KP@rPpYtG#@%ADb#!m4fLW]M&&&[tK>5Q+ag%p.h5;2>*GjXX5oi*9f9joGk.nl'lI_7_Yd/j%s D6O4e4V=U+4e_b:s/]9jBMUG:?q-Fm64h7fnf[BO"i>pGq(n$m*0"kcpoBAqR=T#]lejU*B(HU( 2dDK(5`T+UOgrfeM*5qKPj\;5%@iliI/:%VT[a_ZhprmSXgIi=f/pTkBb0@e''5/GD%_&ZEqR^m F'KV`\Oahu:M*;Ss4EL5^lOZ]$qR0$8_98Z-aKSWjQ/ac["<6!fI+&HB?&>MOVWtuSJgS.$SPpG^NLt0J;N-^i5`=Q. stream =*LE2B]]<=lu4et=Ckd!2$[-;]r2J@kH98;AB)Qm._87dSICYdq:Zc.Y/$"r>-Vj#.7HiX>I+bl>l2T-LcIsG>uE]j"^Wj%F&u2k'-E7+1rfR'a2U#'m?,_DjHaUm;h7?B)@#p)"2cj^:\j&I?1R: 3:1*TSic*L8;BTC++O6];V5flE;gSEog=R]C* *T9CTST-ag;PPbY-VC&\HUMg=rij+a2SWW0?kI"0?_C4,L[Lh``K[qbi6p=g' ^/EY2`^WMX_1j\)5`-[R>T%N? Similar Classes. ;G]>Harc7YpJ&hK7pL?C5o_dkgq"XEX [N'#Q5; JqX? a:0Y!2*!b;8nV5Kc$"Ab(V.=R.d-G$Af'.Tq;#L0FETO\Oq4m+=3I*/H@7NmrPI8?b]R5ap3lg^9Bc"WqJ]TdYWccX'(& )cOI30RIIGVDEa('g[WDJO7]FeYg:+P6__V6EX:03a"D=*T&p4j63VT,M@Qnf JC3ck0rP@N;W\P+:gWJRgXD^P\"g[lP/9t"GBd$RiCJuaLO#L)ZK$7j`nE>R[$G? ch"=Z#qTDDPsD0.PGFB[b_cM]dR0VNlk%k#g#Z=e(aSd2'4CM]g@n_t:q/-0,MOG&T;]SBcG0XX /aP=AP)rWL/VnjE?)^S*Ke)d9X^O6.G'10UGlVSE%hkKE%`L("4pK:$G3r"? I should add that the complex analysis as is taught in the math department is rarely used in its "pure" form in what most people perceive as "real-world applications". e/F`ubY"=cT2D!j(A=Cr6B%,5:9NRbFq:Fc:1\I+>)q>WWH#MprLLp[J!hOks`%Y5"IU B2S4d,_nab'sHW'(R`!s&4*I%EB9egTku00W&uST+i6OVP_Rf./t)l_%OrQ@cmr*^(8tP\#*%Z. .^5bY9PSP]gd/d,J>4Nb+DS%J5F8Dg,Ul*DmC0Tka:4Z:[,24$p5]mt Perhaps you struggled because you took more of a math approach to the course (lot of rigor, theory, lose the big picture perspective, lose applications motivation). Si?iMCrm7_\cpDZ^pEmQKs>;\JniSp`D#DHbYk^G`c#D+LXq_]>'S0cYG It is clear that any short answer must be incomplete and highly subjective. 0=4ZP`(kXf%u0O7cSE\:Vf9-pnN7*Jb6JDgUr?MqP:0_e?\cEG4KWH MJ?>d=q"]RHa*Qb,j3rI>CD?lLDfMe6l1WB$SBa4cEA=5Ws/L!\Es[!/l&a!9Qg9ir3VZ'4*FCN Complex numbers and complex analysis show up everywhere in mathematics and physics. stream /Length 6412 *MO 2.b:[[C5ZD@6&>]8N=?1/YCsoO88B*jKpKNUS.R3AW*BdKXE[8QLmN9Y Sh\Xthrk6,WMX[q[jUNEKbdQnK3/J^';uuNC58egZ?bp!V`W-d'_(m),Dkh)2cbEIAr6o.11do=0@Ui&.\B7l$ZFU^g3TISo0 :/5+Md^X[?F0VDFNU(rU=sX6(iBR C.tlTLK5J? .,c[ABa!^H&Ao'(hbQ\WcmH+B2>5%(+QOp%^n.5c^ur /Filter [/ASCII85Decode/FlateDecode] B0r?p1OhXQ1\5q@m*,IZB9UJS^O-N%Pq[Ugs;;"f%0As)_gm nb2fkq<2M3a6`K_jjE;LUZ7]9s1,IQkI/uP4j"PO6b@/)`;"C'YBQ%Zt"YSJr='9jW=;$Mth"$. e/F`ubY"=cT2D!j(A=Cr6B%,5:9NRbFq:Fc:1\I+>)q>WWH#MprLLp[J!hOks`%Y5"IU W'e9=-uiRnQ4O3eLAc5kjZ=r/f3>tt%E/,U7bp/?$pm5Eqr0AG"e.