Complex Variables. Chapter 19 Series and Residues. March 26, 2013 Lecturer: Shih-Yuan Chen
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1 omplex Vrble hpter 9 Sere d Redue Mrch 6, Lecturer: Shh-Yu he Except where otherwe oted, cotet lceed uder BY-N-SA. TW Lcee.
2 otet Sequece & ere Tylor ere Luret ere Zero & pole Redue & redue theorem Evluto o rel tegrl
3 Sequece & Sere Sequece { }: A ucto whoe dom the et o potve teger; other word, we g complex umber to ech teger,,, Ex. The equece { } lm Sequece { } coverget. 5 L
4 Sequece & Sere Or { } coverge to L, or ech potve umber ε, N c be oud uch tht L < ε wheever > N. A { } coverge to L, ll but te umber o t term re wth every ε -eghborhood o L. Ex. Sequece lm,,,, 5,
5 Sequece & Sere rtero or covergece Thm. { } coverge to complex umber L Re( ) coverge to Re(L) d Im( ) coverge to Im(L). Ex. Sequece Im Re 5
6 Sequece & Sere Sere A te ere o complex umber coverget the equece o prtl um {S }, where S, coverge. I S L, we y tht the um o the ere L. 6
7 Sequece & Sere Geometrc ere For the geometrc ere the -th term o the equece o prtl um S S For For S <,, S the ere dverge. ( ), 7
8 Sequece & Sere Two pecl geometrc ere I ddto, we hve vld or < 6 5 8
9 Sequece & Sere Ex. The geometrc ere < 9
10 Sequece & Sere Necery codto or covergece lm Thm. I coverge, the The -th term tet or dvergece lm Thm. I, the the ere dverge. 5 Ex. dverge ce. Ex. The geometrc ere () dverge whe.
11 Sequece & Sere Abolute covergece De. Ite ere coverget d to be bolutely coverge. Abolute covergece mple covergece. Ex. Sere bolutely coverget ce & the rel ere coverge. Alo, rel ere coverge or p > & p dverge or p.
12 Sequece & Sere Rto tet Thm. Suppoe complex term uch tht ere o oero lm L 7 () L <, the ere coverge bolutely. () L > or L, the ere dverge. () L, the tet cocluve.
13 Sequece & Sere Root tet Thm. Suppoe uch tht ere o complex term () L <, the ere coverge bolutely. () L > or L, the ere dverge. lm L 8 () L, the tet cocluve.
14 Sequece & Sere Power ere A te ere o the orm ( 9) where re complex cott, clled power ere. (9) d to be cetered t & the complex pot reerred to the ceter o the ere. From (9), oe c dee eve whe
15 Sequece & Sere rcle o covergece Every complex power ere h rdu o covergece R or h crcle o covergece deed by R or < R <. The power ere coverge bolutely or ll tyg < R & dverge or > R. A power ere my coverge t ome, ll, or oe o the pot o the crcle o covergece. R c be ero, te umber, or. 5
16 Sequece & Sere Ex. oder the power ere lm lm Thu the ere coverge bolutely or <. The crcle o covergece & the rdu o covergece R. O the crcle, the ere doe ot coverge bolutely, ce the ere o bolute vlue the dverget ere 6
17 Sequece & Sere At, the ere coverget. Ideed, the ere coverge t ll pot o the crcle except t. From bove, or power ere the lmt (7) deped o oly the coecet. Thu, lm lm α R α R lm R ( )
18 Sequece & Sere Ex. The power ere ( ) ( ) ( )!! lm lm!! R The power ere wth ceter coverge bolutely or ll. 8
19 Sequece & Sere Ex. The power ere The crcle o covergece /; the ere coverge bolutely or < / lm lm 5 6 R 9
20 Tylor Sere Here, we wll ume tht power ere h ether potve or te rdu R o covergece. otuty ( ) Thm. A power ere repreet cotuou ucto () wth t crcle o covergece R.
21 Tylor Sere Term-by-term tegrto Thm. A power ere ( ) c be tegrted term by term wth t crcle o covergece R, or every cotour lyg etrely wth the crcle o covergece. Term-by-term deretto Thm. A power ere ( ) c be deretted term by term wth t crcle o covergece R.
22 Tylor Sere Tylor ere Suppoe power ere repreet ucto () or < R ; tht
23 Tylor Sere The orgl power ere repreet deretble ucto wth t crcle o covergece. From bove, we coclude tht R, power ere repreet lytc ucto wth t crcle o covergece. Reltohp betwee & the dervtve o () ( ), ( )!, ( )!, ( ) ( ) ( )! ( ) ( )! or!
