9 Lectue Sigulities of omplex Fuctio A poit is clled sigulity of fuctio f ( z ) if f ( z ) is ot lytic t the poit. A sigulity is clled isolted sigulity of f ( z ), if f ( z ) is lytic i some puctued disk 0 z, i.e. if f ( z ) does ot hve y sigulity i 0 z, except t the poit. Exmples: (i) Evey poit o egtive el xis is o isolted sigulity of Log z (ii) The poits 0 d e isolted sigulities of the fuctio. z z We e iteested hee i studyig the tue of fuctio f ( z ), i puctued disk ceteed t isolted sigulity of f ( z ). (fo exmple, (i) existece o oexistece of lim f ( z ) z (ii) boudedess o uboudedess of f ( z ), etc.) Fo this pupose, we eed the followig esult:
0 Luet Theoem Let f be lytic i the closed ulus z. The, fo ech poit z{ z }, it c be expded s the Luet s seies K z d f( z) c z (i) 0 z whee, f ( w) c dw; 0,,,...( ii) i w : w is oieted couteclockwise, f ( w) d dw;,,...( iii) i w : w is oieted couteclockwise. Notes.. If f is lytic i 0 z, (i) is vlid i 0 z. If f is lytic i z d o (iii) = 0 f ( w) w, the fuctio is lytic iside The Luet s expsio (i) educes to Tylo s expsio ( 0) i this cse.
. The Luet s expsio (i) c lso be witte s f( z) z, f( w) whee, dw; 0,,,... i w beig y closed p.w. smooth cuve oieted ticlockwise, lyig iside the ulus z d suoudig the poit (follows by usig the coolly to uchy Theoem fo Multiply oected Domis becuse the itegds o RHS itegls i (ii) d (iii) e lytic o cuves, & d i the domis lyig betwee the cuves, d, ).
Poof of Luet s Theoem: Let z be y poit i z. By uchy Theoem fo Multiply oected Domis d uchy Itegl Fomul, f ( w) f( w) f( w) dw dw dw wz wz wz K i f( z) 0 K z f ( w) f( w) f ( z) dw dw i wz i wz (*) Fo w, f ( w) f( w)[ ] wz w( z) f( w) z z (( z)/( w)) [... ( ) ] w w w z w q (sice, ( q) qq..., fo q) q
3 f ( w) f( w) f( w) ( z ) f( w) ( z)... ( z) w ( w) ( w) ( w) ( wz) f ( w) dw i w z f( w) dw i w K z c 0 f( w) ( )( )... dw z i ( w ) c f( w) ( z ) f( w) ( dw)( z ) dw) () i ( w) ( w) ( wz) i c - R whee, fo M mx f( w), w R z f( w) M. dw ( ) w wz ( wz w z ) M 0 s ( ).
4 Fo w, f ( w) f( w)[ ] wz w( z) f ( w) w = [ ] z z f( w) f( w) w w (( w)/( z)) [... ( ) ] wz z z z w z f ( w) f( w)( w)... f( w)( w) z ( z) ( z) f ( w)( w ) ( z ) ( z w) f ( w) dw ( f ( w ) dw ) i wz i z d d d ( f( w)( w) dw)... i ( z ) ( f( w)( w) dw) i ( z ) ( w ) f( w) dw i ( z) ( zw) () * R
5 whee, fo M * mx f( w), w R * * f( w)( w ) M dw ( ) i (- ( z) ( zw) ) 0 s,( ). ( zw z w ) K z Theefoe, the equtio (*) f ( w) f( w) f ( z) dw dw, i wz i wz togethe with () d () gives the desied Luet s expsio.
6 Popositio. If z coveges to the fuctio f(z) fo ll the poits i z, the it is Luet s seies expsio of f(z) i this ulus. Poof. Let be y simple, closed, p.w. smooth ticlockwise oieted cuve lyig i z d eclosig the poit. The, fo ll w,. f ( w) w. f ( w) i w i w dw m m m, m 0,,,.... dw 0, if m ( dw m ) w i, if m.
7 Exmple: Fid Luet seies expsio of fo ( z )( z) () z (b) z > (c) z < (d) 0 < z <. Solutio. () z : z Wite ( ) ( ) d expd RHS s z z z z biomil expsio. (b) z > : Wite ( ) ( ) d expd RHS s z z z z z z biomil expsio. (c) z < : z Wite ( ) ( ) z d expd RHS s z z biomil expsio. Note tht the Luet s expsio i this cse is othig but the Tylo s expsio sice the fuctio is lytic i z <. (d) 0 < z < : Wite ( ( z )) z z z z z expd RHS s biomil expsio. d
8 lssifictio of Sigulities. Let the poit be isolted sigulity of fuctio f(z) d let 0 f ( z) c z d z be the Luet s expsio of f(z) i 0 z R. The secod seies o RHS (cotiig egtive powes of (z )) is clled the Picipl Pt of the Luet s expsio). (i) If d 0,,..., the poit is clled emovble sigulity of f. (ii) If d 0 0 but of ode 0 of f. d 0, the poit is clled pole 0 (iii) If 0 d fo ifiitely my s, the poit is clled essetil sigulity of f.
Behviou of f ( z ) i the eighbouhood of Removble Sigulity: Popositio. The poit is emovble sigulity of fuctio f iff f is bouded i 0 z fo some 0. Poof. (i) Let f be bouded i 0 z fo some 0. Let, 0 f ( z) c z d z be the Luet s expsio of f. The, fo some with0, 9 d f ( w) dw. i w w d.. M, wheem is the uppe boud of f ( z ) i 0 f( z). M 0 s 0. d 0 is emovble sigulity of f.
(ii) Let be emovble sigulity of f. The, d 0,,.... Theefoe, 0 f( z) c z, 0 z fo some. 0 Defie, gz ( ) f( z), if 0 z c0, if z The, g(z) is bouded i z fo some. 0 ( g( z) c z i z, hece is lytic ) f(z) is bouded i 0 z.
oolly. The poit is emovble sigulity of f iff lim( z) f( z) 0. z Poof. (i) is emovble sigulity of f f is bouded i 0 z, fo some. lim( z) f( z) 0. z (ii) Suppose lim( z) f( z) 0. z fo evey 0, thee exists 0 such tht f( z) fo 0 z. z f ( w) Theefoe, d dw,),0. This gives, i ww d.. 0 s 0, if d, sice by usig the bove estimte gi with, d d is bity, d = 0 f(z) hs emovble sigulity t the poit.