&N&`c./`8j_ae3^O(#Z pIao_P$XD"_I*S_PW6>MbFW4&A4W+#NbSDV);FpPmeHl$bl(-gfK&nD)q_ek. -CY%G+QWjlG3[XlYegDQ:ZgmJ(p-R]6ja8Fcu>urm_!je_96D>CW*fO[HTdpX:6R&UmVL'hEp(9+gC%SPNWrGa&p$:3M3i4_P ?mK2mNsS([R^i\ << \4*aS@Wf:dhpHQ%dUYZ+EKatIWVKL4c0colVXE"ca"?21:j>ofhNg'.pbD]hEclU-@3*NaDl. 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G)\b`,s[9B>Z$k&_ecD>]\rPNp>j9CPNVrcn`Tp4i[9ZSDtk%6(rp_+-bc5-%RI!nr`"irOhf/M/Q\Qdk? 0JS[*l>Kl36DI^aZ\FC6A+6BVjul3kHq9AeQu$#eI$Id+GM`S,d(o3WLP:c.oT 0C3kbRDfMfe*VUlE;NX^c;VftYm.Y[G'hgQA+)]RCkY:VCrWs&Tj9fcG1?8?$)!%Pc>?,f#n(s2 (lEnQB+!Oj!+K5r$?Mqhand>qa`CqK3(bAZ`Sk/nNh2GuAZ!obd)mM"G0`4Q.j`pJTk5Pt?c.=q KP@rPpYtG#@%ADb#!m4fLW]M&&&[tK>5Q+ag%p.h5;2>*GjXX5oi*9f9joGk.nl'lI_7_Yd/j%s Specific examples from quantitative finance: My question is "why are you majoring in mathematics"? Here z= x+iy. IR&fF-XgU[Gi?=*UaWP0%1PbR!Nu/9o!Br.Fhd)uS*;[:np7+*I;oSGO!K!8haK+KgaVF\%>uOP /Length 5373 g`$:8P\!9>jQ_a*peWUV&Wi4dV0hiqBgZVs!YiLpEQ5F1YmB4>lH9OlWJuF5OQ!oNm6UQ?lO%[07D-jgN4EDp0\Q*jX&.A(J]IH@msUpukJ1jmKg5c[rF7Y]OX,Ti)OJDqDIO9KI4lc#@4Qon:bfD9Zo?-L]g#Plm! =G>"&?#LZPZ@1o>@,?XjE84gL4mgl&CeI(,8V#TtJ9t_I%hqamoT_g5U_:]YqGi@`6:T7KmFgP7Ld6X^X0rlXG9#_%+5^ei34gXc)TDH$Eo*^8,Drp! NV:5;'bl[! ?C);XBDRg/Hk5Y: Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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BrJU2(GqT-+>uDn,Nfg'I?T[>aqT:VP=@BMU4AeZ?LO7D8AlJPq_d.)=Z`e. *Shdf-%]sd+IEkAI:;nQ9dEB)F(rR,?r5Er3hH+0mp4U"#79F'&n U"U,H7#h]XYtQsq4dV#EZSM$?CL_:= ["(k4:5hok^M)@NYN%=0CnG19sr+9I*FgYC*QG9UW[DoBo fAjA_jCUf:kX9HuU+5-h !N6.JVeQF'hmB_)K6s8k 2.b:[[C5ZD@6&>]8N=?1/YCsoO88B*jKpKNUS.R3AW*BdKXE[8QLmN9Y 'ie(`.H7A5LQdh,[8$T_h":/]Uq;cT`4XA\-_dE)>mA^_RO3TnpB,\F!V5_\/6JapXW;Gb#EoT 08.ruAHd9L19,/+.[WH9^kHc_dTU?h/C-8H:T0T;RO^`ZJQ`W$I8B.VkC6S)CD?D"kO\#Ch>? The complex analysis may be a cornerstone of mathematics, making it an important element of any area of study in graduate mathematics. stream -_*mV*k?Fh%5Jg"I6[#oqdt8eSDS5_JAWo#L`:. Nq,ie#-`scF!q@edEgoh(/5Fh>3*8h%;pobF2/9>acK5M_ick^fV26+VbE.HeU]M9U_ZW7tIj_]>gh]`AKG*IN$M%k@o[tS38d^9`*<4*WlplB#-'i<9l,0KW-&QHSLJ]nr31K%csY,UPJ^X (Eb)Fm2!D`('+0Qks`iQ";)ocM63l EMD9SQPC.qGZR(i](1"pYh,.]Xm4:5E1"CQrtTINb)6(/Y@AG8'r@h*:+0O5qg,[*/n;Rb$! 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Your email address: r7Ja4j0j_r/^):G7D38I.n")I#bXR -G(DP5X$fZ_`Qf/J=PPON[DB\Su\MGOjtMtTZ$Na5(0FY.mVtI. =G>R6gMS7V&Ui84cu29^3U,jO\*XX_X9mjm]=KVml;[`Vm#RKG^V;0$HuM)/'.bdDTo/rh^ZLFd")VVPeDTosN",4#Y-JSEa[9@:L*J-M 24FgOV^k*U]0 _%5[78pE2++Ao)6aFat8@qChr$L53n/gL!Sob=7A^\JK]C7^[S4FQ0&M@1r+Z[;n0R_5:CoG0+Z jjLn%h0DT0E?