An effective Caratheodory Theorem

AN EFFECTIVE CARA TH ´ EODOR Y THEOREM TIMOTHY H. MCNICHOLL Abstract. By mean s of t he property o f effe ctiv e lo cal c onnectivit y , the com- putabilit y of fin ding the Carath ´ eodory extension of a conformal map of a Jor- dan domain ont o the unit disk is demonstrate d. 1. Introduction In 1851, B . Riemann prov ed his pow erful Riemann Ma pping Theorem whic h states tha t every simply connected subset of the plane is conformally equiv alent to the unit disk, D . E . Bishop gav e a co nstructive pro of of this theorem in 196 7 (now in [1]), and in 1999 P . Hertling prov ed a unifor mly effective version o f this theorem [9]. In classical complex analysis, the next step up from the Riemann Ma pping The- orem is the Carath´ eo dory Theorem which states that a confor mal map of a simply connected Jor dan domain onto the unit disk can be extended to a homeomor phism of the closure of the do main with the closure of the disk. Such an extension is called the b oundary ext ension or Car ath´ eo dory extension . This result is the ba- sis for demonstrating the existence of so lutions to Diric hle t problems on simply connected Jor dan domains. Here, we will extend Her tling’s work b y proving a uniformly effectiv e version of the Ca rath´ eo do ry Theorem. That is, roughly sp eaking, we show that from sufficiently go o d approximations to a para meterization o f a Jordan curve γ , we can compute arbitrarily go o d a pproximations of a conformal map o f the interior of γ on to the unit disk as w ell as arbitarily go o d a pproximations of its b oundar y extension. This will b e a chiev ed b y means of effe ctive lo c al c onne ctivity . This concept, which first appear ed in [1 1], is the sub ject of a previo us in vestigation by Daniel and McNic holl [5] and also plays an important r ole in [2]. The resulting construction o f the Car ath´ eo dor y extension ca n b e considered as a new pro of of this result. The pap er is organized as follo ws . In Sections 2 and 3, we summarize ba ckground information fro m complex ana lysis and co mputable analysis . In Section 4, we head straight for the pr o of of the main theor em. 2. Back ground from complex anal ysis Most o f our ter minology a nd notation from complex analysis can b e found in [3] and [4]. W e will use C [0 , 1] to denote the set o f contin uous functions from [0 , 1] into C . 2010 Mathematics Subject Classific ation. 03F60, 30C20, 30C30, 30C85. Key wor ds and phr ases. Computab le analysis, constructiv e analysis, complex analysis, confor- mal mapping, effectiv e l ocal connectivit y . 1 2 TIMOTHY H. MCNICHOLL When X ⊆ C , define D ǫ ( X ) = [ p ∈ X D ǫ ( p ) . Let k k ∞ denote the L ∞ norm on C [0 , 1]. Hence, if k f − g k ∞ < ǫ , then ra n( f ) ⊆ D ǫ (ran( g )). An f ∈ C [0 , 1] is p olygonal if there is a partition of [0 , 1], 0 = a 0 < a 1 < . . . < a k − 1 < a k = 1, such that f is linear o n each of [ a j , a j +1 ]. f is r ational p olygonal if each a j is a rational num b er and ea c h f ( a j ) is a r a tional p o int . Suc h a function is completely determined by the tuple ( a 0 , f ( a 0 ) , . . . , a k , f ( a k )). W e say that an arc A 2 extends an arc A 1 if A 2 ⊃ A 1 and A 2 , A 1 hav e an endp oint in common. A p ar ameterization of an a rc A