Tsuji, M.: Potential Theory in Modern Function Theory. Maruzen, Tokyo 1959.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....t ) HD(J t ) HD(Y ) The proof is complete. We have already seen that hyperbolic rational functions with connected Fatou set provide good examples of conformal expanding repellers. Another natural class of examples is given by the limit sets of Kleinian groups of Schottky type. Here (see [Be] [Ts]) a nite set of generators fg 1 ; g k g can be found along with nitely many mutually disjoint geometric disks D 1 ; D k covering the limit set and such that min i k finffjg 0 i (z)j : z 2 B i gg 1: Hence, as an immediate consequence of Theorem 6.12 we get the following. ....

M. Tsuji, Potential theory in modern function theory, Maruzen Co., Tokyo 1959.

....in this area progress has been made in the last 15 years (see [N1] Po] S1] S2] and [GR] but also because this notion is essential in the definition of spurious poles at the end of the present section. In the sequel by cap( we denote the (logarithmic) capacity (for a definition see [La] [T], or [ST] Appendix I] Definition 2. A sequence of functions f n , n = 1, 2, is said to converge in capacity to f in a domain D # C if for every # 0 and every compact set V # D # we have lim n## cap z # V ( f n f ) z) # = 0. 2.1) The big advantage of ....

M. Tsuji (1959): Potential Theory in Modern Function Theory. Tokyo: Maruzen.

....d h ae (A) t 1 (d)m ae (A) 4) and m ae (A) t 2 (d; l; ae)h ae (A) 5) By considering for example a ball of radius p l, one sees that the dependence of t 2 on l cannot be removed. A possible choice is t 2 = 8 d l d Gammaae : 6) Analogously to Theorem 1 in Carleson [3] p. 7, see also [9], Chapter III.4) we have the following Lemma: Lemma 1 There are constants t 3 and t 4 , depending only on d, such that for every bounded set A ae Z d there is a discrete measure supported on A with (B) t 3 jBj ae=d for all balls B ae Z d (7) and (A) t 4 h ae (A) 8) Proof. Start the ....

M. Tsuji, Potential Theory in Modern Function Theory. Maruzen, Tokyo, 1959. 16

....in D # , and D 0 # C a Jordan domain. Assume that f maps #D # C to #D 0 , Then D equipped with the pullback of the spherical metric is regularly exhaustible; one can take an exhaustion by D # z : z # r . It follows from Ahlfors theory that f can omit at most two values from D 0 [52], Theorem VI.9. This is how the result about countability of the set of direct singularities for a parabolic surface (3.3) can be proved. The situation when D = D 0 (and they are not necessarily Jordan) occurs frequently in holomorphic dynamics, namely in the iteration theory of meromorphic ....

....see, for example [13] where D is a periodic component of the set of normality of a meromorphic function. In this case it follows from Ahlfors theory that the Euler characteristic of D is non positive [16] Here is another variation on the same topic, due to Noshiro and Kunugui, see for example [52]: let f be a meromorphic function in the unit disk U such that lim z## f(z) exists and belongs to the unit circle #U for all # # #U E, where E is a closed set of zero logarithmic capacity, f(0) 1. Then f(z) a has solutions z # U for all a # U with at most two exceptions. If f is ....

M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1953.

....k=1 kp k kE M n E kpkE (2.1) for arbitrary algebraic polynomials fp k (z)g m k=1 with complex coecients, where p(z) Q m k=1 p k (z) and n = deg p, as before. In order to give a solution of the above problem, we have to introduce certain notions from the logarithmic potential theory (cf. [24]) Let cap(E) be the logarithmic capacity of a compact set E C . For E with cap(E) 0, denote the equilibrium measure of E (in the sense of the logarithmic potential theory) by E . We remark that E is a positive unit Borel measure supported on E, supp E E (see [24, p. 55] De ne dE (z) ....

....Ch. 4] One can see that the regularity of is essential in the above theorem, by considering the following example. Let E = D [ fag, where a 3 is real, so that a is the isolated irregular point. Then cap(D [ fag) cap(D) 1 and D[fag = D = 1 2 d (see Theorems III.31 and III.37 in [24]) where d is the arclength on D. This gives M D[fag = exp 1 2 Z 2 0 log d D[fag (e i )d = exp 1 2 Z 2 0 log je i ajd = a: 6 IGOR E. PRITSKER For the polynomials Qn (z) z n 1, n 2 N, we have that n ( Qn ) 1 2 d = D = D[fag ; as n ....

