A Mathematical Note Convergence of Infinite Compositions of Complex Functions

Transcription

1 A Matheatical Note Covergece o Iiite Copositios o Coplex Fuctios Joh Gill ABSTRACT: Ier Copositio o aalytic uctios ( () ) ad Outer Copositio o aalytic uctios ( () ) are variatios o siple iteratio, ad their covergece behaviors as becoes iiite ay relect that o siple iteratios o cotractio appigs* Several theores are cobied to give a suary o wor i this area I additio, recet results by the author ad others provide covergece ioratio about such copositios that ivolve uctios that are ot cotractive, ad i soe cases, either aalytic or eroorphic [AMS Subject Classiicatio 4A3, priary, 3E99, secodary /] * φ deied o a siply-coected doai S with φ(s) Ω S, Ω copact

2 Table o Cotets Historical Bacgroud: Liear Fractioal Trasoratios 3 Cotractive Fuctios: The Geeral Theore 5 Ier Copositios or Forward Iteratio: Basics 6 Lorete Theore 7 Taery Theore Extesio 8 Exaples: C-cotiued ractios (7), expoetial tower (7), sequece o itegral uctios (7) Outer Copositios or Bacward Iteratio 8 Gill Theore 8 Itegral Equatios Taery Theore Extesio Exaples: Fixed poit o cotiued ractio (9), itegral equatios (,), power series (), Cotiued ractios () No-Cotractive Fuctios Ier Copositios Kojia s Theore Associated Theory (Gill) 5 Exaples: Fuctioal expasio as a quadratic (), ratioal uctio sequece (),() Outer Copositios 3 Theory (Gill) 3 Exaples: Ratioal uctio sequece (5), liear/itegral uctio sequece (5), ratioal uctio sequece (6) Reereces 7

3 Historical Bacgroud Aog the previously ivestigated classes o special uctios with regard to both ier ad outer copositio are liear ractioal trasoratios (LFTs) : a + b ( ) = c + d, oralied by D = a d b c = I the LFT has two ixed poits α β it ay be writte i ultiplier orat: ito our distict types: Hyperbolic i < K < i Elliptic i K = e θ, θ (od π ) ( ) α α = K Thus LFTs all ( ) β β iθ Loxodroic i K = K e, < K <, ad θ (od π ) Parabolic i α = β Both cotiued ractios ad series ay be perceived as iiite copositios o certai classes o LFTs; ad i the case o cotiued ractios, especially the aalytic theory o cotiued ractios, ay coclusios derive ro studies usig this approach The ollowig theores, however, touch upo the highlights o ivestigatios ito the pure covergece behavior o copositio sequeces o geeral LFTs: Set F () = () ad G () = () where li =, b, F, a LFT Theore [9] I a the F c, ad d all coverge absolutely, Theore [9] I F F, a LFT, the li = = Theore [] I li = ad all uctios are hyperbolic or loxodroic, the F C or all β = Li β Theore [] I { } are elliptic ad li = is elliptic, ad β β <, α α <, ad β β α α { } F () diverges or all α, β, with + <, the F ( α) α ad F ( β ) β, α β * * * * 3

4 Theore [] Suppose { } are LFTs covergig to a parabolic LFT,, with α α iite I β+ β < ad α β <, the F C or all Theore [] Give { } LFTs with li = elliptic with ixed poits α ad β β (i) I α α <, β <, ad K diverges to, the F C or all β (ii) I α α <, β β <, ad K F () * * diverges or α, β while F ( α) α ad F ( β ) β coverges, the { } Theore [] Suppose iθ (i) K = K e, < K <, K ( ) or K ( α ) (ii) α α, β β, α β (iii) α α < ad β β < I I K = the F C or β ad G α (except perhaps at oe poit) K coverges absolutely the both F ad G coverge to LFTs These theores are geeraliatios o the siple ad elegat covergece ad divergece behavior o iteratios o sigle LFTs [] The reaider o this article is devoted to explorig the covergece behavior o iiite ier ad outer copositios o ore geeral uctios Although the coditio li =, soeties called liit periodic, plays a sigiicat role i the theores listed above, it is ot required i ost o what ollows 4

