9.5 Complex variables

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1 9.5 Cmpl varabls. Cnsdr th funtn u v f( ) whr ( ) ( ), f( ), fr ths funtn tw statmnts ar as fllws: Statmnt : f( ) satsf Cauh mann quatn at th rgn. Statmnt : f ( ) ds nt st Th rrt statmnt ar (A) nl (B) nl (C)Bth& (D)nthr nr. If f( ) u v, thn nsdr th fur slutn fr f ( ) () u () v v () u v () u v Th rrt slutn fr f ( ) ar (A)& (B)& (C)& (D)&. If f() s ln (A) (C), thn f ( ) st at all pnts n th (B) (D). Th njugat f th funtn u ( ) s (A) (B) (C) (C) ( ) (D) ( ). If u snh s thn th analt funtn f( ) u jv s (A) sh (B) sh (C) snh (D) snh 7. If v, thn th analt funtn f( ) u vs (A) (C) (B) (D) 8. If v ( ) s, thn analt funtn f u v (A) (B) (C) ( ) (D) ( ) sn 9. If u sh s, thn th analt funtn f( ) u v s (A) t (B) s (C) snh (D) sh. Th ntgratn f f( ) frm A(, ) t B(, ) alng th straght ln AB jnng th tw pnts s (A) 9 (B) 9 (C) 5 (D) 5. d? whr s th rl f ( ) 5. If f( ) u v s an analt funtn f and v v (s sn ), th f( ) n trms f s (A) ( ) (B) ( ) (A) 9 (C) (B) 9 (D) 8

2 58 Cmpl varabls Chap 9.5. d? whr s th rl 5. ( )( ) (A) (B) (C) (D). ( ) d? whr s th uppr half f th rl (A) (C) (B) (D) s. d? whr s th rl (A) (B) (C) (D) sn 5. d? whr s th rl ( )( ) (A) (B) (C) (D). Th valu f s d arund a rtangl wth vrts at ±, ± s (A) (B) (C) 8 (D) Statmnt fr Q. 7 8: f( ). 7 d ( ) 7. Th valu f f ( ) s (A) (B) (C) (D) 8. Th valu f f ( ) s, whr s th rl (A) 7( ) (B) ( ) (C) ( 5 ) (D) Statmnt fr 9 : Epand th gvn funtn n Talr s srs. 9. f( ) abut th pnts (A) (...) (B) (...) (C) (...). f( ) abut (A) ( ) ( )... (B) ( ) ( )... (C) ( ) ( ).... f( ) sn abut (A)...! (B)...! (C)...!. If <, thn s qual t n (A) ( n )( ) n (B) ( n )( ) n (C) n ( ) n (D) ( n )( ) n n n n Statmnt fr Q. 5. Epand th funtn ( )( ) n Laurnt s srs fr th ndtn gvn n qustn.. < < (A)... (B) 8 8

3 Chap 9.5 Cmpl varabls 59 7 (C).... > (A)... (B) (C) < 7 5 (A)... (B) (C) If <, th Laurnt s srs fr ( ) (A) ( )! ( ) (B) ( )! ( ) 5! 5 ( ) 5! (C) ( ) ( ) ( )... 5 (D) ( ) ( ) ( ) ( )... 5 s ( )( ) 7. Th Laurnt s srs f fr < s ( ) (A)... 7 (B)... 7 (C)... 7 (D) Th rsdu f th funtn (A) (C). Th rsdu f s (A) (C) (B) (D) at s (B) (D). d? whr s 5. ( )( ) (A) (B) (C) (D). (A) (C) s d? Z whr s. d? whr s (A) C). (A) dθ? s θ (B) at ts pl s (B) (B) (C) (D) 8. Th Laurnt s srs f f( ) ( )( ) s, whr < (A) (B) 5... (C) (A) ab a b (C) a b a b d? ( )( ) (B) ( a b) ab (D) ( a b)