24 Tylor Sere Thu we hve ( ) whch clled Tylor ere or cetered t. A Tylor ere wth ceter, ( )! reerred to Mclur ere. ( ) ( )!
25 Tylor Sere Tylor theorem Let be lytc wth dom D & let be pot D. The h the ere repreetto ( ) ( )! vld or the lrget crcle cetered t & rdu R tht le etrely wth D. 5
26 Tylor Sere (proo) Let be xed pot wth & let deote the vrble o tegrto. decrbed by R. Ue the uchy tegrl ormul to obt the vlue o t : d d d 6
27 Tylor Sere Ug (6), we hve d d d d d 7
28 Tylor Sere Ug uchy tegrl ormul or dervtve, ( ) where R ( ) ( ) ( ) ( ) ( ) R ( )!! ( )( ) () clled Tylor ormul wth remder R.! d 8
29 Tylor Sere Sce lytc D, () h mx vlue M o. Bede, ce de, we hve < R, & coequetly, The ML-equlty the gve d R R d d R MR R R d R M d d R 9
30 Tylor Sere Thereore, the te ere ( ) ( ) ( )! coverge to (). I other word, the reult vld or y pot teror to.! ( ) ( )!
31 Tylor Sere Note tht R the dtce rom the ceter o the ere to the eret olted gulrty o (). A olted gulrty pot t whch () l to be lytc but lytc t ll other pot throughout ome eghborhood o the pot. Ex. 5 olted gulrty o () /( 5). I etre, the the rdu o covergece o Tylor ere cetered t y pot ecerly te.
32 Tylor Sere Ug () & the lt ct, we c y tht the Mclur ere repreetto e ( )!!! 5! 5! co!! re vld or ll. ( 5)! ( 6)!
33 Tylor Sere The power ere expo o ucto wth ceter uque. It me tht power ere expo o lytc ucto cetered t, rrepectve o the method ued to obt t, the Tylor ere expo o the ucto. Ex. Oe c lo obt (6) by mply derettg (5) term by term.
34 Tylor Sere Ex. Fd the Mclur expo. Note tht or <, The rdu o covergece o th lt ere the me the orgl ere, R.
35 Tylor Sere Ex. Expd ceter., ( )! ( ) ( )! ( ) ( ) ( ) Tylor ere wth, ( ), The crcle o covergece or the power ere 5 5
36 Tylor Sere Altertvely, we c wrte From (), we hve
37 Tylor Sere The ucto c be expded to The rt ere h ceter ero & R. The d ere h ceter & R. 5 7
38 Luret Sere I l to be lytc t, the th pot d to be gulrty or gulr pot o the ucto. Ex. () /( ) & Ex. L & ll egtve rel umber Iolted gulrte Suppoe tht gulrty o (). d to be olted gulrty o there ext ome puctured ope d < < R o throughout whch lytc. 8
39 Luret Sere A ew d o ere I gulrty o, the c ot be expded power ere wth t ceter. However, oe could repreet bout olted gulrty by ew ere volvg both egtve & oegtve teger power o ; tht, 7 9
40 Luret Sere The ere prt o the RHS (7) wth egtve power, tht ( ) ( ) clled the prcpl prt o the ere (7) & wll coverge or /( ) < r* or > /r* r. The prt cotg o the oegtve power, ( ) clled the lytc prt o (7) & wll coverge or < R.
41 Luret Sere Hece, the um o thee prt coverge whe both > r & < R, or whe pot ulr dom deed by r < < R. (7) c be wrtte compctly ( )
42 Luret Sere Ex. () ( )/ ot lytc t & hece cot be expded Mclur ere. However, etre ucto, & rom (5) coverge or ll.! 5 5! 7 7!! 5! 7! ( 8) Th ere coverge or ll except t, or or <.
43 Luret Sere Luret theorem Thm. Let be lytc wth the ulr dom D deed by r < < R. The h the ere repreetto ( ) ( ) d or ( 9) vld or r < < R. The coecet re, ±, ±, where mple cloed curve tht le etrely wth D & h t teror.
