Beyond simple iteration of a single function, or even a finite sequence of functions, results

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A Primer o the Elemetary Theory of Ifiite Compositios of Complex Fuctios Joh Gill Sprig 07 Abstract: Elemetary meas ot requirig the complex fuctios be holomorphic Theorem proofs are fairly simple ad

1 A Primer o the Elemetary Theory of Ifiite Compositios of Complex Fuctios Joh Gill Sprig 07 Abstract: Elemetary meas ot requirig the complex fuctios be holomorphic Theorem proofs are fairly simple ad are outlied This is a brief compilatio of such basic results with examples prefaced by three fudametal theorems about holomorphic fuctios Beyod simple iteratio of a sigle fuctio or eve a fiite sequece of fuctios results appearig here refer to the followig: Ier compositios: F ( ) = R f ( ) = f f f ( ) = F( ) = lim f ( ) or R F ( ) = R f ( ) = f f f ( ) F( ) = lim R f ( ) Outer compositios: G ( ) = L g ( ) = g g g ( ) G( ) = lim L g( ) or = G ( ) = L g ( ) = g g g ( ) G( ) = lim L g ( ) = = As bacgroud we begi with three fudametal theorems for aalytic fuctios By requirig aalyticity Lipshit cotractio a frequet coditio for iteratio covergece ca be replaced with simple domai cotractio Schwar s lemma is ey to these results

2 Theorem (Herici [][974]) Let f be aalytic i a simply-coected regio S ad cotiuou o the closure S of S Suppose f( S ) is a bouded set cotaied i S The F( ) = f f f( ) = R f( ) α the attractive fixed poit of f i S 3 f( ) = + S = < 4 The F( ) α = Example: ( ) For Ier compositios there is: Theorem : (Lorete[][990]) Let { f } be a sequece of fuctios aalytic o a simplycoected domai S ad cotiuous o the closure of S Suppose there exists a compact set Ω S such that for each f( S) Ω The F( ) = f f f( ) = R f( ) α a costat uiformly o S aζ a ζ Example: The simple cotiued fractio is geerated by + + aζ f( ) = F( ) = f f f( ) If ζ < < R < the + a < ρr( R) f ( ) < ρr ρ < For example R = implies F ( ) ( ) α ζ for ζ < For Outer compositios there is: Theorem 3 :(Gill[3][99]) Let { g } be a sequece of fuctios aalytic o a simplycoected domai D ad cotiuous o the closure of D Suppose there exists a compact set Ω D such that g( D) Ω for all Defie G( ) = g g g( ) The G () α uiformly o the closure of D if ad oly if the sequece of fixed poits { α } of the { g } i Ω coverge to the umber α

3 e e e Example: let G() = 4 8 where We solve the cotiued fractio equatio G( α ) = α i the followig way: e 4 Set t ( ξ ) = ; let g () = t t t (0) Now calculate 3+ + ξ G () = g g g () startig with = Oe obtais α = to te decimal places after te iteratios Additioal theorems o etire fuctios appear i the addedum The results that follow do ot require aalyticity Some of these appear i [4] Theorem 4: Let { t } be a family of complex fuctios defied o S a simply-coected compact set i C havig the properties t( S) S ad t( ) t( ) < ρ ρ < for all uiformly o S Set G( ) = t t t( ) ad F( ) = t t t( ) The F( ) β S uiformly o S If t( α) = α the uique fixed poits of t G ( ) α uiformly o S if ad oly if α α = ε 0 Setch of proof : F + m( ) F( ) < ρ t+ t+ m( ) ρ DiamS ( ) 0 implies F ( ) 0 β Also F( ) F( 0) 0 uiformly o S Next Set σ = ε + ε + The ( ) α ρ α ε ρ σ G ( ) < To show G ( ) α uiformly o S implies α α α = assume there exists { } such that α α > r > 0 Now suppose is sufficietly large that G( ) α < ε for ρ ε < r For > + + ρ G ( ) α < ρg ( ) α + ρ α α < ρε + ρ α α ad G ( ) α > α α G ( ) α Therefore G ( ) α ( ρ) r ρε ε # > > ( )

4 4( + ) 8 Example : t( ) = x + i y S = { : x < y < } The F( ) i = 0 ad Thus ρ = G( ) α = i slowly 6 7 t( ) < 8 Theorem 5: Let S be a bouded simply-coected domai ad t ( S) S t ( α ) = α S α α S Furthermore assume t ( ) α < ρ α ρ < Set T( ) = t t t( ) The T( ) α uiformly o S Setch of proof: T α α α ρ α ( ρ ε ) ( ) ε = α α + # Example : t( ) = i + i T( ) i + + Theorem 6: Suppose g ( ) = + ρ ϕ( ) where there exist R > 0 ad M > 0 such that < R ϕ( ) < M Furthermore suppose ρ 0 ρ < ad R > M ρ The there exists 0 < R* < R such that G( ) g g g( ) G( ) for { : < R*} Setch of proof: Assume mometarily that R * exists The g( ) + ρ M < R* + ρ M g g( ) + M( ρ + ρ ) < R* + M( ρ + ρ ) m + m g g g( ) + M ρ < R* + M ρ R* + M ρ + m + m with R < R M ρ Next * =