4 5 Cmpl varabls Chap 9.5 d.? (A) (C) (B) (D) *************** Slutns ( ) ( ). (C) Sn, f( ) u v ; u ; v Cauh mann quatns ar v and u v B dffrntatn th valu f u v v,,, at(, ) w gt, s w appl frst prnpl mthd. At th rgn, u h u h h lm (, ) (, ) lm h h h h u k u k k lm (, ) (, ) lm v h k k k v v h v h h lm (, ) (, ) lm h h h h v v k v k k lm (, ), (, ) lm k k k k Thus, w s that u v and u v Hn, Cauh-mann quatns ar satsfd at. f f Agan, f ( ) lm ( ) ( ) lm ( ) ( ) ( ) ( ) Nw lt alng, thn f ( ) lm ( ) ( ) ( ) ( ) ( ) Agan lt alng, thn ( ) f ( ) lm ( ) S w s that f ( ) s nt unqu. Hn f ( ) ds nt st. df Δf. (A) Sn, f ( ) lm d Δ Δ Or f ( ) Δu Δv lm Δ Δ Δ...() Nw, th drvatv f ( ) ts f th lmt n quatn () s unqu.. t ds nt dpnds n th path alng whh Δ. Lt Δ alng a path paralll t ral a Δ Δ Δ Nw quatn () f ( ) Δu Δv Δu Δv lm lm lm Δ Δ Δ Δ Δ Δ u f v ( )...() Agan, lt Δ alng a path paralll t magnar as, thn Δ and Δ Δ Thus frm quatn () φ ( ) Δ Δv lm Δ Δ Δu Δv lm lm u Δ Δ Δ Δ v u v f ( )...() Nw, fr stn f f ( ).H.S. f quatn () and () must b sam.., v v u v v u f ( ) u u v v. (A) Gvn f( ) sn, f( ) u v Hr u and v Nw, u u and u and v v v w knw that u f u ( )...()

5 Chap 9.5 Cmpl varabls 5 and v f v ( )...() Nw, quatn () gvs f ( )...() and quatn () gvs f ( )...() Nw, fr stn f f ( ) at an pnt s nssar that th valu f f ( ) mst b unqu at that pnt, whatvr b th path f rahng at that pnt Frm quatn () and () Hn, f ( ) sts fr all pnts l n th ln.. (B) u ( ) ; u ; u...()...() u u, Thus u s harmn. Nw lt v b th njugat f u thn v dv d v d u u (b Cauh-mann quatn) dv d ( ) d On ntgratng v C 5. (C) Gvn f( ) u v...() f ( ) v u...() add quatn () and () ( f ) ( ) ( u v) u ( v) F ( ) U V whr, F ( ) ( f ) ( ); U ( uv); V u v Lt F ( ) b an analt funtn. Nw, U u v (s sn ) U (s sn ) and U ( sn s ) U Nw, dv d U d...() (sn s ) d (s sn ) d d [ (sn s )] n ntgratng V (sn s ) F U V ( ) (s sn ) (sn s ) (s sn ) (s sn ) F ( ) ( ) ( ) ( f ) ( ) ( ) f ( ) ( ) ( )( ) ( ) f( ) ( ). (C) u snh s sh s φ (, ) and u snh sn ψ(, ) b Mln s Mthd f ( ) φ(, ) ψ (, ) sh sh On ntgratng f( ) snh nstant f( ) w snh (As u ds nt ntan an nstant, th nstant s n th funtn and hn.. n w). 7. (A) v h(, ), v g(, ) b Mln s Mthd f ( ) g(, ) h(, ) On ntgratng f( ) 8. (D) v ( ) ( ) ( ) g (, ) ( ) v ( ) ( ) h (, ) ( ) ( ) B Mln s Mthd f ( ) g(, ) h(, ) ( ) On ntgratng f d ( ) ( ) ( ) 9. (A) u s (sh s ) sn (sh s ) s sh φ(, ) (sh s ) sn snh ψ(, ) (sh s ) B Mln s Mthd f ( ) φ(, ) ψ(, ) s ( ) s ( s ) s On ntgratng