is a homeomo r phism of [0 , 1] with A . A pa- rameteriza tio n of a Jordan c urve J is a ho meo morphism of ∂ D with J . W e will follow the usual conven tion of identifying an arc or Jo rdan curve with any of its parameteriza tions. A domain is a n open, connected subset of the plane. A domain is Jor dan if it is bo unded by Jordan curves. A function u on a doma in D is harmonic if ∂ 2 u ∂ x 2 ( z ) + ∂ 2 u ∂ y 2 ( z ) = 0 for all z ∈ D . Suppo se D is a Jo rdan domain and that f is a bounded piecewise co n tin uo us function on its b ounda ry . The resulting Dirichlet pr oblem is to find a harmonic function u on D with the prop er ty that lim z → ζ u ( z ) = f ( ζ ) for all ζ ∈ ∂ D a t which f is contin uous. Solutions to Dirichlet problems always exist a nd ar e unique. The function f is said to provide the b oundary data for this problem. 3. Back ground from comput able anal ysis and comput ability An informal summar y of the fundamentals o f Type- Two E ffectivit y app ears in [5]. W e identify C with R 2 . Hence, we can also use the na ming systems in [5 ] for C and its h yp e rspaces. W e shall iden tify ob jects with their names where ver this results in simplicity o f exp osition while not creating misc o nceptions. While w e identif y arcs and Jordan curves with their para meterizations, such curves will alwa ys b e named b y names o f their parameter izations. W e also r efer the reader to [5] fo r the definitio ns of lo c al c onn e ctivity witness and uniform lo c al ar cwise c onn e ctivity witness and rela ted theorems. In addition to these theorems, we will need the following. Lemma 3.1. F r om a name of a home omorphi sm f of ∂ D with a J or dan curve J , we c an c ompute a lo c al c onne ctivity witness fo r J . Pr o of. It follows by essentially the same arg umen t a s in the pr o o f o f Theorem 6.2.7 of [1 3] that we can compute, uniformly in the given data, a mo dulus of contin uity for f , m . Compute f − 1 and a mo dulus of contin uity for f − 1 , m 1 . Let h = m 1 ◦ m . AN EFFECTIVE CARA TH ´ EODOR Y THEOREM 3 W e c la im that h is a lo cal connectivity witness for J . F o r, let k ∈ N , and let w 1 ∈ J . Let z 1 be the unique pre image of w 1 under f . Let I b e the in ter v al ( z 1 − 2 − m ( k ) , z 1 + 2 − m ( k ) ). Let C = f [ I ]. W e claim that J ∩ D 2 − h ( k ) ( w 1 ) ⊆ C ⊆ D 2 − k ( w 1 ) . F or, let w 2 ∈ C . Let z 2 be the unique preimag e of w 2 under f . Then, z 2 ∈ I , and so | z 1 − z 2 | < 2 − m ( k ) . Hence, | w 2 − w 1 | < 2 − k . Thus, C ⊆ S 2 − k ( w 1 ). Now, supp ose w 2 ∈ J ∩ D 2 − h ( k ) ( w 1 ). Again, let z 2 be the unique preimage of w 2 under f . Then, | z 1 − z 2 | < 2 − m ( k ) . Hence, z 2 ∈ I . Th us, w 2 ∈ C .  Lemma 3.2. Given n ames of ar cs γ 1 , . . . , γ n such that ∂ D = γ 1 + . . . + γ n , and given names of c ont inu ous r e al-value d functions f 1 , . . . , f n such that γ j = dom ( f j ) , we c an c ompute a name of the harmonic funct ion u on D define d by the b oundary data f ( ζ ) =  f j ( ζ ) ζ ∈ γ j , ζ 6 = γ j (0) , γ j (1) max j max f j otherwise . In addition we c an c ompute the extension of u to D exc ept at the endp oints of t he ar cs γ 1 , . . . , γ n . Pr o of. Let u b e the solution to the res ulting Dirichlet pr oblem on D . F or z ∈ D , we use the Poisson Integral