5 G ( ) G( ) = g ( G ) G < ρ G ( ) G ( ) g ( G ) G + g ( G ) G < ( ρ + ρ ) M G ( ) G ( ) < M ρ 0 as + m M There is o requiremet that ϕ be aalytic or eve cotiuous merely that it cotract as described # = Example : g ( ) ( xcos( y) iysi( x) ) Therefore * ρ = < R = ϕ( ) < M = R = Hece G ( + 4) i = i = 0 Theorem 7: Suppose f ( ) = + ρ ϕ( ) where there exist R > 0 ad M > 0 such that < R ad ζ <R ϕ( ) < M ad ϕ( ) ϕ( ζ) < r ζ Furthermore suppose ρ 0 ρ < ad R M ρ > The there exists 0 * < R < R such that F( ) f f f ( ) F( ) for { : < R*} Setch of proof: (similar to the previous proof) f ( ) < Mρ f ( ) < R* + Mρ < R + m + m + m + m f f ( ) f ( f ( )) f ( ) + f ( ) < Mρ + Mρ + m + m + m + m + m + m + m + m f f f ( ) < M + + m f f ( ) < R* + M( ρ + ρ ) < R + m + m + m + m + m ρ

6 Next for R < R M ρ * = + m ( ) ( ) F ( ) F( ) < + rρ f f f ( ) < M + rρ ρ = M S ( m) + m + m + m + With S ( m) 0 as # f( ) = + Cos( x) + isi( y) < R = r = 3 Example : ( ) ϕ( ) < R = F( ) i = 0 * Theorem 8: Suppose t ( ) ( ρ ϕ ( ) ) () ( ) = + The set 0 G( ) = + ρ ϕ ( G ( )) G ( ) = G ( ) = t t t ( ) ad () ( ) F( ) = + ρ ϕ ( F + ( )) F + ( ) = F + ( ) = t + t + t ( ) with = < R ϕ ( ) < M ρ < The there exists 0 < R* < R such that S = { : < R*} G ( ) ( ) G If i additio ϕ( ) ϕ( ) < r the F( ) F( ) Both uiformly o S R t * * < R + M < R R < + ρ M Setch of proof: ( ) ( ρ ) * * ( ) ( ) ( ρ ) G F < R + M < R R < G ( ) G ( ) < MR ρ 0 Ad + m = m ( ρ ( )) + m = + ( + ρm ) R Thus F ( ) F( ) < + M + rr MR ρ 0 # < R = ϕ( ) < M = Example : g ( ) = + ( xcos( y) + iysi( x) ) * + M < 4 R < 4 ; G ( + 3) i i 00

7 Example : Expadig fuctios as ifiite compositios [6] frequetly ivolves a special case of theorem 8 For istace F( ) = e ca be expaded by writig F( ) = F( )( F( ) + ) This leads to e = + R + + = + R + + = Example : Example : Example : Example : 4 ( + ) = L + + L 4 ( ) Ta( ) = R + 4 = 4 ArcTa ( ) = L + 4 ( ) = ± + ( ) R Si Theorem 9: Let ρ( ) t( ) = ρ ( ) + Set P( ) ρ( ) = For G( ) = t t t( ) set S( ) = ρj( ) = j= S ( ) + P( ) The G( ) = S ( ) P( ) ( ) ( ) + + ( ) For F( ) = t t t( ) set T( ) = ρj( ) j= T( ) + P( ) The F( ) = T( ) P( ) ( ) ( ) + + ( )

8 Commet: Whe the ρ ( ) = ρ are costats ad = 0 this formula reduces to Euler s equivalet (reverse ad forward) cotiued fractio The proof is purely computatioal As a reverse cotiued fractio the expasio is self-geeratig Example : ρ ( ) ρ ρ < ρ LimF( ) = LimG( ) = = Example : ρ ( ) ρ ρ < ρ LimG( ) = = Lim ρj LimF( ) = = = j = Example : ( ) = ( ) Cos( x) + i( ) Si( y) = 3 4i ρ + + G ( ) = i F ( ) = i Theorem 0: Let { f } be a family of complex fuctios defied o S a simply-coected compact set i C havig the properties f( S) S ad f( ) f( ) < ρ ρ < for all uiformly o S Set F( ) f f f( ) = Cosider ow a sequece { } that Limf ( ) = f( ) uiformly o S Write F ( ) = f f f ( ) The LimF ( ) = LimF( ) = λ for S f such Setch of Proof: The covergece of F( ) = f f f( ) is easily determied ( ) F( ) F ( ) = f f ( ) f f f f ( ) ρ f f ( ) 0 + p + + p + + p F( ) F( ) ρ 0 Set Z = f f f ( ) The Ad p p+ p+ F ( ) λ = F ( Z ) λ F ( Z ) F( Z ) + F ( Z ) λ p p p p p p p p ε For the secod term i the iequality choose ad fix p sufficietly large that Fp( ) λ < for ε all S For the first term choose sufficietly large to isure Fp ( ) Fp( ) < This is true sice a fiite compositio of fuctios of the type described above covergig uiformly o S will also coverge uiformly o S #