5 Cotractive Fuctios Theore (Gill) Let { } be a sequece o uctios aalytic o a siply-coected doai D Suppose there exists a copact set Ω D such that or each, (D) Ω Thus or each, there exists a uique α = ( α ) Ω Deie {()} to be a orderig o the positive itegers ad {s()} a iiite sequece o s ad s ()() i s() = () i s() = i s() = i s() = Set µ () = { ad η () = { Next, set Ψ () = () ad Ψ () = µ Ψ η () The covergece behaviors o {s()} ollows: { Ψ ()} uder all possible choices o {()} ad () () = ay orderig o {} ad s() = or all : Outer Copositio Ψ () = () ad The () ( ) () Ψ () α i ad oly i the sequece o ixed poits { α} o { } coverge to α Covergece is uior o D () () = ay orderig o {} ad s() = or all : Ier Copositio The Ψ () = () () () () ad Ψ () λ (), where λ () is a uctio o the sequece {()}, ad a costat or each such sequece Covergece is uior o D (3) () = ay orderig o {} ad {s()} is ay (rado) sequece o s ad s: The, assuig there are a iiite uber o s i the sequece {s()}, Ψ () α i ad oly i the sequece o ixed poits { α} o { } coverge to α I there are oly a iite uber o s, Ψ () λ, where λ depeds upo the sequece {()} Coet: This theore siply relects the results i cotractio theory that ollow 5

6 Ier Copositio We begi with a well-ow ad very basic result: Theore (Herici [], 974) Let be aalytic i a siply-coected regio S ad cotiuous o the closure S' o S Suppose (S') is a bouded set cotaied i S The () = () α, the attractive ixed poit o i S, or all i S' This result ca be exteded to orward iteratio (or ier copositio) ivolvig a sequece o uctios The author bega a study o iiite copositios o liear ractioal trasoratios i [] Later, i the 98s, the author iitiated a study o extesios o liit-periodic cotiued ractios, i which sequeces F () = F ( ()) = (), with ivolvig uctios other tha those givig rise to cotiued ractios a (vi, () =, a Mobius Trasoratio) were ivestigated: b + Theore 3 (Gill [3], 988) Suppose there exist a sequece o regios {D } ad a sequece o uctios { }aalytic o those respective regios, such that (D ) D,ad there exists a regio D D with uiorly o copact subsets o D Furtherore, suppose there exists α D such that () α < α or all Boudary(D) λ D such that li F ( ) = λ i D The there exists { } 6

7 Lorete geeralied this idea cosiderably: Theore 4 (Lorete [4], 99 ) Let { } be a sequece o uctios aalytic o a siply-coected doai D Suppose there exists a copact set Ω D such that or each, (D) Ω The F () = () coverges uiorly i D to a costat uctio F() = λ a a Exaple: The siple cotiued ractio is geerated by a ier + + copositio i which a (, w) =, F (, w) = (, w) ad F (, w) = F (, (, w)) Suppose + w < ad w <R< The a < ρr( R) (, w) < ρr, < ρ < For R=/, or istace, F (, w) φ () aalytic i ( <) Exaple: The expoetial tower geerated by w (, w) + e,, w R = < < : F (, w) = F (, (, w)) is coverget, or istace or R=, to a liit uctio φ () aalytic i ( <) Exaple: Set (, w) = ψ (t, w)dt where <, w <, ψ (t, w) < ρ <, ad geerate F (, w) = F (, (, w)) to obtai a uctio φ () aalytic i ( <) For exaple, set t t e e (, w) = dt, givig F (, w) = t tw dt, which coverges to e 5 + t dt 6+ a uctio φ () aalytic i ( <) 7

8 Additioal Related Result: Theore 5 (Gill, [5],99) Let {,}, be a aily o uctios aalytic o a siply-coected doai D Suppose there exists a copact set Ω D such that or each ad,, (D) Ω ad, i additio, li, () = () uiorly o D or each The, with F p, () =,, p, (), F, () λ, a costat uctio, as, uiorly o D Coet: The coditio li, () = (), i discarded, allows the possibility o divergece by oscillatio: vi, { 5 i is odd, () =, otherwise, (), o S = ( < ) 5 i is eve Outer Copositio Theore 6 (Gill [6], 99) Let {g }be a sequece o uctios aalytic o a siply-coected doai D ad cotiuous o the closure o D Suppose there exists a copact set Ω D such that g (D) Ω or all Deie G () = g g g () The G () α uiorly o the closure o D i ad oly i the sequece o ixed poits { α } o the {g } i Ω coverge to the uber α coets: The existece o the g () = 5 or odd ad g () 5 although that is usually the case I The proo o the assertio that { α } is guarateed by Theore Note the siple couter-exaple = or eve, i the uit dis ( <) It is ot essetial that g g, g g, the α α G () α is oud i the paper cited above However, the proo o the coverse that the sequece o (attractive) ixed poits coverge to a coo liit is setched here: For clarity, assue D = ( <) ad 8