F ormula u ( z ) = 1 2 π Z 2 π 0 u ( e iθ ) 1 − | z | 2 | e iθ − z | 2 dθ z ∈ D . (3.1) (See, for example, Theo r em I.1.3 of [6].) In the cas e under consider ation, we have u ( z ) = X j 1 2 π Z γ j f j ( ζ ) 1 − | z | 2 | ζ − z | 2 | dζ | . Since integration is a computable o pe r ator, this shows we can compute u on D . Since w e a re given f 1 , . . . , f n , it might seem immediate that we can now compute the extension of u to D except at the endp oints of γ 1 , . . . , γ n . How ever, it is not po ssible to determine from a name of a p oint z ∈ D if z ∈ ∂ D . T o see what the difficult y is, and to lead the wa y to wards its solution, we delve a little mor e deeply int o the formalism. Supp ose we are g iven a name of a z ∈ D , p . As we read p , it ma y be that at some p oint we find a subbasic neighbor ho o d R whose clo sure is contained in D . In this ca se, w e can just use equation (3.1). Ho wev er, if w e keep finding subba s ic neig hborho o ds that intersect ∂ D , then at some p o int w e must commit to a n estimate of u ( z ). If we g uess z ∈ ∂ D , then la ter this guess a nd this resulting estimate ma y turn out to b e incorrect. W e face a simila r problem if w e guess z ∈ D . The heart of the matter then is to estimate the v alue of u ( ζ ) when ζ is near z and in D . This can b e done b y effectivizing one of the usual pro o fs that lim ζ → z u ( ζ ) = f ( z ) when z is betw een the endp oints of a γ j . T o b egin, fix rationa l nu m ber s α ∈ [ − π , π ], 2 π > δ > 0, and 0 < ρ < 1 . Let S ( ρ, δ, α ) = d f { re iθ | ρ < r ≤ 1 ∧ | θ − α | < δ / 2 } . W e now wr ite the solution to the Dirichlet problem on the disk in a slightly different wa y that co nsidered previously in this pro of. T o this end, let P r ( θ ) b e the Poisson kernel , Re  1 + r e iθ 1 − r e iθ  . 4 TIMOTHY H. MCNICHOLL It is fa irly well-known that if ζ ∈ ∂ D and z ∈ D , then 1 − | z | 2 | ζ − z | 2 = Re  ζ + z ζ − z  . A t the same time, if z = r e iθ and ζ = e iθ ′ , then Re  ζ + z ζ − z  = P r ( θ − θ ′ ) . Let f b e a function on ∂ D such that for each ar c γ j f ( ζ ) = f j ( ζ ) whenev er ζ is a po in t in γ j bes ides one of its endpo int s. It then fo llows that when 0 < r < 1, u ( re iθ 1 ) = 1 2 π Z π − π f ( e iθ ) P ( θ 1 − θ ) dθ. W e c a n compute a rationa l n umber M suc h that M > max k max ran( f k ) . Suppo se α is such that for s ome j , e iα is in γ j but is not an endpoint. F rom the given data, we can enumerate all such α, j . W e cycle through all suc h α, j a s w e scan the name of z . Fix a ra tional num b er ǫ > 0 . F ro m ǫ a nd the g iven data, one can compute a ra tional δ α,ǫ > 0 such that the arc { e iθ : | θ − α | ≤ δ α,ǫ } is contained in ra n( γ j ) and such that ǫ 3 > max {| f j ( e iθ ) − f j ( e iα ) | : | θ − α | ≤ δ α,ǫ } . W e c la im we ca n then compute ρ α,ǫ such that ǫ 3 M > max { P r ( θ ) : | θ | ≥ 1 2 δ ∧ ρ α,ǫ ≤ r ≤ 1 } . W e p ostp one the computation of ρ α,ǫ so tha t we can reveal our in tent. Namely , we claim that | u ( ζ ) − f ( e iα ) | ≤ ǫ when ζ ∈ S ( ρ α,ǫ , δ α,ǫ , α ). F or , let ζ ∈ S ( ρ α,ǫ , δ α,ǫ , α ), and write ζ as re iθ 1 . Hence, ρ < r ≤ 1, and w e can c ho o se θ 1 so that | θ 1 − α | < δ α,ǫ / 2. If r = 1, then there is nothing mor e to do. So , suppo s e r < 1. F or conv enience, abbrev