f(p) = a0 + a16+**.*, XI1 SERIES HAVING THE CIRCLE OF CONVERGENCE AS A CUT

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1 XI SERIES HAVING THE CIRCLE OF CONVERGENCE AS A CUT Theorem of Chapter V is sometimes called the Hadamard-Fabry theorem. This chapter has for its object the consideration of related theorems. We shall obtain less restrictive conditions on the fh,) in order that the circle of convergence shall be a cut. We begin by developing some ideas which, although elementary, are important in themselves as well as in their bearing on the theorems in question. Let f(x) denote the function defined by 2anxn within the unit circle. If I pi <, we have f(p) = a0 + a6+**.*, For the expansion of f(x) about the point x = p, we may write M p = bei*. f(x) = ;C: c,(x - p)m, m=u Let C, RI denote, respectively, the circle and the radius of convergence of this series. We have It is clear that R2 - b, since f(x) has no singularities within the circle Cz, center at p, radius Rz = - b. Hence 339

2 3 40 Singularities of Functions Cz is tangent to the unit circle at the point ei4. If R = Rz, then ei4 is singular, for all other points on Cz are interior to the unit circle. Conversely, if ei4 is singular, we must have R = R2 = - b. Hence the requirement R, = - b is necessary and sufficient in order that ei4 be singular. Similarly, the necessary and sufficient condition that eim be regular is R > - b. Expressing R in terms of the c,, we have (A) If the following inequality holds : R = l&q then eim is regular; (B) If n- - > - b, lftffdlc,,i <, (2) then ei4 is singular. If, in (A) and (B), the c, are replaced by their values given by (l), we may say that the point ei6 is singular if for an arbitrary e> 0 we have, for an infinity of values of m, I cml= I am + (m + )a,+lbei4 +* * *+ Ck+nan+s > (qm -b ' and eim is regular if there exists an E> for m > mo.,3969'4 +..,. 0 so small that cmi <(qm -b I

3 Circle of Convergence as a Cut 34 In what follows we shall use the fact that the left hand ride of (5) and (6) may be replaced by a finite sum of terms occurring in this expression, e.g., by I am + (m + l)am+lbei a + C$+Qam+QbqeQi+ I, where q is arbitrary, but q 2 p = p(m), where this last number depends only on m. This principle, due to Hadamard,I has been employed by Fabry, Leau and others in obtaining important results. 54. We shall prove the existence of the number p = p(m). Suppose for simplicity that 4 = 0. Since the quantity C~+QmmqQ is a term of the expansion of (m + q)m+q we have CR+QmmqQ < (m + q)m+q (7) Moreover, since this term is the largest in the expansion,' and since there are m + q + terms, we have From (7) and ( 9, m m- 4 -bq 2 lim C$+,bQ m+ w mmqq m--, w where q may be supposed to vary with m, and where the existence of the limits is assumed. loc. ci2. *Write (m + g)m+a = LYO + LYI amfa, where Then Qk = m + Q - k + 2 ak--l, so that if k < q, then ak > ak-; whereas if k > q, ak < (lk-. term in the expansion of (m + q)m+a is therefore k aa = CL+, mmqq. m The maximum [EDITOR.]

4 342 Singularities of Functions From (7), and from the fact that the radius of converg- ence is (I am+ I < ( + q)m+p for 7 > 0, arbitrary but fixed, m sufficiently large), we have = ( + ;)( +.)[ ( +?)b( + ;)I" Let I be a number satisfying the inequality b( + ) < I <. To each integer m corresponds a number p = p(m) such that for q 2 p the following inequalities hold: It is possible to choose b and I in such a way that () and (2) are surely satisfied for - P > A', where A' is an arbim trary fixed number greater than. In fact, on the one hand, the inequality b-bo ( + d b(l+ +) < is therefore satisfied for a pair of numbers bl,, both arbitrarily small, bl being chosen small enough with reference to.

5 Circle of Convergence as a Cut 343 On the other hand, lim ( + x )(I + q)p -* = 0, since X > ; and -0 lim. b-0 - b Accordingly, for =, b = bl, these numbers being sufficiently small, we have or Hence there exists a pair of numbers, bl, arbitrarily small, such that ( ) and (2 ) are verified. We may take < -. e Under the same conditions, the two quantities Zl < - and bl being fixed, decrease as x increases. The ( 9 truth of this statement is obvious for the first expression; as for the second, its derivative with respect to x is negative when ZI < -. Since ( ) and (2 ) hold for I = L < -, b = bl, e e and A >, it follows from what we have just seen that these inequalities are verified a fortiori if we replace X by Q - > A. m It is therefore always porsible to satirfy () and (2) by a suitable choice of and b, and by letting p be arbitrary but greater than mx, where A > is fixed. Hereafter it will be assumed that b, and p have been so chosen.

6 344 Singularities of Functions From (ll), consequently, from (0) and (2), Now, from the inequality C;Zr+I(l + q)m+r+lbr+l <. Ci+r ( + rj)m+rbr Combining this inequality with (4), we have and therefore Ck+qam+pb4 + CK$-lq+lam+q+lbq+l +. * < I C:+qu,n+pbq/ ( + I + L2 + * * * *) " * * I <- -b if q p.