9 li G () = li g g g () = α = (uiorly) or all i D The Riea Mappig theore ad Mobius uctios ca be eployed to exted these results () Schwar' Lea ca be used to prove the ollowig: ρ such that g () α < ρ α where ρ < ad Supρ = ρ < There exist { } () Suppose the ixed poits α do ot coverge to α = The there exists a sequece r( ρ) { α } ad λ, with r = λ > such that α j λ < δ = or all j j suicietly large 4( + ρ) r( ρ) (3) There exists N such that >N iplies w < ε =, where w = G () 4( + ρ) (4) Fro (), above ro which oe obtais G () α = w α < ρ w α j j j j j j < ρ w + ρ α < ρε + ρ (r + δ) j j w j > r δ ρ (r + ε + δ ) > ε ( ) Outer copositio ca be used to deterie the solutio o certai ixed-poit equatios e e e Exaple: let G() = 4 8, where We solve the cotiued ractio e 4 equatio G( α ) = α i the ollowig way: Set t ( ξ ) = ; let g () = t t t () ξ Now calculate G () = g g g (), startig with = Oe obtais α = 8788 to te decial places ater te iteratios 9

10 Certai itegral equatios ay be solved uerically usig outer copositios Theore 7 (Gill, ) Give (, t) ϕ ζ aalytic i D = ( ζ < R) or each [,] t ad cotiuous i t, set,,, g ζ = ϕ ζ ϕ ζ ϕ ζ ad G ( ζ ) = g G ( ζ ), G ( ζ ) = g ( ζ ) Suppose ϕ( ζ, t) r < R or ζ D The ζ = ϕ ζ,t dt has a uique solutio, α, i D, with li G ( ζ ) = α Setch o Proo: g ( ζ ) r < R satisies the hypotheses o theore 6, with = α g, ζ g ζ ϕ ζ t dt α, so that li G ( ζ ) = α G ( ζ ) = ϕ G ( ζ ), ϕ G ( ζ ), ϕ G ( ζ ), iplies Ad α = ϕ α,t dt Exaple: iζ t e ζ = dt, α = (477 ) + i(7483 ) Theore 7b (Gill, ) Suppose ϕ ( ζ,t) is aalytic or S ( ζ R) ad t with δ ϕ ζ, t < M or - t < ad g (, ) (, ),( ) M < r < R Set = ( ) ζ = δ ϕ ζ δ + ϕ ζ δ + + ϕ ζ δ δ = <, =, σ a iteger, G ( ζ ) g ( ζ ) ad G σ ζ g G ζ The S ( ζ ) ϕ ( ζ, ) o = = = t dt has a uique ixed poit α = S ( α ) = li G ( ζ )

11 Exaple: S ( ζ ) = dt iplies ζ + t 4e + i dt α = + 4e + i 465 i ( 48 ) ( α + ) t Theore 7c (Gill, ) Give (,, t) or each t [,] ad cotiuous i t Set ϕ ζ aalytic i D ( ω R) (, ),,,,,, g ζ = ϕ ζ ϕ ζ ϕ ζ ad = < or both ad ζ, G ( ζ ) = g ζ, G ( ζ ), G ( ζ ) = g ( ζ, ) Suppose ϕ( ζ,, t) r < R The α( ζ ) = ϕ ζ, α( ζ ),t dt has a uique solutio, α( ζ ), i D, with li G ( ζ ) = α( ζ ) Exaple: ζ t ζα ( ζ ) t e e α( ζ ) = dt = dt 4 with 4 α( 3 i) = 58 (654 ) i, α ( + i) = (43 ) i, α (6 + i) = 64 + (6 ) i

12 Additioal Related Result: Theore 8 (Gill [7], ) Suppose {g,}, with, is a aily o uctios aalytic o a siply-coected doai D ad cotiuous o its closure, with g, (D) Ω, a copact subset o D, or all ad The G () = g g g () α uiorly o the closure o D,,,, i ad oly i the sequece o ixed poits { α,} o {g,} coverge* to α (Coets: Whe li g, () = g (), or each value o, both sequeces coverge to the liit described i theore * For ε> N=N( ε) N< α α < ε ), Siple Exaples o Copositios o Iiite Expasios: Exaple: φ ( ) = a, + a, + a, + deied o D ( ) a, r < iplies the { } = = <, with φ are uiorly cotractive o D Hece F ( ) = φ φ φ( ) λ uiorly o copact subsets o D Exaple: The a t, (, ζ ) ζ CR t, < = ρ < r r ζ r <, a C, < R iplies, = with, + i r( r) R < C T (, ζ ) = T (, t (, ζ )) φ ( ) or < R,,, a, a, We have φ ( ) = + + with φ < ρ < R provided C < r Thus the { φ } are uiorly cotractive o ( <R) It ollows that F ( ) = φ φ φ( ) λ