iate δ α,ǫ by δ . It then follows that | u ( ζ ) − f ( e iα ) | ≤ 1 2 π Z | θ − α |≥ δ | f ( e iθ ) − f ( e iα ) | P ( θ 1 − θ ) dθ + 1 2 π Z | θ − α | <δ | f ( e iθ ) − f ( e iα ) | P ( θ 1 − θ ) dθ. Suppo se | θ − α | ≥ δ . Since | θ 1 − α | < δ / 2, | θ 1 − θ | ≥ δ / 2 and so P r ( θ 1 − θ ) < ǫ/ 3 M . Hence, the first term in the preceding sum is at most 2 ǫ/ 3. At the same time, by our choice of δ α,ǫ , it follows that the second term is no larger tha n 1 2 π Z π − π ǫ 3 P r ( θ 1 − θ ) dθ ≤ ǫ 3 . Hence, | u ( ζ ) − f ( e iα ) | ≤ ǫ . Hence, as we s can the name of z , if w e encounter a rational r ectangle R and ǫ, α such that S ( ρ α,ǫ , δ α,ǫ ) contains R , then we can lis t any subbas ic neighborho o d that contains [ f ( e iα ) − ǫ, f ( e iα ) + ǫ ]. It follows that if we only read neighborho o ds that int ersect the b oundary , then we will write a name of f ( z ) on the o utput tap e. AN EFFECTIVE CARA TH ´ EODOR Y THEOREM 5 W e conclude by s howing how to compute ρ α,ǫ . Let δ ′ = 1 2 δ . The key inequality is P r ( θ ) ≤ P r ( δ ′ ) when δ ′ ≤ | θ | ≤ π . This is justified by P r op osition 2.3 .(c) on page 257 of [3]. Since e iδ ′ 6 = 1, P 1 ( δ ′ ) is defined, and in fac t is 0. W e can compute P r ( δ ′ ) as a function of r on [0 , 1]. W e can thus c o mpute ρ α,ǫ as req uir ed.  4. Proof of the main theo rem Theorem 4.1 ( Eff ectiv e Cara th ´ eo dory Theorem ). F r om a name of a p ar am- eterization f of a Jor dan curve J , and a name of a c onformal map φ of t he interior of J onto D , it is p ossible to c ompute a name of the Car ath´ eo dory ex tension of φ . Pr o of. It follows fr om the main theorem of [7] that we ca n compute, uniformly from the given da ta, a na me of the interior of J . It then follows that from the given data, we can compute a rational point z 0 in the in terior of J . W e can also compute a uniform lo cal a rcwise connectivit y witness for J , h . W e can assume h is incre a sing. Finally , from the given data, we c a n uniformly compute a sequence of r a tional po lygonal J ordan cur ves { P t } t ∈ N such that k P t − J k ∞ < 2 − t . See, for example, L emma 6.1.10 o f [13]. Hence, J ⊆ D 2 − t ( P t ). Let D denote the in ter ior o f J . Let us allow φ to denote the Carath´ eo dor y extension of φ . The outline of our pro of is as follows. W e first compute the r estriction o f φ to the bo undary of D . It then follows that we can compute φ − 1 on ∂ D . It then follows by applying Lemma 3.2 to the rea l and imag inary parts of φ − 1 that w e can compute φ − 1 on the closure of D and hence φ on the closur e of D . T o compute φ o n the b ounda ry of D , we app eal to the Principle of Typ e Co n ver- sion. Namely , we supp ose a re a dditionally given a name o f a ζ 0 ∈ J a nd c ompute φ ( ζ 0 ). The key to this is to co mpute an arc Q from z 0 to ζ 0 such that Q ⊆ D and Q ∩ J = { ζ 0 } . W e do this by computing a sequence of r ational p olygona l arcs { Q t } t ∈ N as follows. T o compute Q 0 , we fir st compute a rational point e 0 6 = z 0 in the in terior of J whose distance from ζ 0 is less than 2 − h (0) . Now, D is a n open connected set. It follows that there is a rational poly g onal arc from z 0 to e 0 which is co nt ained in the interior of J . Suc h an ar c can now b e discovered thro ugh a