M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publ. Co., New York, 1975.

....Remark 1.4. In the case W (z) # 1 of Theorem 1.1, the result that the approximation property (1.3) holds is a known classical result in complex approximation theory (cf. 14, p. 26] This also follows from Theorem 1. 1 because the measure (G, 1) exists by Theorems III.12 and III.14 of Tsuji [13], and is the classical equilibrium distribution measure (in the sense of logarithmic potential theory) for G. The topic of weighted approximation by W n (z)Pn (z) # n=0 , on the real line, has been extensively and thoroughly treated in the recent books of Sa# and Totik [10] and Totik [12] ....

....with respect to# ; here, # 1 and # 2 are, respectively, the balayages of # 1 and # 2 from C G to G. Furthermore, if of (2.4) is a positive measure, then (cf. Theorem 1.1) G, W ) and supp (G, W ) # #G. 2. 5) We point out that the harmonic measure #(#, cf. Nevanlinna [8] and Tsuji [13]) is the same as the equilibrium distribution measure for G, in the sense of classical logarithmic potential theory [13] For the notion of balayage of a measure, we refer the reader to Chapter IV of Landkof [5] or Section II.4 of Sa# and Totik [10] In the following series of subsections, we ....

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M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo, 1959. MR 22:5712; MR 54:2990

....3.2) Remark 1.5 In the case W (z) j 1 of Theorem 1.1, the result, that the approximation property (1.3) holds, is a known classical result in complex approximation theory (cf. 13, p. 26] This also follows from Theorem 1. 1 because the measure (G; 1) exists by Theorems III.12 and III.14 of Tsuji [12], and is the classical equilibrium distribution measure (in the sense of logarithmic potential theory) for G. The topic of weighted approximation by fW n (z)P n (z)g 1 n=0 , on the real line, has been extensively and thoroughly treated in the recent books of Saff and Totik [9] and Totik [11] ....

....; here, 1 and 2 are, respectively, the balayages of 1 and 2 from I CnG to G. Furthermore, if of (2.4) is a positive measure, then (cf. Theorem 1.1) G; W ) and supp (G; W ) ae G: 2. 5) We point out that the harmonic measure (1; Delta; Omega Gamma (cf. Nevanlinna [7] and Tsuji [12]) is the same as the equilibrium distribution measure for G; in the sense of classical logarithmic potential theory [12] For the notion of balayage of a measure, we refer the reader to Chapter IV of Landkof [5] or Section II.4 of Saff and Totik [9] In the following series of subsections, we ....

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M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.

....with respect to Omega , and and Gamma are, respectively, the balayages of and Gamma from I CnG to G. Furthermore, if of (2.4) satisfies (2.5) then (see Theorem 1.1) G; W; fl) 2. 6) We point out that the harmonic measure (1; Delta; defined in Nevanlinna [8] or Tsuji [18]) is the same as the equilibrium distribution measure for G; in the sense of classical logarithmic potential theory [18] For the notion of balayage of a measure, we refer the reader to Chapter IV of Landkof [6] or Section II.4 of Saff and Totik [13] In the following series of subsections, we ....

....Furthermore, if of (2.4) satisfies (2.5) then (see Theorem 1.1) G; W; fl) 2. 6) We point out that the harmonic measure (1; Delta; defined in Nevanlinna [8] or Tsuji [18] is the same as the equilibrium distribution measure for G; in the sense of classical logarithmic potential theory [18]. For the notion of balayage of a measure, we refer the reader to Chapter IV of Landkof [6] or Section II.4 of Saff and Totik [13] In the following series of subsections, we consider various classical weight functions and we find their corresponding signed measures, associated with the weighted ....

M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.