6 5 Cmpl varabls Chap 9.5 f( ) s d t. at b, t d On A, and On B, Lt rrspnds t t and rrspndng t t thn, t b, d b, d and t a b, d a, a, AB s, t d dt ; d dt f ( ) d ( )( d d) [( t ) ( t )( t )][ dt dt] t [( t t ) ( t t )]( ) dt ( ) t ( ) 9 t t t t t. (D) W knw b th drvatv f an analt funtn that n! f( ) d f ( ) n ( ) f( ) d Or n f n n ( ) ( )! Takng n, Gvn f f( ) d f ( ) ( ) d d ( ) [ ( )]...() Takng f( ), and n (), w hav d f ( )...() ( ) Nw, f( ) f ( ) 8 f ( ) 8 quatn () hav d 8 ( )...() If s th rl Sn, f( ) s analt wthn and n d ( ) 8 d I I I...() ( )( ) Sn, s th nl sngulart fr I d and t ls nsd 5., thrfr b Cauh s ntgral Frmula I d...() f( ) d f( ) [Hr f( ) f( ) and ] Smlarl, fr I d, th sngular pnt ls nsd 5., thrfr I...() Fr I d, th sngular pnt ls utsd th rl 5., s th funtn f( ) s analt vrwhr n.. 5., hn b Cauh s ntgral thrm I d...() usng quatns (), (), () n (), w gt d ( ) ( ( )( ) ). (B) Gvn ntur s th rl θ d θ dθ Nw, fr uppr half f th rl, θ ( ) d θ ( θ θ ) dθ θ θ θ ( ) dθ θ θ ( ) ( ). (B) Lt f( ) s thn f( ) s analt wthn and n, nw b Cauh s ntgral frmula f( ) f( ) d tak f( ) s,, w hav s d f () f ( ) d f ( ) s. (D) Sn, ( )( ) ( ) 5. (D) sn d ( )( )

7 Chap 9.5 Cmpl varabls 5 sn sn d d f ( ) f ( ) sn, f( ) sn f ( ) sn and f () sn. (D) Lt, I s d s d s n s n Or I d 7. (D) f ( ) 7 d, sn s th nl sngular pnt f 7 and t ls utsd th rl..,, thrfr 7 s analt vrwhr wthn. Hn b Cauh s thrm f ( ) 7 d 8. (C) Th pnt ( ) ls wthn rl (... th dstan f.., (, ) frm th rgn s whh s lss than, th radus f th rl). Lt φ ( ) 7 thn b Cauh s ntgral frmula 7 d φ( ) f( ) φ( ) f ( ) φ ( ) and f ( ) φ ( ) sn, φ ( ) 7 φ ( ) 7 and φ ( ) f ( ) [ ( ) 7] ( 5 ) 9. (C) f ( ) f ( ), f () f ( ) f ( ) ; ( ) f ( ) f ( ) ; ( ) f ( ) f ( ) ; and s n. ( ) Nw, Talr srs s gvn b ( ) f( ) f( ) ( ) f ( ) f ( )! ( ) f ( )...! abut f( ) ( )! ( )! ( ) f( ) (...). (B) f( ) f () f ( ) f () ( ) f ( ) f () ( ) f ( ) f () and s n. ( ) 8 Talr srs s ( ) f( ) f( ) ( ) f ( ) f ( )! ( ) f ( )! abut ( ) f( ) ( )! ( )! ( ) ( ) ( )... 8 r f( ) ( ) ( ) ( ).... (A) f( ) sn f f ( ) s f f ( ) sn f sn f ( ) s f Talr srs s gvn b and s n. ( ) f( ) f( ) ( ) f ( ) f ( )! ( ) f ( )...! abut