9 ρaζ ρaζ ρa3ζ ρaζ Example : F( ) = ρaζ ρ = f ( ) = + aζ f ( ) f( ) = Suppose + < R < a < R ζ < R ad The f ( ) < R f ( ) f ( ) < r r < Also f ( ) f( ) < R 0 ad Therefore LimF ( ) = LimF( ) = λ( ζ) Theorem : Let { } g be a family of complex fuctios defied o S a simplycoected compact set i C havig the properties g ( S) S ad g ( ) g ( ) < ρ ρ < Write G ( ) = g g g ( ) ad let { } α be the uique fixed poits α = g ( α ) Suppose α α ad α α+ < ε 0 The for S G( ) = G ( ) α Proof: ( ) + G α ρ α + ρ α α + α α ρε ρ α + + α α ρ 0 as # Observe that it is ot ecessary that α α : Example : g ( ) = + The R R g ( ) R Ad α α+ = = ε 0 ad α = 0 fixed α = fixed whereas α Computer evaluatio gives G 0 ( + 3) i i ad G 00 ( + 3) i 99

10 Theorem : Let { } f be a family of complex fuctios defied o S a simplycoected compact set i C havig the properties f ( S) S ad f ( ) f ( ) < ρ ρ < Write F ( ) = f f f ( ) ad let { } α be the uique fixed poits α = f ( α ) Suppose α α α+ < ε 0 The for S F ( ) α α ad Proof: ( ) + F α ρ α + ρ α α + α α ρε ρ α + + α α ρ 0 as # Example : f ( ) = + R R f ( ) R α α+ = = ε 0 α = fixed whereas α = 0 fixed ad α α = Theorem 3: Suppose g ( ) = + ϕ( ) ϕ C( S I) I = [0] S g S Set G0 ( ) = G ( ) = g g g ( ) G( ) = G ( ) The = = 0 G ( ) = ϕ( G ( ) ) = ψ ( ) ψ ( t) dt = λ( ) d Commet: This is simply a discrete aalogue of = ϕ( t) 0 t havig a exact dt solutio Z( t ) by ispectio or uder certai coditios described i the Picard Lidelöf theorem The λ( 0): = ψ( 0 t) dt = ϕ ( Z( t) t ) dt = Z() 0 0 0

11 Example : Let g ( ) ( ) = + 0 The d = ϕ( t ) = t Z( t) = ( ) = dt t 0 0 λ Addedum & Refereces: Theorem (Kojima [5][00]) Cosider etire fuctios coefficiets Set { } r C sup cr = r= 34 f () = R f () exists ad is etire f () c r = + r with complex r= = The the covergece of the series C implies = Theorem (Gill [4][0]) Cosider etire fuctios whose liear coefficiets approach oe: Let f () = a + a + + a + where a ad a ρ with ρ < Set ε = a with ε < ad α = a ( ) The lim f f f () = F() etire with uiform covergece o compact sets i the complex plae Theorem (Gill [4][00]) Cosider fuctios coefficiets Set { } r sup cr r= 34 g () = a + c + c + with complex 3 3 ρ = ad ε = a with ε < If δ ρ < where δ < + δ < M + ε < M ( ) ( ) RMM The G() lim ( g g g ()) = exists ad is aalytic for <R Covergece is uiform o compact subsets of ( <R)

12 Image o page oe: From Theorem 9 ρ( ) t( ) = ρ ( ) + ( ) xcos( y ) iysi( x ) G( ) G( ) [ 55] 50 ρ + = [] P Herici Applied & Computatioal Complex Aalysis Vol 974 [] L Lorete Compositios of Cotractios J Comp & Appl Math 3(990) 69-78] [3] J Gill The Use of the Sequece i Computig Fixed Poits of Appl Numer Math 8 (99) 469-4] [4] J Gill Covergece of Ifiite Compositios of Complex Fuctios Comm AalTh CF Vol XIX(0) Researchgateet [5] S Kojima Covergece of Ifiite Compositios of Etire Fuctios arxiv:009833v [6] J Gill Expadig Fuctios as Ifiite Compositios Comm AalTh CF Vol XX(04) Researchgateet

Informal Notes: Zeno Contours, Parametric Forms, & Integrals. John Gill March August S for a convex set S in the complex plane.

Informal Notes: Zeno Contours, Parametric Forms, & Integrals. John Gill March August S for a convex set S in the complex plane. Iformal Notes: Zeo Cotours Parametric Forms & Itegrals Joh Gill March August 3 Abstract: Elemetary classroom otes o Zeo cotours streamlies pathlies ad itegrals Defiitio: Zeo cotour[] Let gk ( z = z + ηk