44 Luret Sere (proo) Let & be cocetrc crcle wth ceter & rd r & R, where r < r < R < R. Let be xed pot D tht lo te r < < R. By troducg cro cut betwee &, we d rom uchy tegrl ormul ( ) ( ) d d
45 Luret Sere Ug (6), we hve,,, or where d d d d d 5
46 Luret Sere,,, d R d R d or where 6
47 d Luret Sere Let d & let M deote the mx vlue o () o the cotour. Sce r, The ML-equlty the gve Thu, r d d r r d Mr r r d r M d d R 7
48 Luret Sere ombg the two term yeld ( ) ( ) ( ) ( ) ( ) where or, ±, ± Note tht we hve replced & by y mple cloed cotour D wth t teror. d, 8
49 Luret Sere Note tht Luret expo geerlto o Tylor ere. Some poble ulr dom: r, R te: the ere coverge the ulr dom < < R. r, R : r <. r, R : <. Regrdle o how Luret ere o ucto obted peced ulr dom, the obted Luret ere uque.
50 Luret Sere Ex. Expd Luret ere ( ) vld or () < < (b) < (c) < < (d) <. For () & (b), oe c repreet ere volvg oly egtve & oegtve teger power o. ( ) 5
51 Luret Sere < / < For (c) & (d), oe c repreet ere volvg egtve & oegtve teger power o. 5 5
52 Luret Sere coverge or < <. Smlrly, [ ]
53 Luret Sere coverge or /( ) < & lo <. 5 5
54 Luret Sere Ex. Expd Luret ere or () < < (b) < <. () 6 8 5
55 Luret Sere (b) [ ] 8 6!!! 55
56 Luret Sere Ex. Expd Luret ere vld or < <. The geometrc ere coverge or <. Thu, the reultg Luret ere vld or < <
57 Luret Sere Ex. Expd or < <. ( ) Luret ere The dom ceter pot o lytcty o. Oe c d two ere volvg teger power o : covergg or < & <, repectvely. 57
58 Luret Sere th ere coverge or ( )/ < or <. 58
59 Luret Sere coverge or /( ) < or <. ombg the reult o () & (), vld or < <. 59
60 Luret Sere Ex. Expd vld or <. From (), e e!! e!! Th ere vld or <. Luret ere or 6
61 Luret Sere Replcg the complex vrble wth, we ee tht whe, () or the Luret ere coecet yeld ( ) d or, ±, ±, d d 6
62 Zero & Pole lcto o olted gulr pot Deped o whether the prcpl prt (PP) o t Luret expo cot ero, te umber, or te umber o term. () I PP ero, removble gulrty. (b) I PP cot te umber o oero term, the pole. For the lt oero term wth ( ), pole o order. A pole o order commoly clled mple pole. (c) I PP cot tely my oero term, the eetl gulrty. 6
63 Zero & Pole Ex. Sce! 5! removble gulrty o (). I h removble gulrty t, the we c properly dee the vlue o ( ) o tht become lytc t the pot. Sce the RHS o () t, t me ee to dee (). Wth th deto, () ( )/ lytc t. 6
64 Zero & Pole Ex. From! 5! or <, we ee tht & o mple pole o () ( )/. Ex. For the exmple p.6, e or <!! PP o () e / cot te umber o term. Thu eetl gulrty. 6
65 Zero & Pole Ex. For prt (b) o the exmple p.5, The Luret expo o ( ) 5 or < However, NOT eetl gulrty o. From prt () o the exmple, we w tht or < < Hece, mple pole. 65
66 [ ] but,,,,, Zero & Pole Zero ero o ucto ( ). Alytc ucto h ero o order t Ex. For () ( 5) 5 ero o order. I lytc ucto h ero o order t, the Tylor ere expo o cetered t
67 Zero & Pole Ex. () h ero t. By replcg by (5), we obt 6!! 5! Hece, ero o order. 5! 8 67
68 Zero & Pole A ero o otrvl lytc ucto olted ce there ext ome eghborhood o or whch () t every pot tht eghborhood except t. Thereore, ero o otrvl lytc ucto, the the ucto / () h olted gulrty t. Pole o order Thm. I & g re lytc t d h ero o order t & g( ), the the ucto F() g()/ () h pole o order t. 68
69 Zero & Pole Ex. For the ucto F 5 ( )( 5)( ) F() h mple pole t & 5 d pole o order t. Ex. From the exmple p.67, we coclude tht the ucto F() /( ) h pole o order t. Note tht ucto h pole t, the () rom y drecto. 69