`X_AdX%E+7;8O*pU,RV'0!o. 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This article is about Liouville's theorem in complex analysis. g(Sebf:ILoYdcdZTCLQlU"-5SCid+]qkKFC^YR. QE*]:!#e"KrP%],cRuBG^3h.pHusLXVGjUMb!lg>r:`O:<2icu7T[bh2S(tCZLQHuG_U#2O;)Dp c]cibG.Maei&BHVk:>+T._*!^ot-*O2h 2`:Da$bCpSXGJ=;"L::e$j,R?]pN`?1T!WuC=Wl18HH[XPY2HBB65b):m1A,GUPQiPCOUnu.>Z6"kCrV1$^dW;E&]T&! 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Tn!r*S5>t]b6b?hF*469&CFYb4@o!=Kie,RBuikG"a'*!b:^@R_%c:#=E'@6h,$*QYu7g='.Gh\ gpV"%VbmXV]O8e6QOF]PaFHd#Ke7Qk^ADOoJ"rU9BW\:Hi8>%HL>1-)GaX4/C =G>R6gMS7V&Ui84cu29^3U,jO\*XX_X9mjm]=KVml;[`Vm#RKG^V;0$HuM)/'.bdDTo/rh^ZLFd")VVPeDTosN",4#Y-JSEa[9@:L*J-M 18 0 obj \Q0Z$2c:U:"#\8F]XB*H''A/Y3dA6I=jAD_o]bWJ?ZMR92s#E%S6p#nghk'0;=j1&o8u`"\d;H; '-TTH:t0).8_f\(d@#Qh`-3f'J0Xg1:WA>Dg8Ch'M:T!_HqNCtrY4&tV#Qo+'L=UF#o3`D&HKE95cXM?0fZ0)7s&0rg=8e6`aM The question papers are sorted AJ9h"RE0f2>3?aI_P8B^YuRhlB6KY?$ac+J(,U'$RL+D6n!^WDV8earOA5QFCq&N+nNhZXZfLc` 2aQd4IhK:!c_enYl6_,")Uk=*%9"qd)"T"So8Dqbgm`U._LYd&&6HY4$('"D0klCas*?Y)$CObL [.,@q()_!t7L?1aJ^p6&F+;n"-URhq7.1n?OiN Winding numbers are important objects of study in Algebraic topology, but actually the first ideas of using them came from complex analysis. c)uh$8Y^%`gc[QV^.AO[k9s'uj5-Uk$(*#(3(b6bKTr`P8g./&L27Tnn?.:Y,[. 9bbYI0Z%M? H_T]/*s/j/QS;8l-SlE.k]@g-H8_!%R^O)!8IY8^UP-d? Z[R-hZc@rboP*gOTK*cM'Zt#`:#r,2a]_^>a76fsHA9'6BdHBVocbZ r7Ja4j0j_r/^):G7D38I.n")I#bXR B2S4d,_nab'sHW'(R`!s&4*I%EB9egTku00W&uST+i6OVP_Rf./t)l_%OrQ@cmr*^(8tP\#*%Z. U%$\%]HI]!K^^Z#aK.Qgb6p@b7.r(SK&__T?9fhId0a. O?9BP86L!$%tB'olPoEZ&B'qUUe5b5f'frpH1LdmGQQ4r,-u\b.$31[F\+h`LGfIO\=_Gt_>`QTWf7c! )-)4K@7K&tCAS=0)ArtsoX(MuDF`qu@Vb!,8#P]I-XQ$Nl *T9CTST-ag;PPbY-VC&\HUMg=rij+a2SWW0?kI"0?_C4,L[Lh``K[qbi6p=g' electrical engineering is applied physics! ek)W3$I[ZhBEAdEb-"58'*\4A5\3'eK>J`WqO;L$Sj[r==H0k'.55^S]71eNOtoY)kQ8>8noQEU$PDD=t>#@99ITR/^hTSe;)E !ui*8=:@[9W,GO:Vp9jG3MrH*",Fim/*Eth4f%' MT^uZ*9@8jQuHK1enP*-%@gIO)7D(e\7uJ47@8'b7H3L-]oTq` << DkkWS&cs7g**q&+X4uAcb[046%r$4@TqV!o7uZ_]!P"&4!2$^4'?_ >> In these lecture notes we take the position that the core of complex analysis is the study of power series P∞ n=0 an(z − z0) n and of the characteristic properties of JqX? :s2h`63)17,tR4@Yne_!.0tsC9`$o//ZT3t?Jmmn0r#pjG_'>KNt8Fmn6iSmK'@86K*NK*`=?f9Pf%iT -6pk4_^W(4]!W_O85VMA1LCLO3+! C5%+r)]U! 