Department of Mathematics and Statistics Indian Institute of Technology Kanpur MSO202A/MSO202 Assignment 3 Solutions Introduction To Complex Analysis

Transcription

1 Dpartmt of Mathmatcs ad Statstcs Ida Isttut of Tchology Kapur MSOA/MSO Assgmt 3 Solutos Itroducto To omplx Aalyss Th problms markd (T) d a xplct dscusso th tutoral class. Othr problms ar for hacd practc.. (a) d, whr s th coutrclockws ortd smcrcular part of th crcl lyg th scod ad thrd quadrats. (T)(b) d, whr s th clockws ortd boudary of th part of th aulus lyg th thrd ad fourth quadrats. (c) ( a) d, whr s th smcrcl a R, arg( a). Soluto: (a) A paramtrc quato of s (b) 3 / d8 d 8. d /, d d d d AB BE D DFA x dx d x dx d (c) ( a) ( ) d= R R d R d () R [( ) ], for. Th rght had tgral () s obvously for +=. 3. Thrfor th gv tgral s A B D E F

2 . Evaluat th tgral d, ach of th followg cass: (a) s th coutrclockws ortd smcrcular part of th crcl th uppr half pla ad s dfd so that. (T)(b) s th coutrclockws ortd smcrcular part of th crcl th lowr half pla ad s dfd so that. (c) s th clockws ortd crcl ad s dfd so that. (d) s th coutrclockws ortd crcl ad s dfd so that. Soluto: (a) Th two dstct valus of, r, ar gv by l ( ) xp( ),,. t. Th paramtrc quato of s,t t / t [ ] d dt ( ). ( t/) / t (b). Th paramtrc quato of s, t t/ t [ ] d dt ( ). t (/) / t (c). Th paramtrc quato of s, t. t / t [ ] d dt. ( t/) (/) (d). Th paramtrc quato of s t,t. t/ t [ ] d dt. t (/) (/) 3. Wthout actually valuatg th tgral, prov that (T)(a) d, whr s th arc of th crcl from = to = lyg th frst 3 quadrat. (b) ( ) d R( R ), whr s th smcrcl of radus R > wth ctr at th org. Soluto: ( a) Lgth of, ad o,. d (by ML Estmat) 3 (b) Lgth of = R, ad o, R. osqutly, th dsrd qualty follows by MLstmat

3 3. (T)Dos auchy Thorm hold sparatly for th ral or magary part of a aalytc fucto f()? Why or why ot? Soluto: No, auchy Thorm d ot hold sparatly for ral or magary part of a aalytc fucto. osdr, for xampl f() = ad :. Th, R d cos d Im d s d 5. About th pot =, dtrm th Taylor srs for ach of th followg fuctos: (T) () ( ) Log( 3 ) Soluto: () = / ( ) / / ( ) [ ] ( 3) ( )[ ( ). () Log ( 3 + ) = Log ( ) + Log ( ) + m, for som m. = Log ( ) + Log + log ( ) + m, for som m / = Log + m Thrfor th Taylor srs of Log ( 3 + ) s gv by log ( 3 + ) = log + m ( )

4 6. About th dcatd pot =, dtrm th Taylor srs ad ts rgo of covrgc for ach of th followg fuctos. I ach cas, dos th Taylor srs cssarly sums up to th fucto at vry pot of ts rgo of covrgc? (), Soluto: () (T) ( ) cosh, (T) ( ) Log, = ( ). Th Taylor srs sums up to. Th rgo of covrgc of ts Taylor srs s pot of. () cosh = cosh ( + ) = cos (( ) ) = cos (( )). Thrfor, ( ) cosh at vry pot of, sc s aalytc at vry. Th rgo of covrgc of Taylor srs s whol complx pla. Th Taylor srs sums up to cosh at vry pot of th complx pla, sc cosh s aalytc at vry pot of th complx pla. d () Log ( ), for,,... d ( ) ( ) Log Log ( ) ( ). Th rgo of covrgc of th Taylor srs o RHS s. Howvr, th Taylor srs dos ot sum up to Log at vry pot of, sc Log s ot aalytc o gatv ral axs, a part of whch s cotad th dsk, whl th sum fucto rprstd by th abov Taylor srs s aalytc at ach pot of ths dsk.

5 d 7. (T)Evaluat th tgral, for all possbl chocs of th cotour that dos ot pass ( ) through ay of th pots =,. Soluto: Lt curv b ortd coutrclockws th followg cass. For clockws ortd th valu of th tgral wll b gatv of th valu obtad ths cass. d as ( dos ot clos ay of th pots, ): I ths cas, I = =, by auchy Thorm. ( ) as Wh closs oly th pot, ( / ) I = d =, by auchy Itgral Formula. Smlarly, wh closs oly th pot, ( / ) I = d = ad wh closs oly th pot, ( / ) I = d =. as3: Wh closs oly th pots,, ( /( )) (/ ( )) I d d, whr ad ar suffctly small crcls aroud ad rspctvly., by auchy Itgral Formula. Othr cass,.. wh closs oly th pots, or, ar tratd smlarly. as ( closs all of th pots,, ): I ths cas (/( )) (/ ( )) (/ ( )) I d d d, 3 whr, ad 3 ar suffctly small crcls aroud, ad rspctvly.. 5

6 8. (T) Us auchy Thorm for multply coctd domas ad auchy Itgral Formula to valuat th tgral cos d, : th crcl 3 ortd coutrclockws. ( ) Soluto: Lt, b th coutrclockws ortd crcls of suffctly small radus ctrd at ad rspctvly. Th tgrad s a aalytc fucto th rgo lyg btw th crcls ad, Thrfor, by auchy Thorm for multply coctd domas, cos d = ( ) 9. Evaluat (T)(a) (b) (cos / ( )) (cos / ( )) d d ( ) ( ) cos cos ( ) ( ). (by auchy Itgral Formula) d, : th crcl ( ) ( ) ortd clockws s d, : th crcl ortd coutrclockws Soluto: (a) Usg auchy Thorm for multply coctd domas, ( /( ) ) ( / ) Gv Itgral d d, ( ) whr, ar suffctly small clockws ortd crcls. 3 d [ ( ) { ( )} 3 ] (by auchy Itgral Formula for th drvatvs) ( ) 3 d d (b) Gv Itgral = ( s ) ( ) d. If u s a harmoc fucto R ad r R, show that u() u( r ) d. Soluto: Lt v b th harmoc cojugat of u R so that f() = u + v s aalytc R. By auchy Itgral Formula f( ) f () d f ( r ) d. r Now takg th ral part o both th sds gvs th dsrd rsult. 6