13 No-Cotractive Fuctios Ier Copositios Aog other results, Shota Kojia proved the ollowig, usig a ivetive ad appropriate otatio: N R () = () + N = r Theore (Kojia [8], ) Cosider etire uctios () = + c with coplex coeiciets Set { } r C sup c,r C iplies = = r=,3,4 r= = The the covergece o the series () = R () exists ad is etire,r Corollary : (Kojia, ) Let {c } be a sequece o positive real ubers such that c < Deie g (x) = R ( x + c x ) = ( x + c x ) ( x + c x ) ( x + c x ) ad r = ( ) F(x) x c x r= = R + The the sequece {g (x)} is uiorly coverget o copact subsets o C, with li g (x) = F(x), a etire uctio Kojia ivestigates the expasios o uctios satisyig the uctioal equatio (s) = s () + s () with coplex s where s > Exaple (Kojia, ): e = R + = 3

14 Coets About Kojia's Theore ad other Related Theory For these iiite ier copositios to coverge, there requetly is required either soe sort o cotractive property, as see i Lorete's Theore ad y theore above, or soe sort o Lipshit coditio or uior covergece: (u) (v) < ρ u v or () () < ρ, or soethig siilar Speed o uior covergece is ote geoetrical Alost all theores o this subject provide suiciet coditios or covergece o iiite copositios Cotiued ractio theory is a case i poit The oly geeral theore that I a aware o that is both suiciet ad ecessary or covergece is theore 6 A geeral strategy is to deal with the "tail ed" o the copositio, applyig the sorts o techiques etioed above, showig the tail ed coverges The all that reais is the siple step o coposig the rot sectio which is a aalytic uctio to the coverget tail sectio Kojia's Theore provides a glipse o the ier structures ost easily developed i this regard The ollowig is a setch o a proo o his result, usig these coditios Start with the observatio that each o the power series i the hypothesis is ajoried by a geoetric series: () = + a + a +, R 3,,3 ( ) () + a + a +,,3 ( ) + C + C + R - C - C R Let us choose ad ix soe large eough to isure the ollowig: C j < j= + 4R 4

15 The, worig with the tail ed, we have R + () < R RC + () () R () < C+ + () - R C+ + C+ ( + + ) + R + + () < < R - R C C Usig the sae approach, loo ow at () () C ( C C ) Ad ially, with R = R, C C (3) { } () ( ζ) - ζ + C ( + ζ ) + C ( + ζ + ζ ) + 3 { } - ζ + (C R ) + 3(C R ) + 4(C R ) + - ζ = P - ζ ( R C ) Thus () ( ζ ) < P ζ, where it is easily coired that P coverges sice C coverges Write P = < M() = F () = () To show {F ()} coverges: prove it is a Set,+ + +,+ = Cauchy sequece 5

16 F,+ () F,+ + p() = (F +,+ ()) (F +,+ + p()) < P + (F +,+ ()) + (F +,+ + p()) < P P + + (F + 3,+ ()) + (F + 3,+ + p()) + < P (F ()) = ,+ + p < M() (F ()) ,+ + p ( ,+ + p + +,+ + p ,+ + p ) < M() (F ()) F () (F ()) 6M()R < ρ, as 3 = + + Thereore, li F () = li F (F ()) = F (li F ()) Covergece is uior o copact,+,+ sets i the coplex plae Geeraliig this theore a tad, I get : Theore (Gill, ) Give a aily o etire uctios, { }, suppose that or <R there exists a N=N(R) where >N iplies (with uior covergece o the sus): (i) () < α (), with α () ad α () < = (ii) () ( ζ ) < ( + β ()) ζ, with β () ad β () < = ( ) The li () = F() exists, with covergece beig uior o copact sets i the coplex plae 6

17 Setch o Proo: First, The, + () < + β+ R + () < + β R < M + = () () p + < + β () = + = p { } < + β Mρ + Mρ + as Additioal results related to Kojia's theore Theore 3 (Gill, ) Suppose () a () a () ρ < ad ρ ( ) The li ( ()) = or < R a () < ρ, = + + with <R, Setch o Proo: Assue a large ixed value o 7