....radius of G l with respect to a l . Similarly, there exists a conformal mapping 8 : 0#D of the unbounded component 0 onto the exterior of the unit circle D = z: z 1] normalized by 8(#) # and lim z ## 8(z)#z=1#C, where C : cap K is the logarithmic capacity (transfinite diameter) of K (cf. [19]) We shall keep the same notation , z) for the extension of the conformal mapping , l : G l #D onto the boundary #G l in the sense of Carathe# odory s theory of prime ends [6] Thus, for each l=1, 2, n, the mapping , l 60 PAPAMICHAEL, PRITSKER, AND SAFF File: 640J 302704 By:DS ....

....measures e (B) #,B,0) 2.1) and l (B) a l ,B,G l ) l=1, n, 2. 2) for any Borel set B C, where (#, B, 0) is the harmonic measure of the set B at the point # with respect to 0, and (a l , B, G l ) is the harmonic measure of B at the point a l with respect to the domain G l (cf. [13, 19]) We remark that e is the same as the equilibrium measure for K in the sense of logarithmic potential theory. Another convenient way to describe the above harmonic measures is to interpret them as preimages of the normalized arclength measure on the unit circle [z: z =1] under the corresponding ....

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M. Tsuji, Potential Theory in Modern Function Theory," Maruzen, Tokyo, 1959.

....CHEBYSHEV POLYNOMIALS WITH INTEGER COEFFICIENTS 9 for quasi every z 2 [a; b] Omega by (3.11) 3.9) 3.16) and the basic properties of Green functions (see [18, p. 14] Using the uniqueness theorem for the solution of the Dirichlet problem in Omega (cf. Theorem III.28 and its Corollary in [18]) we conclude that Fw Gamma U w (z) j h(z) z 2 C: Thus (3.12) follows from (3.14) Proof of Theorem 2.1. Suppose that fq n g 1 n=0 is a sequence of polynomials with integer coefficients, satisfying (3.2) 3.4) We also assume that (3.5) 3.6) hold for this sequence, as before. It follows ....

M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publ. Co., New York, 1975.

....to Omega , and and Gamma are, respectively, the balayages of and Gamma from C nG to G. Furthermore, if of (2.4) satisfies (2.5) then (see Theorem 1) G; W; fl) 2. 6) We point out that the harmonic measure (1; Delta; Omega Gamma (defined in Nevanlinna [8] or Tsuji [18]) is the same as the equilibrium distribution measure for G; in the sense of classical logarithmic potential theory [18] For the notion of balayage of a measure, we refer the reader to Chapter IV of Landkof [6] or Section II.4 of Saff and Totik [13] In the following series of subsections, we ....

....if of (2.4) satisfies (2.5) then (see Theorem 1) G; W; fl) 2. 6) We point out that the harmonic measure (1; Delta; Omega Gamma (defined in Nevanlinna [8] or Tsuji [18] is the same as the equilibrium distribution measure for G; in the sense of classical logarithmic potential theory [18]. For the notion of balayage of a measure, we refer the reader to Chapter IV of Landkof [6] or Section II.4 of Saff and Totik [13] In the following series of subsections, we consider various classical weight functions and we find their corresponding signed measures, associated with the weighted ....

Tsuji, M., Potential Theory in Modern Function Theory , Maruzen, Tokyo, 1959.

....(1.2) and l (B) a l ; B; G l ) l = 1; n; 1. 3) for any Borel set B ae C; where (1;B; Omega Gamma is the harmonic measure of the set B at the point 1 with respect to Omega Gamma and (a l ; B; G l ) is the harmonic measure of B at the point a l with respect to the domain G l (cf. [2, 7]) It is well known that [2, p. 37] 1;B; Omega Gamma = m( Phi(B Omega Gamma3 (1.4) and (a l ; B; G l ) m(OE l (B G l ) l = 1; n; 1.5) where dm = d =2 on fz : jzj = 1g. Clearly, e and l , l = 1; n, are compactly supported unit Borel measures, i.e. k e k = k l k ....

M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.

....g(z) 0 for z E; 1b) g(z) log jzj for z 1: 1c) In (1a) Delta denotes the Laplacian operator, and thus g is harmonic throughout the complex plane exterior to the polygons P j . Standard results of potential theory ensure that there exists a unique function g satisfying these conditions [12,13,29,32]. The solution to (1) can be constructed by conformal mapping. What makes this possible is that the problem is symmetric with respect to the real axis, so it is enough to find g(z) for the part of the upper half plane Imz 0 exterior to E; the solution in the lower half plane is then obtained by ....