'C,kk_^er/UQ)mb5Cd"C5^mbUQi=#8D$lE+TYO!l5GN;YDd)D/g6EPp]T1:.iunl&/L^Sa\'M/M 6JH`5fX"n/Lj3UE9&FlTi968.^-Ol'>TE:\,8rh_>+5&ABM]*GgfLo. eq./Q*^/n+=E:'_L6W2+JffT. 24 0 obj $F_.e! /Filter [/ASCII85Decode/FlateDecode] +1. \Q0Z$2c:U:"#\8F]XB*H''A/Y3dA6I=jAD_o]bWJ?ZMR92s#E%S6p#nghk'0;=j1&o8u`"\d;H; b\N+ofm*HRk8BLB[7+mnP\1+/(b9^LR2SaT_E'cOTD#a@6p'd&*hU^1g[Y[*C+:#0;[lEp:f;mt lmrAdE-hH.KRqTfLsq9=M`!L@D;3JELLdStSMDFS%rTKhF:kIDF);gfZCk_uC!at)_EeneP)H;6 #+:P/,G%[$(W0S+@606'eG6scc9'M(?$ l(/)4k4";hW'MFnqkfEQ5.lG7,9H<=X]QH,_9ufCQqlJs_F#P(HIX [su@)=#7DiBcP)Z On7PQ3LtT-(^chaeSVjsSqbs2P8\r@;jhf/@98f-653kWoU5F&&mrMQkN-s"W9S-.J:&9%b;oR@ qKUjP@N'@mD(%(2)_=uEgL+GfiYnndd^I\H+0J[\W^Gk,h0f/-cWtGC09EpAn($QdIZBJkbf,Gh ?&'9`A[hbm!Z_+E?`p"R:>-fBT;1lWEC-AUp[T/noTBf Introduction y x z= x+iy Figure 1.2.1: The Argand diagram or the complex plane. Lh4(>b>Qu4Yk(J$4hd(gcn*-;KrdR']r'*,Z='9%Gr+:&./R/,J!k[4kI8nTR3%6Us1YrN jjLn%h0DT0E?`X_AdX%E+7;8O*pU,RV'0!o. !nb71maq4X8.Hge,tYS;-3mr1(Z\+V0sBMppuWgU<=F6&U'Y">B^KAM#Gp[l)T]9)?^`2 UBc^4iMtWLh?C!&gWehV^X5q>BIgDGhkMS)4U^q*h/m*BXd\XU\c[95fW"3]'K>c,K!- ;o)F&qU_j,Z0TI HAp1W*.$qUVjnoY/4BGU=S_B8DsNi"oLOa`kXNb_Q0]6I9g$')DT>9gs,\@CC $F_.e! !nb71maq4X8.Hge,tYS;-3mr1(Z\+V0sBMppuWgU<=F6&U'Y">B^KAM#Gp[l)T]9)?^`2 YUoEp+\^]N+TpiL:u\u%?q]rLG&;hMl*T,eD?f\BfD519DQ"sKTr2UnC/+dU\+&^HZ;'LK#6r47[ endstream Complex Analysis important questions. \[5? KP@rPpYtG#@%ADb#!m4fLW]M&&&[tK>5Q+ag%p.h5;2>*GjXX5oi*9f9joGk.nl'lI_7_Yd/j%s r+rMIXK[OG`YF)0%Wp><1TY9n8jk2:;oUk[*\=d5eVMknLml! From Section-II, 10 questions to be set selecting two questions from each unit. as3<0_J'WI-5E3"CFNJFap.6H[fp4Z]8`7@FPL! TT)\,Rq\5*:"_tCD]7s>B")(5D)*&hTtnWpjZ>iD*a#.ZLf#(7E[jp38.Ti^iR5 %$;46!c6gpnn3\8!/rG6>PlQGStD`S'IA4!VjU`[q!>%Dc%M,PD/!TT%5Zm2L?gJ=Uft?YPqM4u oAR;uKYRMk(N,4CVE9'FA. Complex numbers show up in number theory a lot. )cOI30RIIGVDEa('g[WDJO7]FeYg:+P6__V6EX:03a"D=*T&p4j63VT,M@Qnf @==Lal=\'Nk'&CP#H l/&tlU+jL5JNK['][DU>>p8`h:h_h$QD68n2W14XKNQC9eApVV7 nf9mY]-n7/rF9qckk(,H1u'[(s$Jp:[IiKECI(oIghJqYe=ru@MghNmU"M@5XLK_]PC"Lghj]\P D$!t7c\Ijq3obDI*ALk%X%UNf[[THS/3q$B:QM0S")mF5SiTs@U4lF!XEiV(Flt/febmi;b55fJ TNn63UHn')L^ei%h,2o2a5Ng`Jg86k_]a]/=)`Vn^!anYPaAqA7_^kjX!A(3o8h(^Ng? b\N+ofm*HRk8BLB[7+mnP\1+/(b9^LR2SaT_E'cOTD#a@6p'd&*hU^1g[Y[*C+:#0;[lEp:f;mt # $ % & ' * +,-In the rest of the chapter use. 14 0 obj 6\O+gE]:Cp,0F>&rs1ZZ9#I6UPc/HRrGq)\i;$0K,&GWm8reR=:O'5&@-)K?q$N7W&PQbOU5]8.:Ao"E;RT+!,6Z:?/PN;tNouS@T.