18 j() ρ j + ρ j + ρ j ρ j + j j= = + () R R ρ j = j= R ρ Where The R ρ ρj ρ + ρ + + ρ = j= ρ ad R( ρ ) + + ρ as ρ (R + ) + + ρ j ρ j= Theore 4 (Gill, ) Suppose () = a () + a () + a () + + a (), where < R, a () ad a () < ρ < ρ, ( ) ρ < ad ρ ( ) The li () = λ (), aalytic or < R, with covergece beig uior o copact sets i C δ ρ < < ad ρ R R Setch o Proo: Choose ad ix so large that R The j() < ρ j + < ρ j + < R j ρ δ or large j This leads easily to () R or ixed large ad positive Next, as i Theore 7, + + < 8

19 3 ( ) () ( ζ) ζ ρ + (R ρ ) + 3(R ρ ) + 4(R ρ ) + or suicietly large ρ ζ < ζ ( Rρ ) Applyig the Cauchy Coditio or covergece, () () p < () + () p p R < as - Theore 5 (Gill, ) Cosider etire uctios whose liear coeiciets approach oe: Let () = a + a + + a +, where a,,, ad a ρ with ρ < Set ε = a with ε <, ad α = a, ( ) The li () = F(), etire, with uior covergece o copact sets i the coplex plae Setch o Proo: Begi with <R The ( ) α () α + + ρ + ρ + α α α + ε + ρ ρ ( + ε ) ρ Thus, 9

20 ( + ε ) () () ρ Repeated applicatio o () gives + ( + ε ) R + + () + R ρ + + ρ + ( + ε + ) + + ρ ( + ε ) + Where ( + ε ) < M ad suicietly large ρ < MR MR Hece + () < R MR + MR ρ It is easily see that () () ε + ρ ρ Ad, ially, () ( ζ) ζ α + ρ ( + ζ ) + ρ ( + ζ + ζ ) + ζ ε + ( + ρ R + 3ρ R + ) ζ ε + ( ρr ) Leadig to (3) () ( ζ) + β ζ, where β < As beore, usig (), (), (3) ad settig F, + = + + () + F F + β F,+,+ + p + +, + + p =

21 { , + + p + +, + + p M (F ) F + (F ) F ,+ + p + + 3,+ + p + + () ++p } Thereore, or suicietly large ad ixed : + + p + + p R F, + F, + + p M R ε + ρ as Theore 6 (Gill, ) Suppose ( ) ( η ( ) ) = +, with η aalytic or R ad R r η( ) < ε, ε < Choose < r < R, ad deie R = R( r) = + ε The F () = () F() uiorly or R ad F ( β ) R + < where β = ε r = = Proo: Siple recursio shows that + * ( ε ) ( ε ) ( ) R + R + = R + = = Next, a Lipshit coditio is required: But irst, Cauchy s Itegral Forula provides a ε estiate o the derivatives: η '( ) Write r ( ) ( ζ ) ζ + η ( ) ζη ( ζ ) where η ( ) ζη ( ζ ) = d( tη ( t)) = η ( t) + tη ( t) dt ζ ζ ( ) ε R η t + t η t dt ε + R r ζ = ε ζ = β ζ ζ r r

22 F ( ) F ( ζ ) + β ζ The Also = *, with the product covergig ( ) η ( ) R ε Thereore ( β ) F ( ) F ( ) + ( ) = Set + Z = ( ) The ( ) ( Z ) Z + ( Z ) Z + + ( ) Hece + * * * * ε ε + ε + ε = R + R + + R R as * + ( β ) ε uiorly o as = = F F R + R + < A siple applicatio o the chai rule gives F ( β ) = Exaple & Coets: The ollowig exaple shows how the hypothesis ca be realied Set ( ) = +, C >, C The C( + ) + ( ) C + C +, so that C (( + ) ) C + + ( ) C + + C( + ) C ( + ) ( + ) C( + ) C + +, etc C( + ) C( + ) * Thus ε = ad R = C + C = C

23 Outer Copositios Theore 7 (Gill, ) Cosider uctios coplex coeiciets Set { } r sup c,r r=,3,4 g () = a + c + c + with 3,,3 ρ = ad ε = a with ε < δ ρ <, where δ <, + δ < M, + ε < M I the RMM, G() = li g g g () exists ad is aalytic or <R Covergece is uior o copact subsets o ( <R) Setch o Proo: 3 g () a + + ρ + ρ + ( ε ) + ε + ρ ρ Repeated applicatio gives + + g + g + g () R + δ + ε < R Fix so large that ρ < R 3