70 Redue & Redue Theorem Redue Recll, the complex ucto h olted gulrty t, the h Luret ere repreetto whch coverge or < < R. The coecet clled the redue o t the olted gulrty wth the otto Re (, ) 7
71 Redue & Redue Theorem I PP o the Luret ere vld or < < R cot te umber o term wth the lt oero coecet, the pole o order. I PP o the ere cot te umber o term wth oero coecet, the eetl gulrty. Ex. For the exmple p.6, eetl gulrty o () e /. The redue o t (, ) Re 7
72 lm, Re Redue & Redue Theorem Redue t mple pole Thm. I h mple pole t, the (proo) The Luret ere expo o bout the mple pole h the orm, Re lm 7
73 lm!, Re d d Redue & Redue Theorem Redue t pole o order Thm. I h pole o order t, the (proo) The Luret ere expo or < < R 7
74 Redue & Redue Theorem Derette tme d d d d d d lm!! lm!! 7
75 Redue & Redue Theorem Ex. h mple pole t ( ) & pole o order t. Fd the redue. Ue () or the mple pole t, Re(, ) lm( ) lm Ue () or the pole o order t, Re! d d ( ), lm lm ( ) lm d d 75
76 Redue & Redue Theorem For () g()/h(), where g & h re lytc t. I g( ) & h h ero o order t, the h mple pole t d g Re(, ) lm lm h Re (, ) lm g h h g h( ) ( ) ( ) ( 5) g h ( ) ( ) 76
77 Redue & Redue Theorem Ex. Ue (5) to compute the redue o e e e e e e e e, Re, Re, Re, Re,,, where
78 Redue & Redue Theorem Redue theorem Thm. Let D be mply coected dom & mply cloed cotour lyg etrely wth D. I lytc o & wth, except t te umber o gulr pot,,, wth, the d Re(, ) ( 6) (proo) From (), & () o h.8, d Re(, ) d d
79 Redue & Redue Theorem Ex. Evlute wth :. d 6 d d, Re d d 6 6 lm, Re
80 Redue & Redue Theorem Ex. Evlute ( ) where () the rectgle deed by x, x, y, y & (b) :. ( ) ( ) d d Re d [ Re(,) Re(, ) ] ( ) (,) ( b) 8
81 Redue & Redue Theorem Ex. Evlute wth :. e d 5 e e d 5 Re lm ( 5) (, ) lm! d d d ( 8 7) ( 5) ( 5) e e 7 5 8
82 Redue & Redue Theorem t d Ex. Evlute wth :. t d co d Re, Re ( ) ( ) [ ] ( ) ( ), 8
83 Redue & Redue Theorem e d Ex. Evlute wth :. From the exmple p.6, eetl gulrty o the tegrd. So ()-(5) re ot pplcble to d the redue o t. e d (, ) Re 6 I () g()/h(), where g & h re lytc t, g( ), h( ), & h ( ), the g g lm h h ( ) ( ) 8
84 Evluto o Rel Itegrl Itegrl o the orm The bc de to covert (7) to complex tegrl wth cotour the ut crcle cetered t the org. (: coθ θ e θ, θ ) Ue d dθ e θ d (7) become F, coθ dθ, coθ F e θ ( coθ,θ ) dθ ( 7) e θ, θ e θ ( ) (, θ ) d ( ) (, ) where e θ
85 Evluto o Rel Itegrl Ex. Evlute We c let where 85 ( coθ ) d dθ, d Re ( ) ( ) d ( ) (, )
86 Evluto o Rel Itegrl Sce pole o order & rom (), 6 lm lm lm, Re d d d d co 6 θ θ d d 86
87 Evluto o Rel Itegrl Itegrl o the orm Recll tht whe cotuou o (, ), both lmt ext, the tegrl d to be coverget; otherwe, the tegrl dverget. For coverget tegrl, we c evlute t by The lmt clled the uchy prcpl vlue o the tegrl & wrtte P.V. ( x) dx ( 8) ( x) dx lm ( x) dx lm ( x) r r R ( x) dx ( x) dx ( 9) lm R R R ( x) dx lm ( x) R R R R dx dx
88 Evluto o Rel Itegrl Note tht the ymmetrc lmt (9) my ext eve though the mproper tegrl dverget. Ex. However, ug (9) Thu, x dx dverget ce lm R R P.V. x dx R x dx R R lm x dx lm R R R lm R ( R) 88
89 Evluto o Rel Itegrl To ummre, whe tegrl o the orm (8) coverge, t uchy prcpl vlue equl the vlue o the tegrl. I the tegrl dverge, t my tll poe uchy prcpl vlue. To evlute ( x) dx wth (x) P(x)/Q(x) cotuou o (, ), replce x by & tegrte () over cloed cotour tht cot o [R, R] o the rel x & emcrcle R o rdu R lrge eough to ecloe ll the pole o () the upper hl-ple Im() >.