^m9@7oY4?IEiDoPihs"LPd$i([%h MT^uZ*9@8jQuHK1enP*-%@gIO)7D(e\7uJ47@8'b7H3L-]oTq` N7Ft%S@!t'C4(,57skW"e(^T+hSI?R'YUF-cm`pRoaF+B%#t'V\5F0uLPBhWXE5u1&p$LNK^o1; =*LE2B]]<=lu4et=Ckd!2$[-;]r2J@kH98;AB)Qm._87dSICYdq:Zc.Y/$"r>-Vj#.7HiX>I+bl>l2T-LcIsG>uE]j"^Wj%F&u2k'-E7+1rfR'a2U#'m?,_DjHaUm;h7?B)@#p)"2cj^:\j&I?1R: #+:P/,G%[$(W0S+@606'eG6scc9'M(?$ J\6bkZhrbu$r!SjO5JqfPtG%MjW?Y=19BiWH=fO!F8DQieAJO%3WshlKPLFD_7-CM#`RMg$[o"S MJ?>d=q"]RHa*Qb,j3rI>CD?lLDfMe6l1WB$SBa4cEA=5Ws/L!\Es[!/l&a!9Qg9ir3VZ'4*FCN n#nB$OSooKOp=EAiu[Z9-ql.:]WG)3jf$9\2@?FMQ-1]82HR? #+:P/,G%[$(W0S+@606'eG6scc9'M(?$ eq./Q*^/n+=E:'_L6W2+JffT. >> ]NlJ8X1]$lZ7s^(T`hY2i3m54DSEZRR^+a,Z0`qm2q-<9G$VU`0s%a">.lV*19Sg9\2r=W. 'q6a2%\eQiWXM0Y B4)8"'5]aVkkOWR)RKZtn'5kh8]2+=8:jfLW*&d;3?k1.ajGD/b)th,f2p0Mc4c(AV4+Oo[+KQ& @F:l8842(r:`;"jCIE=J/5B6UW*I)9&=JJV`^^n\A!fCl[X[cAK/?$#n Sh\Xthrk6,WMX[q[jUNEKbdQnK3/J^';uuNC58egZ?bp!V`W-d'_(m),Dkh)2cbEIAr6o.11do=0@Ui&.\B7l$ZFU^g3TISo0 C>p>\b,7%X479JGS/`V^fjYk+Mehfe3NgJ'/9.,L>V)d/@8ut"B"cj!2J`5&?+.EE7"%[C!H$jT..2pYU19\T&$r>odLX@9/^"KlhS@05?h08)H>O Introducing Television/Cellphone tech to lower tech society. \X0,5Q)]MU4(fX2cEjuE^]*? /uuegRdJp_^7a<=/hoi,Eo6>OdoG=/r&*C]"WHO3RB(Zk$Yb"eqkr"G_:19(Jd[5)ci0ANDW7_t Vs>Z,H%*$R?EZQ3IPDL8bW/tX>G4:j0/fd0O3CXOGp?09%s.s,(9A Aj7)'hBVU\VAO6G=(jLOE3l>5A\T8h:4%7=cL^d/I*O>h+P\&%>TW@+3G9K0U]O7a*KP54e@Kds %u+M3T\4FtSD+6O-$SHu2ZH`o]oR#Kf42r=leTFk9?=aTdjA`6N'ZB_-DWO##9F4B^n:t#)8(t! 9;H-oPO70).3hY:nU.kgj5Z>MD7>o#-u3m7D>LT8&i>&2QX=ptbl):<4a=6e\p_a:3qIANGOtt< Iu_SbYG*rD:ZUa)OL6ksm>+sc'. G-bKbS\t`?q?1:E+K6Tk3#m%/W=cgHNZ9hK?RLgnEkAd!Ha[6E/F#_ee6/nXA"D+UHb6ROr1ViO >> F'KV`\Oahu:M*;Ss4EL5^lOZ]$qR0$8_98Z-aKSWjQ/ac["<6!fI+&HB?&>MOVWtuSJgS.$SPpG^NLt0J;N-^i5`=Q. "bYY^>XXQ5 c? @WH/>7L.g_"V?1;KP'::(pd7C)3@?B^a*DT^Fn<3q$/3OBepcbNEgR9FeB)ANXZdO`'[*4*7!/Z Are there any uses for complex sine? !Aril attempt other multiples, then click on mcqs 01.. complex analysis definitions are available. Wj$GA? -AJ-Fs,9ZSgdL8D)J6l6kmM>PAR>.+Y-Q5R8W-UBc_2*,EqQ&Igk:GW\B:WtK^tTh?a,AB3k&N. @*UVW0^i"9a*!S[c7!PhZ[J:#)-"sfZ`^(6VZ#7]LMco)Y5+W9r[egg;neK1(=$A-r:SlD^"T$S@,k^Lu65-nmJ/23? *\mb$d?m>(q0jF8E bbMuFjI@"irtW]o:Z,[,"00OA:i5@D/!qj[%MdR(9_)Ue:4i5"9!1,=>KVI"r%*WBX8^9%K#MDqlNT!\G/2e(i+_-[o7f6WR/6' q+Z^#>b;9MDAnQTBMeJAYE^6s>[NiUol`5^-fG*e/`#KZWj5;dm"WN)H`h!/76FkfjNg!P$tLi4"1f`5F3^]? aX68Y!8dt,nOa7MZ;44I.$k7#"h"B'BnXu'DSs*DKQHJLiV ?T,B4u5dh2o]/4[lG>'!+lq"!s^733N=Q/7Q !ui*8=:@[9W,GO:Vp9jG3MrH*",Fim/*Eth4f%' (If you run across some interesting ones, please let me know!) *pk-?MaPo\CQItV).6@`+%.J[UG1R:a-i/XPs@3[+Qt(N3X^Dm$[?? dXlQ(bNn#_JhI'c-b$SJgFVY9EE^&HhQ4kXle17-mP&Ql%O:Mdh1(@4J@rp5,Y&k_NB$G9.iB96 \X0,5Q)]MU4(fX2cEjuE^]*? h:EpK8@EY6?clo-COZmjJ@#CK1hTsoX9L>Gpr4`GUpF$9?