8 5 Cmpl varabls Chap 9.5 f( )!! f( )...!!. (D) Lt f( ) [ ( )] f( ) [ ( )] Sn, <, s b pandng.h.s. b bnmal thrm, w gt f( ) ( ) ( ) ( ) ( n )( ) n r f( ) ( n )( ) n. (B) Hr f( ) ( )( )...() Sn, > < and < < and 9 quatn () gvs f( ).. 9 r f( ) 8 8. (C) < < < < 8 and... Laurnt s srs s gvn b 98 f( ).... n 7 7 f( ) 5. (B) <, ( ) ( 8...) 7 5 f( ) 8. (D) Sn, ( )( ) ( ) Fr < Lt u u and u < ( )( ) ( ) ( u ) u ( u ) ( u) u ( u) [ u u u...] u ( u u u...) ( uu...) u uu u u qurd Laurnt s srs s f( ) ( ) ( ) ( ) ( ) 5 7. (B) Lt f( ) ( )!!!!!!! " "! "! " 8 8 " r f( )......

9 Chap 9.5 Cmpl varabls qurd Laurnt s srs s f ( ) (A) Sn, f( ) ( )( ) ( ) ( ) < < f( ) ( ) (...) 5 5 r f( ) 9. (B) Lt f( ) thn f( ) has a pl at f rdr. sdu f f( ) at m ( )! lm, d m ( ) m f d ( ) m ( )! lm d d 8! lm d ( )! lm d 8 8 ( ). (B) Put t, f( ) s ts t t! t! t t t t sdu f f( ) at s th ffnt f t... Pls f f( ) ar at,, sn and l wthn and ds nt nsd. f( ) d [sum f rsdus at and at ]...() Nw, sdu at s lm f ( ) lm ( )( ) and sdu at s lm ( ) f( ) lm ( ) quatn () gvs f( ) s. (D) f( ) thn f( ) has a pl at f rdr. b Cauh s rsdu thrm f( ) d sdu at Nw, sdu at s d lm ( ) d f lm ( s d d ) lm [s sn ] f( ) d. (C) f( )!! Th nl pl f f( ) s at, whh ls wthn th rl f( ) d (rsdu at ) Nw, rsdu f f( ) at s th ffnt f.. f( ) d. (B) Lt θ d dθ ; θ and s θ

10 5 Cmpl varabls Chap 9.5 dθ s θ Lt f( ) d ; : d f( ) has pls at, ut f ths nl ls nsd th rl : f( ) d (sdu at ) Nw, rsdu at lm ( ) f( ) lm ( ) f( ) d dθ s θ 5. (C) I d f d a b ( ) ( )( ) whr s b sm rl r wth sgmnt n ral as frm t. Th pls ar ± a, ± b. Hr nl a and b l wthn th ntur f( ) d (sum f rsdus at a and b) sdu at a, lm ( ) a a a ( )( )( a a b ) a ( b ) sdu at b b lm ( b) b ( a )( a )( b )( b ) a ( b) f( ) d f( ) d f( ) d r a b ( a b ) ( ) a b Nw r f( ) d θ θ d θ a b θ θ θ dθ θ θ ( a)( b) Nw whn, bd ( ) a b d ( )( ) a b r d. (C) Lt I f d ( ) s th ntur ntanng sm rl r f radus and sgmnt frm t. Fr pls f f( ), ( ) n ) ( whr n,,,,, 5, Onl pls,, l n th ntur sdu at ( )( )( )( )( ) 5 ( ) sdu at s sdu at s ( ) f( ) d f( ) d f( ) d r ( ) r f( ) d f( ) d r Nw f( ) d θ d θ θ θ dθ 5...() θ whr, f( ) d a () r ********

11 Chap 9.5 Cmpl varabls 57

Math 656 Midterm Examination March 27, 2015 Prof. Victor Matveev

Math 656 Midterm Examination March 27, 2015 Prof. Victor Matveev Math 656 Mdtrm Examnatn March 7, 05 Prf. Vctr Matvv ) (4pts) Fnd all vals f n plar r artsan frm, and plt thm as pnts n th cmplx plan: (a) Snc n-th rt has xactly n vals, thr wll b xactly =6 vals, lyng n