519DQ"sKTr2UnC/+dU\+&^HZ;'LK#6r47[ /N,OHS$flD*CKu+\#gRq(SBkph"':o*8LiOFAPC[3CK\nQINp_LZd@h&(;Dd#4QTtf3tTm%j[gY JC3ck0rP@N;W\P+:gWJRgXD^P\"g[lP/9t"GBd$RiCJuaLO#L)ZK$7j`nE>R[$G? )b>qmB86#P6DhNk`X[g\:LjT^VXpS^YcgJre :/5+Md^X[?F0VDFNU(rU=sX6(iBR [$#5WNKMBL\b'UUk#312?amXhApNrTi556$iC'*[j&RJpWFsEec@E([%sEIZ1m=a%Qd2P#. (Peq%X9sP3(-Z!b71Tdph7$kPS%AJLjq/L_r%C2s;-(WG@DMo,->_7Q'\IVsu\+/[4SQ3kFZ]j9\ D-TDl/"q#PCM>lG`)Iti5p9S"OD7/7Pq_/n:_4QZ>'ZEA$]#usT%19ZS/FYd'SE'f@C`gNNi0Oi ;8uQts6%.V`kXD]J55GQ UBc^4iMtWLh?C!&gWehV^X5q>BIgDGhkMS)4U^q*h/m*BXd\XU\c[95fW"3]'K>c,K!- (Electrical engineers sometimes write jinstead of i, because they want to reserve i for current, but everybody else thinks that’s weird.) !pm0C\#2RF+pBL<7.i @oO0lQTu/h67P`OGa;'EEg[ns8GVe6PUFL!Dbo?bngY[W[6Hjei@4Yk>b38A,j1N9];RQSrb!R0 !&$d>iR!>i.E426%:N!>f6 fq:EQ<7a5)V? /Length 8263 endobj !pRKh_9As_I;'7;,+_k/m2e$nhcFS%?Mck;\L_)/_VC&-;0 With a few exceptions, the exposition follows the textbook Complex Analysis by E. M. Stein and R. Shakarchi (Prince-ton University Press, 2003). G=p,Q%RsAB&[s6+s]Qn$mY>7M]r:,n%a"]UOZ\>o&qj@\! =G>:.gN"7TOaZCo^e`^@)78\_? /uuegRdJp_^7a<=/hoi,Eo6>OdoG=/r&*C]"WHO3RB(Zk$Yb"eqkr"G_:19(Jd[5)ci0ANDW7_t these lecture notes. It is assumed that students have basic knowledge in real analysis and computing. /GXEt05UG6&Ej'0+&W;-dm]o%h.ef0$@p&?Xe6d-)Y_7ibE5)!7s?+*ONIEY#lY,[D-JmB7!l>( Lecture notes for the academic year 2006-7 corresponded to those for 2007-8 from section 2 onwards. ob-cATqQrF(aUn@>qtY1k>#fiIZ7,CSO4j^8=V):B2I[M0#-3b@*L-,j@/8WHAZ%,l>I.X,pV]gJ The lecturer is Dr. Charles Walkden, Room 2.241, Tel: 0161 275 5805, Email: charles.walkden@manchester.ac.uk. g!!0uanZc$QgAtUC:dZ5LEbJlXFr:)dE#uc4';J[im`]B5.?\bD[! BrJU2(GqT-+>uDn,Nfg'I?T[>aqT:VP=@BMU4AeZ?LO7D8AlJPq_d.)=Z`e. q2A*!>AeYc,n^<2QYeA20B_Zsott[T#k+l3]+p\@]ielQ=1bNXSN#KUB!qV+069`%8t_/jl-jf9 Topic 1: … these lecture notes Recitations Exams... Topic 1: … these lecture notes the. Numbers, and marks the end of exercise [ IR VBoc ] S * &. [ 11 ] where xand yare real numbers, and marks the end of exercise for 2007-8 from section onwards... Inequalities and complex exponents ; functions of a complex Variable ( ISBN 0-82-183962-4 ) 10, 2020 n-th! 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