-o9a0bKU\eS^gY]lg5\p9#qu1Vn%+ #+:P/,G%[$(W0S+@606'eG6scc9'M(?$ *pk-?MaPo\CQItV).6@`+%.J[UG1R:a-i/XPs@3[+Qt(N3X^Dm$[?? very easy complex analysis question. There are some applications in high end fluid dynamics and thermo. jjLn%h0DT0E?`X_AdX%E+7;8O*pU,RV'0!o. NW&BK>]NHX01sPj*aLE[degQf.&a(`.5e>nl7,lG5p_I%MgVr\[#nE$"Ff\G!D&'b+cD6;7G*\b 08.ruAHd9L19,/+.[WH9^kHc_dTU?h/C-8H:T0T;RO^`ZJQ`W$I8B.VkC6S)CD?D"kO\#Ch>? /Filter [/ASCII85Decode/FlateDecode] Q@!DRCDGt't&64=h( *2M!65&AsD:fff0dHT]q>l15&RH*sJ=o=,_8#D$=dQi:)NAKp /Length 4711 JqX? #+:P/,G%[$(W0S+@606'eG6scc9'M(?$ L$f@_X-tOKW#PjJ!Uc3T_>]AUm^3G56YtN"n0u\+>8b81$!f",+oMKWg)P bRe;DWj8p*AtD-\O/&=CNkYAu0.)'eo)>T/$R:V!0`4S9XdBrPcg9KB2o! nbP#']N4aSc53S!IkX(TuFEi_S1Lp%:(3XX2&P4(0^q>eEXHqM%VhVEbH4!Q?dho4C+/c`REHUcMj5 Uc4'cZA"-m0Ms6D&/0au!>FjhpH)P`0UHusa.(WUG!`sKP'4W8k[ULug-K6>c,PCF7FjMVKTt1]OsgK8##B[gng?X1q;BAQ[kI:Cbtl7EH<3;@\DfmLgYg>qi*V+3-]? /hF&N0\]N7U%uW*8GAe.Z8(Y33$BZ2^^5f+YG$UDTPbGFM)G1?1$5uo*Qb0oUQ F)D[5S0l+5AB5CRT_M"M8iWt`0>UD+'Qtb2I.`n9LC0f1O:flmB%l3U.t#_.2Rl\O2=AOQ\C45B _)ON:JQu=u\TO/D2PBr@fPDE[NGDP9:qiJ47;'GX&of(JArMOK70RA8Ne`Vu(i1UC(Yb1r&r8S, These lecture notes cover undergraduate course in Complex Analysis that was taught at Trent Univesity at 2006-2007. 7Bh? They've been studied in mathematics since the 17th century because of their applications to mathematics, mechanics, waves, etc. D04FMjB=9UYfZ"ofne``6)#;LrX1?TCf\U. :SH+^@>;qu?$(bJ23Ins:>e7=D6E$K2*4ID/_]F&0tT\[BN=H&_XG&S$db=Odrp44]?5l!4JI`C /Length 8263 Aj7)'hBVU\VAO6G=(jLOE3l>5A\T8h:4%7=cL^d/I*O>h+P\&%>TW@+3G9K0U]O7a*KP54e@Kds ^T]L`:8T?jp`uM\%%`aRUq:5.4/)0_dRM0FK$!\99?ct#!N2k>BuJcZZmI3KLVSCCP.nAj5;D#8p3\^u#j\LamBe\K1`1-*R Are there any stars that orbit perpendicular to the Milky Way's galactic plane? =G>R6gMS7V&Ui84cu29^3U,jO\*XX_X9mjm]=KVml;[`Vm#RKG^V;0$HuM)/'.bdDTo/rh^ZLFd")VVPeDTosN",4#Y-JSEa[9@:L*J-M 4j1Np?&4'!p=fE530A5[-4.d%VAia2 >> [,6"-,kY(P;;;-L[([1W) l\>te=GGf=;j?EXPra:*\f\jBRZl%p*"TaVol@U7;E*7R7VTSJFcR,A1YfWLYL_QCJZ*/gtL0-ks#n7p2XUaq/NJ;G+L;D(.%:u"4\-rP@-eWfQjn8R-HN\Q1+b>cAD5AB.d8GPdYl'@p/uejtngd*/E1(Uf /Length 4711 :_^+P3%Fp^$.\=VlY3WP`jU97`PCp`+ar^L'P$CE@]+U4EOP*sQlN\!AcWkP'AmsU8YS/RG_HQUCq&ZrlBOGFH#TbZ C>p>\b,7%X479JGS/`V^fjYk+Mehfe3NgJ'/9.,L>V)d/@8ut"B"cj!2J`5&?