....z = 1, and the capacity C = C(1) can be obtained in a standard manner by Richardson extrapolation. For the example of Fig. 3, C = 4:082273. The ideas of this section can be spelled out more fully in formulas, generally integrals or double integrals involving the charge density distribution; see [13,21,29]. We omit these details here. 5. Further examples. Figures 4 7 present computed examples with K = 2; 3; 4 and 5 polygons. In each case, the critical level curve of g(z) has been plotted together with three level curves outside the critical one. In the case of Fig. 6, a fifth level curve has also ....

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M. Tsuji, Potential Theory in Modern Function Theory, Dover, New York, 1959.

.... (x) oe cap( Gamma1; 1] 2n exp ae Gamma Z 1 Gamma1 log jp(x)jd [ Gamma1;1] x) oe : 8) For a Jordan arc J in the complex plane, we use k Delta k J for the supremum norm taken on J , cap(J) for the capacity of J , and d J for the equilibrium (extremal) measure associated with J (cf. [5]) To generalize Theorem 1 to the case when the norm is taken on Jordan arcs in the complex plane, we formulate the following question: Problem 1. Let J be a Jordan arc with positive logarithmic capacity. Is it true that, for any monic polynomial of degree n, p(x) x n Delta Delta Delta, ....

M. Tsuji, "Potential Theory in Modern Function Theory", 2nd edn., Chelsea, New York, 1958.

....in his result remains unsettled. The interested reader will note that our example in Section 3 shows that his particular methods cannot be extended any further. 2. Proof of the theorems For completeness we give a proof of Theorem 1. 1 and except for an important technical lemma found in [4] we also prove Theorem 1.2. We feel these proofs have a certain geometric character and are easily readable. Let E be any set in the complex plane and let f be meromorphic in a neighborhood of E . Denote by A(E; f) A(E) the area of the image of E under f on the Riemann sphere counting ....

....f(z) g 1 (z)w(z) g 2 (z) g 3 (z)w(z) g 4 (z) Then (2:11) A Gamma D(z 0 ; r) f Delta C Phi A Gamma D(z 0 ; 64r) w Delta log jz 0 j log r Psi ; where C is a constant depending only on w and g k , k = 1; 2; 3; 4 . To see that Lemma 2. 3 follows from [4] in the form just given, let Delta = Delta 0 = D(z 0 ; r) set i = z on p. 279 and use the fact that g k is rational. To prove Theorem 1.2, let C j be as in Lemma 2.2 satisfying (2.7) We will show that 64C j = D(z j ; 64 j jz j j) is the required sequence of cercles de remplissage. Suppose ....

Tsuji, M.: Potential theory in modern function theory. - Maruzen, Tokyo, 1959.

....(B) #(#,B, ## (1.2) and l (B) #(a l , B, G l ) l = 1, n, 1. 3) for any Borel set B # C, where #(#,B, ## is the harmonic measure of the set B at the point # with respect to ## and #(a l , B, G l ) is the harmonic measure of B at the point a l with respect to the domain G l (cf. [2, 7]) It is well known that [2, p. 37] #(#,B, ## = m(#(B # ###3 (1.4) and #(a l , B, G l ) m(# l (B # #G l ) l = 1, n, 1.5) where dm = d# 2# on z : z = 1 . Clearly, e and l , l = 1, n, are compactly supported unit Borel measures, i.e. # e # = # l # = 1, l = ....

M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.

....Moreover, the minimum is achieved by a charge distribution in which all the charge is confined to the boundary #S 0 . The process of finding a charge distribution on #S 0 that is equivalent outside S 0 to a continuous charge distribution on S 0 is known in potential theory as balayage [69] [78]. We have arrived at the following physical picture. Think of S 0 as a collection of conductors in the plane that are electrically connected. Inject a quantity 1 of charge into this system, and let it find an equilibrium. The charge will distribute itself along the boundary #S 0 in such a way ....

