PETER HENR IC I TECHNICAL REPORT NO. CS 137 JULY COMPUTER SCIENCE DEPARTMENT School of Humanities and Sciences STANFORD UN IVERS ITY

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1 cs 137 j FXED PONTS OF ANAYTC FUNCTONS BY * PETER HENR C TECHNCA REPORT NO. CS 137 JUY 1969 COMPUTER SCENCE DEPARTMENT School of Humantes and Scences STANFORD UN VERS TY

2 FXED POXNTS QF ANAZYTC.FUNCTQNS*.. BY Peter Henrc*, t Repraducton n whole or n part s permtted for any purpose of the Unted States Governme& * Ths work was supported n part; by the Natonal Scence Foundaton and the Offce of Naval Research. 'Edgengsssche Technscha Hochschule, Zrch, Swtzerland.

3 r : c Abstract A contnuous mappng of a smpy connected, Qosed, bounded set of the eucldean plane nto tself s known to have at least one fxed pont. t s shown that the usual condton for the fxed pont to be unque, and for convergence of the teraton sequence to the fxed pont, can be relaxed f the mappng s defned by an analytc functon of a complex varable.

4 , , We consder the problems of the exstence and of the constructon of solutons of the equaton (1) Z = F(z) 9.. where the functon F s analytc n some doman S of the complex plane. Such solutons are called fxed ponts of F By standard results n real numercal analyss, t follows mmedately that F has at least one fxed pont f S s bounded and smply connected, F s contnuous on the closure S' of s, and F(S) c S'. f the mappng defned by F s contractng, defned by -. then there s a unque fxed pont, and the teraton sequence (2) Z M-1 = F(zn) 2 n = 0, 1, 2,..., converges to the fxed pont for every choce of ZOES' 0 f S s convex, a necessary and suffcent condton for the mappng to be contractng s that the dervatve F' of F satsfes (3) \F (z)\ 2 k > z E s, where k 1. t s the purpose of ths note to show that the hy-pothess that F s _ contractng can be dspensed wth due to the analytcty of F. The argument provdes an opportunty to apply some basc facts of complex varable theory n a constructve settng. THEOREM. et S denote the nteror of a Jordan curve ' j let F be analytc n S and contnuous on S U Y 9 and let F(S U J?) C S Then F has exactly one fxed pont, and the teraton sequence defned bx (2) con- 1

5 ! verges to the fxed pont for arbtrary z. E S u P. Clearly, there are functons F satsfyng the hypotheses for whch F' s arbtrarly large, e.g., F(z) = $ zloo n \z 6 1. Proof. We frst prove the Theorem n the case where S s the unt dsk. t rl 1 Here the hypothess mples The pont s r:= max ॐ z $1 s a fxed pont f and only f t s a zero of z - F(z). To prove the exstence of a zero, we apply Rouch'e's theorem ([l], p. 124) wth Z n the role of the "bg" functon and F(z) n the role of the "small" functon. =' The essental hypothess of Rouch/e's theorem s satsfed n vew!. e of (4). t follows that z - F(z) has exactly as many zeros nsde 1~1 = 1 as z, namely one. et s denote the unque fxed pont. of the teraton sequence, let t(z) = z - s. 1 - ZF Ths s a lnear transformaton whch maps z\ n order to prove the convergence = 1 onto tself and sends s nto 0. Hence the functon G = t 0 F 0 t -1 has the fxed pont 0. t s contnuous and maps z\ = 1 onto a closed subset of \z 1, hence, k:= sup G(z)\ 1 z 51 We may assume that k > 0, for otherwse G t and consequently F, s constant, and convergence takes place n one step. The functon k'lg vanshes at 0 and s bounded by 1, hence by the emma of Schwarz (Cl], p. llo),! 2

6 k-llg(z)l 5 \z\ (5) and consequently, G(z)\ z k\z\ for all z suchthat \z\ $1. et wn = t(z,>. Snce W n+l = t(zn+l ) = f(f(z,)) = t(f(t-l(w,))) = G(w,), t follows from (5) that lwn+l 5 k \w,\ and hence that w n 2 kn\wol, mplyng that wn + 0. Hence Z n = t-l(w,) -b t-l(o) = s. To prove the Theorem for an arbtrary Jordan doman S, we requre a less --. elementary tool, the Osgood-Caratheodory theorem ([2], p* 92-98) statng the exstence of a functon g that maps S conformally onto \z\ 1 and S U r contnuously and one-to-one onto \z\ = 1. H=g.Fo g-l The functon s easly seen to satsfy the hypotheses of the Theorem for the unt dsk. Furthermore, f the ponts zn are defned by (2) and W n = g(z,) ) then w Thus the valdty of the Theorem for the unt dsk mples the valdty for a general S. n lne wth the paedagogcal nature of ths note, we add some problems amplfyng ts content. 1) Show that k = t&d. 2) n the case where S s the unt dsk, show that z - s\ = n (1 + r) kn j n = 0, 1, 2,.... 3) et F'(s) = F"(s) =... = F(m'l)(s) = 0, F(m)(s) # 0 for some nteger m > 1. f S s the unt dsk, establsh the followng error estmate showng superlnear convergence: 3

7 F 2 n-l & Z - s! = n =1,2,.... Research problem. Can smlar results be establshed for systems of analytc equatons?..

8 Cl a Aklfors, Complex Analyss, st edton. McGraw-Hll, New York [21 Co Caratheodory, Theory of functons of a complex varable, vol. 2 (Englsh edton). Chelsea, New York l$o.