sea rch pro cedure. W e now descr ibe ho w we compute Q t +1 . This is to be a rational p olyg onal ar c with the following pro pe r ties. ((1)) Q t +1 is contained in the interior of J . ((2)) Q t +1 extends Q t and has z 0 as a common endp oint. ((3)) The o ther endp oint of Q t +1 , e t +1 , is such that | e t +1 − ζ 0 | < 2 − h ( t +1) − 1 . ((4)) The points in Q t +1 that are not in Q t are con tained in the disk of radius 2 − t +1 ab out e t . By way of induction, suppose Q t is a r ational p olygonal ar c from z 0 to a p oint lab elled e t and tha t | e t − ζ 0 | < 2 − h ( t ) − 1 . Supp ose also that Q t is co nt ained in the in terio r o f J . W e fir s t co mpute a ra tional point e t +1 in the in ter ior o f J and an in teg er r t +1 such that | e t +1 − ζ 0 | < 2 − h ( t +1) − 1 , e t +1 ∈ C − D 2 − r t +1 ( P r t +1 ), | e t +1 − e t | < 2 − h ( t ) − 1 , and e t +1 6∈ Q t . W e summarize the key claim at this po in t in the pro of by the fo llowing Lemma. 6 TIMOTHY H. MCNICHOLL Lemma 4.2. e t +1 and e t ar e in t he same c onne ct e d c omp onent of D ∩ D 2 − t +2 − h ( t ) − 1 ( e t ) . Pr o of. F or conv enience, let ǫ = 2 − t , δ = 2 − h ( t ) , and B = D ǫ + δ / 2 ( e t ). 1 By way of contradiction, suppo se e t +1 and e t are no t in the sa me connec ted comp onent of D ∩ B . Let E and E ′ be the distinct co mpo ne nts that co n ta in e t and e t +1 resp ectively . Since B is convex, there is a line segment l having e t and e t +1 as endp oints and such that l ⊂ B . The length o f l is of course less that δ / 2. Let P = J ∩ l ∩ ∂ E , and let P ′ = J ∩ I ∩ ∂ E ′ . Each of P and P ′ is clo sed and therefore compact as a subset of the pla ne. Therefore, there exists p ∈ P and p ′ ∈ P ′ such that | p − p ′ | = d ( P , P ′ ). Hence, d ( P, P ′ ) < δ / 2. F or p oints, a and b o f J , let l ( a, b ) denote the minimum diameter o f a n arc in J having a and b as endp oints. Hence, | a − b | ≤ l ( a, b ). Now, let A denote an arc in J such that the endp oints of A are p and p ′ and so that the diameter o f A is l ( p, p ′ ). Note that l ( p, p ′ ) < ǫ . A straightf orward connectivity a r gument shows that A intersects ∂ B . (If not, then there is a piecewis e linear arc with one e ndpoint in A and the other in E ′ - a co n tradiction.) A ∩ ∂ B is a compact subset o f A . Hence, there is a po in t q ∈ A ∩ B such that the diameter of the suba r c of A with endp oints p and q is minimal. Denote this diameter by l A ( p, q ). Then, we have ǫ + δ / 2 = | e t − q | ≤ | e t − p | ≤ | e t − p | + l A ( p, q ) < | e t − p | + l ( p, p ′ ) < δ / 2 + ǫ Whic h is a contradiction, and the Lemma is proved.  It then follows that there is a rational polygo nal a rc P from e t to e t +1 that lies in the in terior o f J , do e s not cross Q t +1 , and that do es not go further than 2 − t + 2 − h ( t ) − 1 ≤ 2 − t +1 from e t . I t now follows that a r a tional p olygona l arc with pr op erties (1) - (4) exists. Such a curv e can be discov ered thro ugh a simple search pro cedure which, in o rder to ensure (1), a lso c omputes s such that Q t +1 ⊆ C − D 2 − s ( P s ). Let Q = S t ∈ N Q t . It follows that one endpoint of Q is z 0 and the o ther is ζ 0 . By (1) and (2), Q ∩ J = { ζ 0 } . W e ha ve until now tho ught of eac h Q t as a set. But, we wish to think of Q as a function. F ur thermore, w e wish to compute a na me of a parameterization of Q from the giv en data. T o this end, w e no w backtrack a little and des crib e in more detail how we pa r ameterize each Q t . Let v t, 1 , . . . , v t,n ( t ) denote the v er tice s of Q t in the o r der in which they are traversed so that z 0 = v t, 1 and e t = v t,n ( t ) . Set a t,j = j − 1 n ( t ) j = 1 , . . . , n ( t ) . 