We show that every analytic function can be expanded into a power series, called the Taylor series of the function.
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1 10 Lectue 8 We show tht evey lytic fuctio c be expded ito powe seies, clled the Tylo seies of the fuctio. Tylo s Theoem: Let f be lytic i domi D & D. The, f(z) c be expessed s the powe seies f( z) b ( z ) (1) whee, whee, b C 0 ( ) 1 f( w) f ( ) dw, f (0) ( ) f( ), 1 2 i C! w D is couteclockwise oieted cicle, of dius d cete t, such tht it ecloses oly poits of D. The epesettio (1) is uique d is vlid i the lgest ope disk with cete, cotied i D. Poof: By usig Cuchy Itegl Fomul d Cuchy Theoem Fo Multiply Coected Domis, w 1 f( w) 1 f( w) f ( z) dw dw 2i C wz 2i C wz, whee, z is y poit eclosed by the cicle C d C is couteclockwise oieted cicle wz with sufficietly smll dius such tht C lies i the bouded domi eclosed by C. C z C D
2 11 Now, z [1 ] wz wz w w Recll tht, 1 1 q 1 q... q 1 q 1 1 q 1 q... q, 1q 1q fo y complex umbe q z Let q w. The, z 1 ( ) 1 1 z z 1 [1... ( ) w ] wz w w w w z 1 w 1 f ( w) 1 f( w) 1 f( w) dw dw ( z ) 2 dw... 2i C wz 2i 2 C w i C w 1 f ( w)... ( z) dw 2 1 i C w 1 1 f ( w) ( z) dw 1 2 i C w ( wz) 1 whee, R ( z )
3 12 1 * z M () * f( w) R ( z)..2, fo M ( ) mx 1 2 wc w z 1 * z z M () 0 s, sice 1. Thus, with b f ( z) b ( z ), 0 1 f ( w) i C w 2 1 dw. C z w Futhe, sice f ( z ) is epeseted C by powe seies, by pevious D popositio o powe seies, f ( z ) is ifiitely my times diffeetible i z d ( ) 1 f ( w) f ( ) b dw. 1 2 i C! w ( f ) ( ) Sice b, it depeds oly o f d ' ', so b s e! uiquely detemied. * (becuse, if f ( z) b ( ) ( ) * f ( ) z, b b).! 0 Thus, (1) epesets f uiquely.
4 13 Popositio: Evey fuctio f ( z ), lytic i domi D, is ifiitely my times diffeetible id. Poof: D { z }. D By Tylo s Theoem, fo evey D, f ( z ) is epeseted by powe seies i z. By elie popositio o powe seies, the fuctios epeseted by powe seies e ifiitely my times diffeetible. So tht f ( z ) is ifiitely my times diffeetible i fo evey D. z Theefoe, f ( z ) is ifiitely my times diffeetible i D.
5 Cuchy Itegl Fomul fo th deivtive If f is lytic i domi D d B (, ) B (, ) { w: w }. The, 14 D, whee ( )! f( w) f ( ) dw, 0,1,2,... (*) 1 2 i ( w ) C it whee, C : w( t) e, 0t 2, is couteclockwise oieted cicle of dius ceted t. Poof: Follows immeditely sice, by the poof of Tylo s ( ) 1 f ( w) f ( ) Theoem, b dw. 1 2 i C w! Fo 0, deotig Itegl Fomul. f (0) ( ) f( ), (*) becomes Cuchy Note: I view of Cuchy Theoem fo multiply coected domis, fomul (*) emis vlid with C eplced by y simple closed piece wise smooth cuve so tht (i) evey poit eclosed by is i D (ii) ecloses the poit. This is 1 becuse the fuctio f( w)/( w ) is lytic i the domi lyig betwee C d.
6 Remk: The fomul (*) gives the vlue of the fuctio d its deivtives t y poit eclosed by simple closed piecewise diffeetible cuve, if the vlues of the fuctio o e kow. This helps i kowig the vlues of the fuctio d its deivtives t sometimes iccessible poits though vlues t ccessible poits. A Computtiol Method, clled Complex Vible Boudy Elemet Method, developed usig (*), is get tool to computtiolly geete the vlues of f ( ), f( ), f ( ),... etc.. 15
7 16 Deductios Fom Tyolo s Theoem: Popositio 1: Evey powe seies with ozeo dius of covegece is the Tylo seies of the fuctio epeseted by it. Poof: Let (*) z R, i.e. b z epesets the fuctio f(z) i 0 ( ) f ( z) b ( z) i z R. The, by the 0 ( f ) ( ) poof of Tylo s Theoem, b. This implies tht the! give seies (*) is the Tylo seies of f.
8 17 Popositio 2 (Cuchy s Estimte): f ( z) M( R) o z R. The, Let f be lytic d f ( ) MR! ( ) ( ). R Poof: By Cuchy Itegl Fomul fo th deivtive (Tke D { z R}, fo y R, ( )! f( w) f ( ) dw, 0,1,2, i ( w ) C R ( )! M( R)! M( R) f ( ).2 (usig ML Estimte) 1 2 Sice < R is bity, the esult follows o lettig R.
9 Popositio 3 (Liouville s Theoem): A etie (i.e. lytic i the whole Complex Ple) fuctio tht is bouded i the whole Complex Ple is costt. Poof: Sice f is etie d bouded i the whole complex ple, f ( z) M o evey cicle C { z: z R}. Now, expd f ( z ) i to Tylo seies s R f ( z) 0 18 z fo z i z R 0. The sme expsio is vlid fo z R fo ll R R0. By Cuchy Estimte, f (0) M 0 s R, fo ll 1,2...! R f ( z) costt, o evey disk z R 0 Cosequetly f ( z ) is costt i the whole complex ple C, sice R R0 is bity.
10 19 Popositio 4 (Fudmetl Theoem of Algeb): A polyomil of degee hs exctly complex zeos (couted ccodig to multiplicity). Poof: Let P(z) be polyomil of degee 1. d it hs o 1 zeos i the complex ple C. The, the fuctio ( z) P ( z ) (i) is etie fuctio (ii) is bouded i C (sice P(z) s z ). Theefoe, by Liouville s Theoem, ( z) is costt. P(z) is lso costt fuctio, cotdictio. Thus, P(z) hs t lest oe zeo, sy 1 of multiplicity m 1. P ( z) Now, the polyomil, is of degee m m 1. A epetitio ( z 1 1) of the bove gumets gives tht it hs t lest oe zeo, sy 2 of multiplicity m 2. Cotiuig the pocess, it follows tht P ( z ) m1m2... mk zeos t 1, 2,..., k. hs
11 Popositio 5. If f is etie fuctio d f ( z) MR 0 i z Rfo evey R,0 R the f is polyomil of degee t most Poof: By Tylo s Theoem, expd f ( z) The sme expsio is vlid fo ll R R0. z i z R0 0. By Cuchy Estimte, f ( ) M! ( R) (0), whee R M ( R) mx f( z) z R MR 0 0 MR 0 s, if 0. R f is polyomil of degee t most 0.
For this purpose, we need the following result:
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