....with S 0 , the unique function defined in the exterior of S 0 that satisfies # 2 g = 0 outside S 0 , g(z) # 0asz # #S 0 , and g(z) log z #C as z ##for some constant C. The quantity e C is the logarithmic capacity of S 0 (or equivalently of S) and C is Robin s constant [45] [78]. We can state the fundamental results about # as follows. THEOREM 1. Let g(z) be the Green s function for the set S 0 . The estimated asymptotic convergence factor of S as defined by (1) and (2) is (9) # = exp( g(0) Moreover, for each n and any p # P n we have (10) #p# S # # n . ....

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M. TSUJI, Potential Theory in Modern Function Theory, Dover, New York, 1959.

....t ) HD(J t ) HD(Y ) The proof is complete. We have already seen that hyperbolic rational functions with connected Fatou set provide good examples of conformal expanding repellers. Another natural class of examples is given by the limit sets of Kleinian groups of Schottky type. Here (see [Be] [Ts]) a finite set of generators fg 1 ; g k g can be found along with finitely many mutually disjoint geometric disks D 1 ; D k covering the limit set and such that min ik finffjg 0 i (z)j : z 2 B i gg 1: 28 MARIUSZ URBA NSKI AND ANNA ZDUNIK Hence, as an immediate ....

M. Tsuji, POtential theory in modern function theory, Maruzen Co., Tokyo 1959.

.... 1, #n (z) # # n (z) 1. Equation (4.8) follows from a similar analysis. Let # 0 be a compact subset of the unit circle and let g #0 be the Green s function associated with the set # 0 (see Section 2) We denote the capacity of # 0 by cap(# 0 ) and we assume that cap(# 0 ) 0. Note ([Tu]) that g #0 has the representation g #0 (z) log cap(# 0 ) Z #0 ln z e i# d#, where # is the equilibrium measure associated with # 0 . We say that lim n## f n (z) h(z) locally uniformly in an open set D if for every z # D and z n # z as n # # we have lim n## f n (z n ) h(z) ....

M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.

....contradicts the hypothesis of Proposition 3.2(b) 2 By slightly modifying the proof of Theorem 3. 1, we may establish convergence almost everywhere for measures satisfying some regularity property, such as the (two dimensional) Lebesgue measure or the ff dimensional Haussdorf measure [21]. Corollary 3.4 Let be a positive measure satisfying ( Delta r ) C Delta r ff for any closed disc Delta r of radius r 0, where ff; C are some positive constants. Then ( n ) n0 converges almost everywhere in Omega 0 to the Weyl function OE, and lim sup n 1 jOE(z) Gamma n ....

M. Tsuji, Potential Theory in Modern Function Theory, Dover, New York (1959).

.... 2d] On the other hand, for d = 1, the density of states is always the equilibrium measure on [ Gamma2; 2] 2) The Green function g(E) of the spectrum oe(L) is related to the electrostatic conductor potential (E) by g(z) Gammau(z) log(fl) where fl is the capacity of the spectrum (see [32]) This shows that a random operator has the equilibrium measure on the spectrum as the density of states if and only if the function (E) is constant on the spectrum (it takes then the constant value log(fl) GammaI (L) on the spectrum) 3) The tails of the density of states can be estimated ....

M. Tsuji. Potential theory in modern function theory. Chelsea publishing company, New York, 1958.

....Moreover, the minimum is achieved by a charge distribution in which all the charge is confined to the boundary S 0 . The process of finding a charge distribution on S 0 that is equivalent outside S 0 to a continuous charge distribution on S 0 is known in potential theory as balayage [69] [78]. We have arrived at the following physical picture. Think of S 0 as a collection of conductors in the plane that are electrically connected. Inject a quantity Gamma1 of charge into this system, and let it find an equilibrium. The charge will distribute itself along the boundary S 0 in such a ....

....S 0 , the unique function defined in the exterior of S 0 that satisfies r 2 g = 0 outside S 0 , g(z) 0 as z S 0 , and g(z) Gamma log jzj C as jzj 1 for some constant C. The quantity e GammaC is the logarithmic capacity of S 0 (or equivalently of S) and C is Robin s constant [45] [78]. We can state the fundamental results about ae as follows. Theorem 1. Let g(z) be the Green s function for the set S 0 . The estimated asymptotic convergence factor of S as defined by (1) and (2) is (9) ae = exp( Gammag(0) Moreover, for each n and any p 2 Pn we have (10) kpk S ae n : The ....