1 I wi sh to express here my gr atitude to m y colleague Dr. Dale Daniel for all o w i ng me to use this pr oof, which is entirely of his own creation, in this paper. AN EFFECTIVE CARA TH ´ EODOR Y THEOREM 7 Define Q t to b e the function o n [0 , 1] tha t linearly maps ea ch in ter v al [ a t,j , a t,j +1 ] onto the line se g men t from v t,j to v t,j +1 , and that maps all of [ a t,n ( t ) , 1] to the p oint v t,n ( t ) . It follows that lim t →∞ Q t ( x ) exists for each x ∈ [0 , 1], and we define Q ( x ) to be the v alue o f this limit. Note that k Q t − Q t +1 k ∞ ≤ 2 − t + 2 − h ( t ) ≤ 2 − t +1 . It now follows that whenever t ′ ≥ t , k Q t − Q t ′ k ∞ ≤ ∞ X j = t 2 − j +1 = 2 − t . So, by Theorem 6.2.2 .2 of [13], Q is computable uniformly from the given data. Let A b e the circ le with center 0 and radius 1 / 2. Let A ′ = φ − 1 [ A ]. Since A ⊆ D , we can compute names of A ′ and its complemen t. W e can then compute R > 0 such that the c ir cle of ra dius R c e ntered at ζ 0 do es not int ersect A ′ . W e now describ e how we co mpute a name of φ ( ζ 0 ). Fix r > 0 such tha t r < R , 1. Compute a p oint w 1 on Q b esides ζ 0 such that every p oint on Q betw een w 1 and ζ 0 inclusive lies in D r ( ζ 0 ). In fact, we ca n take w 1 to b e a vertex of Q . W e can then compute m such that m 2 > (2 π ) 2 log( R ) − log( r ) . Note that the right side of this inequality approaches 0 as r approaches 0 from the right. Let T 2 be the line x = Re ( φ ( w 1 )). Compute α s uch that φ ( w 1 ) = | φ ( w 1 ) | e iα . W e will pro ceed under the assumption that T 2 hits A . F or, we can apply the following construction and arg umen t to ψ = e − iα φ . Result will b e an upper b ound on | ψ ( w 1 ) − ψ ( ζ 0 ) | = | φ ( w 1 ) − φ ( ζ 0 ) | . Le t T 1 be the line x = R e ( φ ( w 1 )) + m , and let T 3 be the line x = Re ( φ ( w 1 )) − m . W e can thus assume m is sma ll enough so that the lines T 1 , T 3 bo th in ters e ct A . Let p 1 be a p oint where one of these lines int ersects ∂ D and p 1 φ ( w 1 ) do es not hit A . Let M = | p 1 − φ ( w 1 ) | . W e c laim that | φ ( w 1 ) − φ ( ζ 0 ) | < 2 M . F o r, suppose otherwise. Then, when φ is applied to w 1 ζ 0 , the r esulting arc hits T 2 and one of T 1 , T 3 . Without loss of generality , suppose it hits T 1 . Let S k = φ − 1 [ T k ] for k = 1 , 2. Hence, S 1 , S 2 hit w 1 ζ 0 . Hence, every circ le centered at ζ 0 and whose ra dius is be tw een r and R inclusive hits S 1 and S 2 . This puts us in position to use the “Leng th-Ar ea T r ic k” as follows. Let C r ′ denote the circle with radius r ′ centered at ζ 0 . Fix r ≤ r ′ ≤ R . The curves S 1 , S 2 hav e pos itiv e minim um distance from each o ther. It fo llows that there a re p oints z 1 ,r ′ and z 2 ,r ′ on C r ′ that b elong to S 1 , S 2 resp ectively and such that no p oints on these curves app ear on C r ′ betw een z 1 ,r ′ and z 2 ,r ′ . (W e do not cla im tha t we can compute such po in ts; this is not nec essary to pr ov e that our estimate is corr e c t.) Let K r ′ denote the ar c on C r ′ from z 1 ,r ′ to z 2 ,r ′ . Hence, for s ome θ 1 ,r ′ , θ 2 ,r ′ | φ ( z 1 ,r ′ ) − φ ( z 2 ,r ′ ) | =      Z K r ′ φ ′ ( z ) dz      ≤ Z θ 2 ,r ′ θ 1 ,r ′ | φ ′ ( z ) | r ′ dθ 8 TIMOTHY H. MCNICHOLL On the other hand, φ ( z 1 ,r ′ ) is o n T 1 and φ ( z 2 ,r ′ ) is o n T 2 . Hence, m ≤ Z θ 2 ,r ′ θ 1 ,r ′ | φ ′ ( z ) | r ′ dθ. A t the s ame time, by the Sch warz Integral Ineq ua lit y (see e.g. Theorem 3.5, page 63 of [1 2]), m 2 ≤ Z θ 2 ,r ′ θ 1 ,r ′ | φ ′ ( z ) | 2 dθ Z θ 2 ,r ′ θ 1 ,r ′ ( r ′ ) 2 dθ. Whence m 2 r ′ ≤ r ′ Z θ 2 ,r ′ θ 1 ,r ′ | φ ′ ( z ) | 2 dθ Z θ 2 ,r ′ θ 1 ,r ′ dθ ≤ 2 π r ′ Z θ 2 ,r ′ θ 1 ,r ′ | φ ′ ( z ) | 2 dθ. If we now integrate bo th sides of this inequality with resp ect to r ′ from r to R , we obtain m 2 [log( R ) − log( r )] ≤ 2 π Z R r Z θ 2 ,r ′ θ 1 ,r ′ | φ ′ ( z ) | 2 r ′ dθdr ′ . It follo ws from the Lusin Area Integral (see e.g. L e mma 13.1.2, page 386, of [8]) that this do uble integral is no larger tha n 2 π . Hence, m 2 ≤ (2 π ) 2 log( R ) − log( r ) . This is a contradiction. So, | φ ( ζ 0 ) − φ ( w 1 ) | < 2 M . As r appro aches 0 from the right , φ ( w 1 ) appro aches φ ( ζ 0 ) and so m, M can b e chosen so as to approa ch 0. It follows that we can now genera te a name of φ ( ζ 0 ). W e hav e now computed φ on ∂ D . Since φ is injectiv e, for each z 0 ∈ ∂ D , φ − z 0 has a unique 0 o n ∂ D . It follows from Cor o llary 6.3.5 of [1 3] that we can co mpute φ − 1 on ∂ D . It no w follo ws as no ted in the introductio n to this pro of that w e can compute φ − 1 on D . F or each z 0 ∈ D , φ − 1 − z 0 has a unique zero . Hence, we can compute φ o n D .  In [9], P . Hertling show ed that to compute a conformal map φ o f a prop er, s imply connected domain D onto D , it is necessa ry and sufficient to hav e a name o f D a nd a name of the b oundar y of D as a closed set. This rais es the questio n as to whether a na me of the bounda r y of D as a closed set is, when D is Jor dan, sufficient to compute the Carath´ eo do ry extension of D . In fact, this is not the case. F or, from a Carath´ eo dory extension of φ , it is p oss ible to unifor mly co mpute a para meterization of the b oundar y o f D . But, it follows from E xample 5.1 of [10] that there is a Jor dan curve that is c omputable as a compact set but ha s no computable parameter ization. Ackno wledgements The author thanks his co lleagues Dr. V a le n tin Andreev and Dr. Dale Daniel for helpful conv ersations and his wife Susa n for supp ort. AN EFFECTIVE CARA TH ´ EODOR Y THEOREM 9 References [1] Bishop, E., Bridges, D.: Constructiv e analysis, Grund lehr en der Mathematischen Wis- senschaften [F undamental Principles of Mathematic al Scienc es] , v ol. 279. Springer-V erlag, Berlin (1985) [2] Brattk a, V.: Plottable real num b er functions and the computable graph theorem. SIAM J. Comput. 38 (1), 303–328 (2008) [3] Con w a y , J.: F unctions of One Complex V ari able I, Gr aduate T exts in Mathematics , vol. 11, 2nd edn. Springer-V erlag (1978) [4] Con w a y , J.: F unctions of One Complex V ariable I I, Gr aduate T exts in Mathematics , vol. 159. 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