90 Evluto o Rel Itegrl By (6) we hve d d ( x) R Re (, ) where deote pole the upper hl-ple. R d I R, the we obt P.V. R R dx ( x) dx lm ( x) dx Re(, ) R R R 9
91 Evluto o Rel Itegrl Ex. Evlute Let be the cloed cotour how the gure. dx x x 9 P.V. 9 [ ] 8 6, Re, Re I I I I d x x dx d R R R
92 Evluto o Rel Itegrl O R, whch how tht I R, & o R R R d I R R R lm R I lm lm 9 lm 9 P.V. I I dx x x dx x x R R R R R 9
93 Re Evluto o Rel Itegrl Behvor o tegrl R P.V. Thm. Suppoe () P()/Q() wth P() o degree & Q() o degree m. I R emcrculr cotour Re θ, θ, d R. R Ex. Evlute P.V. x dx e d e (, ), Re(, ) x dx [ Re(, ) Re(, )] 9
94 Evluto o Rel Itegrl Itegrl o the orm They pper the rel & mgry prt the tegrl ( x) coαx dx ( ) ( x) αx dx ( ) wheever both tegrl o RHS coverge. Whe (x) P(x)/Q(x) cotuou o (, ) we c evlute both Fourer tegrl t the me tme by coderg α e d, where α > & cotg o [R, R] o rel x & emcrculr cotour R wth R lrge eough to ecloe the pole o () the upper hl-ple. αx x e dx ( x) coαx dx ( x) αx dx ( ) 9
95 Evluto o Rel Itegrl Behvor o tegrl R Thm. Suppoe () P()/Q() wth P() o degree & Q() o degree m. I R emcrculr cotour Re θ, θ, α >, the R α e d R. Ex. Evlute x x P.V. dx x 9 x x x x dx dx x 9 x 9 Wth α, we orm the cotour tegrl 9 e d
96 Evluto o Rel Itegrl From the bove theorem, e e e e dx x xe d e R R x R, 9 Re P.V. 9 P.V. 9 P.V. d 9 co P.V. 9 P.V. 9 e dx x x x dx x x x e dx x x x dx x x x e dx e x x R d e x R 96
![Evluto o Rel Itegrl Ideted cotour So r, the mproper tegrl (8), (), () re cotuou o [, ].](/docs-images/92/108616018/images/97-0.jpg "Whe h pole o the rel x, the procedure ued the bove exmple mut be moded. oder whe () h pole t ( x)dx c, where c rel umber.")
97 Evluto o Rel Itegrl Ideted cotour So r, the mproper tegrl (8), (), () re cotuou o [, ]. Whe h pole o the rel x, the procedure ued the bove exmple mut be moded. oder whe () h pole t ( x)dx c, where c rel umber. r deote emcrculr cotour cetered t c oreted the potve drecto.
98 Evluto o Rel Itegrl Behvor o tegrl r Thm. Suppoe h mple pole c o the rel x. For cotour r deed by c re θ, θ, the (proo) lm r r d Re(, c) Sce h mple pole t c, t Luret ere g wth Re( (), c) c & g lytc t c. re re θ I θ r ( θ ) θ d dθ r g c re e dθ I
99 Evluto o Rel Itegrl Sce g lytc t c, t cotuou t th pot & bouded t eghborhood; tht, there ext M > or whch g(c re θ ) M. c d d re re I, Re θ θ θ θ [ ] lm lm lm lm I I I d I I rm Md r d e re c g r I r r r r r θ θ θ θ 99
100 Evluto o Rel Itegrl Ex. Evlute P.V. oder the cotour tegrl h mple pole t & ( ) the upper ple. where R r R Te R & r d recll the two precedg theorem. r x ( x x ) e d r R r x dx e Re r ( ) ( e, ) d
101 Evluto o Rel Itegrl x e ( P.V. ) dx Re e, x x x Re( e, ) Re P.V. P.V. P.V. ( e,), Re x ( x x ) x x e x co ( x x ) x x dx ( x x ) ( e, ) ( ) dx dx e e e e ( co) ( ) [ ( )] e co
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