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M. Tsuji, Potential Theory in Modern Function Theory, Dover, New York, 1959.

.... 2d] On the other hand, for d = 1, the density of states is always the equilibrium measure on [ Gamma2; 2] 2) The Green function g(E) of the spectrum oe(L) is related to the electrostatic conductor potential (E) by g(z) Gammau(z) log(fl) where fl is the capacity of the spectrum (see [27]) This shows that a random operator has the equilibrium measure on the spectrum as the density of states if and only if the function (E) is constant on the spectrum (it takes then the constant value log(fl) GammaI (L) on the spectrum) 3) The tails of the density of states can be estimated ....

M. Tsuji. Potential theory in modern function theory. Chelsea publishing company, New York, 1958.

....every periodic leaf is dense in A f . Let us now show that every leaf L( z) A f accumulates on some periodic leaf. To this end take ve periodic points k and associated periodic leaves L k L( k ) Select ve disjoint topological discs D k 3 k . By Ahlfors Five Islands Theorem (see [52], Theorem VI.8) for any n, each f n L( z) has a univalent local leaf over one of the domains D k . Take a k for which this happens for in nitely many n s. Then by the same argument as above L( z) accumulates on the periodic leaf L k . 8. Convex cocompactness, non recurrence and conical ....

M. Tsuji. Potential theory in modern function theory. Chelsea Publ. Co.

.... H d 1 is ergodic with respect to the Lebesgue measure type. Usually the term Hopf Tsuji Sullivan theorem is applied to the equivalence of (1) 3) and (4) In the case d = 1 the implication (3) 4) was first proved by Hopf [1] 2] and the implications (4) 1) 3) by Tsuji, see [3]. Tsuji s proof is essentially 2 dimensional as it uses complex function theory, whereas Hopf s argument easily carries over to the higher dimensional case. Sullivan [4] used entirely different way for proving the chain of implications (4) 3) 1) 4) for an arbitrary dimension d. ....

M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo, 1959.

....j j X Gamman(n Gamma1) 31) and lim n 1 n Y j=1 jff (n) j j 1 n(n Gamma1) X Gamma1 lim inf n 1 n Y j=1 j ff (n) j j 1 n(n Gamma1) 32) where the constant X = X (T ) is the transfinite diameter of T . REMARK 4.1. Properties of the transfinite diameter are discussed in [31, 52]. Sometimes the transfinite diameter of a set T is referred to as the capacity of T . 2 Proof. It follows from (30) that n Y j=1 jff (n) j j = n Y j=1 n Y l = 1 l 6= j jt (n) j Gamma t (n) l j Gamma1 = n Y j=2 j Gamma1 Y l=1 jt (n) j Gamma t (n) l j Gamma2 : ....

.... (n) j g n j=1 be Fekete points for T , i.e. they satisfy t j 2 T for all j and n Y j=1 n Y l = 1 l 6= j j t (n) j Gamma t (n) l j max t k 2T n Y j=1 n Y l = 1 l 6= j j t (n) j Gamma t (n) l j: 35) Properties of Fekete points are discussed in, e.g. [15, 52], where it is shown that lim n 1 n Y j=1 n Y l = 1 l 6= j j t (n) j Gamma t (n) l j 1 n(n Gamma1) X : 36) In view of that ff (n) j = n Y l = 1 l 6= j ( n) j Gamma (n) l ) Gamma1 ; we obtain from (35) that n Y j=1 j ff (n) j j Gamma1 = n Y ....

M. TSUJI, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.

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Tsuji, M.: Potential Theory in Modern Function Theory. Maruzen, Tokyo 1959.

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M. Tsuji, Potential theory in modern function theory, 2-nd ed., Chelsea, New York, 1975.

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M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959. MR 22:5712

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M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.

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M. Tsuji (1959): Potential Theory in Modern Function Theory, Maruzen, Tokyo.

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M.Tsuji, Potential Theory in Modern Function Theory, Second Edition, Chelsea, 1975.

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