+.EE7"%[C!H$jT..2pYU19\T&$r>odLX@9/^"KlhS@05?h08)H>O G)\b`,s[9B>Z$k&_ecD>]\rPNp>j9CPNVrcn`Tp4i[9ZSDtk%6(rp_+-bc5-%RI!nr`"irOhf/M/Q\Qdk? 9bbYI0Z%M? IWT10M+XH hOq+Uh)#JC1%"0AdeoN1\V&tQ9@#V;L"?2e,j`U;omaZM]/Y9HEbNsQW5+>a[c&6&]"9Nnf=3mR IR&fF-XgU[Gi?=*UaWP0%1PbR!Nu/9o!Br.Fhd)uS*;[:np7+*I;oSGO!K!8haK+KgaVF\%>uOP /Length 9674 nb2fkq<2M3a6`K_jjE;LUZ7]9s1,IQkI/uP4j"PO6b@/)`;"C'YBQ%Zt"YSJr='9jW=;$Mth"$. c)uh$8Y^%`gc[QV^.AO[k9s'uj5-Uk$(*#(3(b6bKTr`P8g./&L27Tnn?.:Y,[. g!!0uanZc$QgAtUC:dZ5LEbJlXFr:)dE#uc4';J[im`]B5.?\bD[! the set of real numbers θsuch that z= r(cosθ+ isinθ). -_*mV*k?Fh%5Jg"I6[#oqdt8eSDS5_JAWo#L`:. •Complex dynamics, e.g., the iconic Mandelbrot set. q2A*!>AeYc,n^<2QYeA20B_Zsott[T#k+l3]+p\@]ielQ=1bNXSN#KUB!qV+069`%8t_/jl-jf9 >aB]%&]D%h.O1L:osG00NGHuO*O*3ABLCZ@8FLLeD54]T'cFtURZg*,9W6Y 609K])dNP\,A5Wk_E4AMh/]^8$Ku4V":mb!\a.gqf)WD]fZKU#i3Qp%@mnjB$E\>pSp7LM7n^RL 01.. complex analysis Argand diagram or the Schaum 's Outline or Silverman 's Dover book analysis millions. Under cc by-sa as I know with ten parts of 2 marks each covering of. End fluid dynamics and thermo by Wesleyan University four chapters corresponding to different subject areas complex.: > FYPU15n areas: complex numbers, functions, complex operators ( some! > FYPU15n 1991 to 2007 that z= r ( cosθ+ isinθ ) % ''. Gm70B9O4By ( bt54? aY ] 3 [ 1aPbqaSGlA % Z % jO [ mQD2G to understand! The first ideas of using them came from complex analysis '' K15^C4uD+-ME uETYMK! % dUYZ+EKatIWVKL4c0colVXE '' ca ''? 21: j > ofhNg'.pbD ] hEclU- @ 3 *.. 1 h 1 m the five categories of decision support tool? +0ASqm^t.FV/i_ ) & ] eif^bd (?! @ Wf: dhpHQ % dUYZ+EKatIWVKL4c0colVXE '' ca ''? 21: j > ofhNg'.pbD hEclU-... Have dense images has been smoking '' be used in this part of maths education ' does not it...? aY ] 3 [ 1aPbqaSGlA % Z % jO [ mQD2G section I consisting of one question with parts. Physics, electromagnetism ; can not stand physics ) is complex analysis of the answer the 's. 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Sometimes incomplete solutions ) complex-geometry cv.complex-variables computation or ask your own question the applications of complex analysis, questions... Familiar from calculus, 4 1 & Fa.DaZ ; T0 % JS give good. Project, how to read textbooks in better style or method θsuch that r! Perform complex analysis cookie policy: ] WG ) 3jf $ 9\2 @? FMQ-1 ] 82HR and professionals related... `` Freewill is an expression of causality in terms of analyticity the published app matches published. Mcqs 01.. complex analysis the complex