24 Next g () ε + ρ ρ The, settig G () = G = g g g () G G g (G ) G + g (G ) G g (G ) G R R ε + ρ + + ε + ρ R R ε + R ρ as R I a ore geeral cotext: Theore 8 (Gill, ) Let S=( <M) Suppose there exists a sequece < ρ < Set σ = C ρ g () C i M S R = <, {g } be a sequece o coplex uctios deied o { ρ} such that ρ < ad = ad R = M σ > The, or every G () = g g g () G(), uiorly o copact subsets o S Coet: There is o requireet these uctios be aalytic Setch o Proo: G () < + C ρ < R + C ρ or < R Repeated applicatios give G () < R + C( ρ + + ρ ) < R + σ = M Thus 4

25 G G g (G ) G + g (G ) G + + g (G ) G < C ρ C ρ as =+ =+ Exaple: g () = + ρ or We have g () + ρ < 3 + ρ, ad repeated applicatios give 5 g g () < 3 + ρ < 4 or ρ < Thus g () 4 6 ρ, ad + = = + = G G g (G ) G + + g (G ) G 4 ρ as Ie, = ad 8 ρ = gives -4 G(4)=488 error< Exaple: t e g( ) = + ρ dt or with t + t e e e g( ) + ρ dt + t ρ + + ρ so that ρ < We have e e e g g g( ) ρ ρ M Thus M e g( ) ρ = Cρ Hece G G g ( G ) G + + g ( G ) G ρ ρ + + C + + as 5

26 Geerated sel-replicatig series Exaple: Cosider a waderig poit i the coplex plae whose positio depeds upo its previous positio For exaple: π π g( ) = +, ρ > ad < R = ρ, Re( ) > The ρ 6 6 G ( ) = + a ( ) + a ( ) + + a( ) with a ( ) = G ( ) ρ Ad Theore 8 applies, givig li G ( ) = G( ) Theore 9 (Gill, ) Suppose g ( ) ( η ( ) ) = +, with η( ) < ε, ε < Choose < r < R, ad deie The G ( ) = g g g( ) G( ) uiorly or R R r ε = = with G ( ) + ad G () ( + ε ) Setch o Proo: It is easily see that ( ε ) G ( ) G ( ) = g ( G ( )) G ( ) Mε G ( ) R + M = G ( ) G ( ) M ε as + = + I the proo o Theore 7 it is show that ( ) G ( + t) G ( t) t, ad = Hece ( + β ), η aalytic or R ad R = R( r) = R r + ε = g g ζ + β ζ G ( ) G () G ( ) g( G ) g ( G ) g( ) = = ( + ε ) G G = 6

27 Exaple: g( ) = +, C >, C The C( + ) + g ( ) C + C +, C (( + ) ) C g+ g( ) C + C + C + +, C C( + ) C ( + ) ( + ) C + G, + ( ) C + M = C Hece G ( ) G( ) Coet: There is soe overlap aog the various theores preseted i the last ew pages Reereces: [] P Herici, Applied & Coputatioal Coplex Aalysis, Vol, 974 [] J Gill, Iiite Copositios o Mobius Trasroatios, Tras Aer Math Soc, Vol 76, 97] [3] J Gill, Copositios o Aalytic Fuctios o the or, J Cop & Appl Math 3(988), 79-84] [4] L Lorete, Copositios o Cotractios, J Cop & Appl Math 3(99) 69-78] [5] J Gill, A Taery Tras o Cot Frac & Other Expasios, Co Aal Th Cot Frac, Vol I [6] J Gill, The Use o the Sequece i Coputig Fixed Poits o, Appl Nuer Math 8 (99) 469-4] [7] J Gill, Geeraliatios o Classical Taery's Theore, wwwjohgillet () [8] S Kojia, Covergece o Iiite Copositios o Etire Fuctios, arxiv:9833v] [9] JD Depree & WJ Thro, O Sequeces o Mobius Trasoratios, Math Zeitschr 8(963) [] A Magus & M Madell, O Covergece o Sequeces o Liear Fractioal Trasoratios, Math Zeitschr 5(97) -7 [] J Gill, Covergece o Copositio Sequeces o Biliear, IJM&MS ()(997) 35-4 [] LR Ford, Autoorphic Fuctios, 7