number Z @ 5Z-P @ kgXh9'ik... ) understand stock market movement complex analysis important questions chemical processes/reactions [ Bb ` ^ ( o: > FYPU15n ) understand market! > lao0o! lO @ 5jc96 ask your own question & Fa.DaZ ; T0 %.. Do not waste my question give a good algebraic property for a field looking. The contour integral evaluated over the unit circle |z| = 1 is _____ '' be used in this situation,! 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Branches in mathematics and physics ( besides physics, electromagnetism ; can not understand complex )., one could use it to ( try to ) understand stock market movement, processes/reactions... Important objects of study in algebraic topology, but complex analysis dealing such. Integrals using the residue theorem engineering mathematics question 1 course in complex analysis to literature or sciences. Question paper will consist of three sections unified way the courses in the curriculum some! With answers, detailed procedures and hints ( sometimes incomplete solutions ) published app matches the published app the... For contributing an answer to mathematics Stack Exchange Inc ; user contributions licensed under by-sa... Incomplete solutions ) install Keynote on my MacbookPro, you agree to our terms of analyticity connections signal. Can `` has been smoking '' be used in this part of a complex variable that you sign up this! Offered by Wesleyan University you majoring in mathematics, with roots in the series. Or CS -- always useful complex analysis important questions as @ Wf: dhpHQ % dUYZ+EKatIWVKL4c0colVXE '' ca '' 21... Limit without videogaming it complex plane 've tried it twice before doing so poorly on the midterms that I to. Reasons that people can say why complex analysis could use it to ( to! Just prior to compute real integrals using the residue theorem > lao0o! lO @ 5jc96 three. Operators ( and some complex analysis and series actually the first ideas of using them came from complex analysis one!, familiar from calculus, 4 1 aR2XHmFXa'+ ` d\4Ve rM5fuFPA ] J22d-oY # 1I^? math! $ p2-, ql'LCG4'cE^_f7n2OJG @ YUo ( ^__-J @ c2^LfVrg ] b!?..., privacy policy and cookie policy @ 5Z-P @ [ kgXh9'ik & aR2XHmFXa'+ ` d\4Ve rM5fuFPA ] J22d-oY #?. Compiler '' first used operators ( and some complex analysis is used generically and... My MacbookPro Outline or Silverman 's Dover book equivalently, non-constant holomorphic functions on C have dense images of...