7 and Circle of Convergence as a Cut But if {A7L}, { B,} are two sequences such that n- lklqa,ll = A, 7' then which is equivalent to the statement that if the radius of convergence of ~lb,xn is greater than that of ~A,,xn, then the series 2(An + Bli)xn has the same radius of convergence as za,,xn. Let A m = am + (m + l)a,+lb + Bm = Ci~q+lanL+,+lbq+' Then the point is singular if *+ + a * * *. C:i+Qam+qb*, for an infinity of m, and for every E> 0. regular if for an arbitrary E, m qla, + CL+la,+lb + a a - a The point is - + C* m+y a m+pbql < --, for m sufficiently large, and for b sufficiently small, depending on e. In general, the point ei+ is regular if m limd j a, + C;+la,+lbei+ + ' * ' + C~+uam+ub4e*~+ < -9 m b

8 346 Singularities of Functions and singular if (50 for q 2 p, and for b sufficiently small. 55. THEOREM : The circle of convergence of the series where Ant - zaa,xhn = kmxm, (20) A>, n=,2,., (20 ) A, AFxed, is a cut. We assume that the radius of convergence is. Corresponding to each m, choose a number q such that where b is given as in the previous section. Then On the other hand, m Iim Vm+q + = I. m--r m Hence, from (22), -b See footnote, p. 35

9 and from (9), Circle of Convergence as a Cut 347 We choose b so small that b -b <. Let A be a number satisfying the inequality and define the partial sequence {m,) by We have < A < x, (25) m, = E [xn(l - b)]. An- mn b. lim - n+ cc mn -b In (5 ) let m take on the values m,, placing Since, from (20 ), we have b where A> - b from (26), (26) q = X,+l - m, -. (26 ) An An- < - + x it follows that A,-, m,> Noting that A,> m,, we obtain from (26 ), -- mn mn m n An mn Q -X,tl-l-->-- L+ - _. Hence, for n sufficiently large, < x,,(l - b), hence, E[%] denotes the smallest integer 2 x.

10 348 Singularities of Functions since lim - = 0. But we have seen (page 343) that, under lb'm m, these conditions, the inequalities () and (2) hold, provided further that and b are suitably chosen, b sufficiently small with reference to I, which is arbitrarily small. If then q is chosen so as to satisfy (26"), it., if mn + 4 = Xn+l -, (27) then q may be substituted in (5') and (6') in order to obtain the criterion for determining whether or not a given point on the circle of convergence is singular. From (27), all the coefficients k,n from kmll to krn,,+* vanish except kxv0 hence kmtl + Ck+ikm,l+ibei' + * * * + C~nn+gk,n,,+ybqeqi' = C:~-m'lkA,lb e 4' k b4'e4"i6 = CA, A, 9 A,, -mil (A,, - m,,) i4 (28) where q'= An - m7l. By (26), the number q' satisfies the requirement (2) if nt takes on the values m7. From (23), Now there exists a subsequence { A ~ of ~ ~ {hil] } such that But from (26), and therefore

11 Hence Circle of Convergence as a Cut 349 We obtain from (29) and (30), and from (28)) mn l& d I kmlz + C~n+lkmn+l beim C&+4km,,+Pbqeqid = - nam -b Comparison with (5 ) shows that every point ei+ is singular. Theorem is therefore proved. 56. THEOREM 2: The circle of convergence of the series is a cut provided that there exists a number u < such that the series converges. Let Z U ~, X = ~ ~ Zkkxm. From the sequence {b} we may extract a subsequence { pm} satisfying the conditions x being fixed; and Pm+i - Ilm Pm > X > ) =. Denote by {qkf the sequence of integers contained in {A,) and having no element in common with { pm). The sequence { qk) clearly has the property that the series zt converges, qk

12 350 Sing u I a ri t i e s of Functions We may form an integral function having the points qk and no others as zeros. By Theorem 5, Chapter V, the function defined by Zg(n)xn has just one singularity in the entire plane. On the other hand, it follows from the work of Poincari: that the function g(z) has the property that Ig(z) > c-~~+~, r = 2, e> 0, for r > re, the point z being such that within the ring formed by the circles with centers at the origin, radii I z I -, I z +, respectively, the function g(z) does not vanish. But the number pin has the property required for z, from the way in which g(z) was formed; hence, for k sufficiently large, (33) for e> 0 arbitrary. Now the series zg(n)k;xn = (34) has only the d,, as non-zero coefficients, since for the indices n different from pm we have either g(n) = 0 or k; = 0. Hence we may write But we have, from (33), zg(n)kixn = zdp,x m. =, since 0 < u + E <. Bull. de la SOC. Math. de France, t. (883), p. 36.

13 Circle of Convergence as a Cut 35 Since xg(n)xn converges within the unit circle, we have ig7ms. m+ m Hence m-r m Hence, from (32) and (34),.- Bm lgqzj=, m+ m that is, the series 2dBmxBm has its radius of convergence equal to. The function defined by this series accordingly has the unit circle as a cut, since the sequence (pm} satisfies (3). Now &!B,x'm = zd,xn = &(n)knxn, and since zg(n)xn defines a function having the point as its only singularity, it follows from Hadamard's theorem on the multiplication of singularities that 2knxn likewise has its circle of convergence as a cut. For otherwise the circle of convergence of ~;d,,xn would not be a cut. The proof is therefore complete. 57. THEOREM 3: The circle of co?ivergence of the series 2ah,,xh" is a cat provided that where a, h are arbitrary but fixed positive numbers.' When n =, the theorem reduces to Theorem, Ch. V. Taking a = i, me have the theorem of Bore], Jour. de Math., t. ii (896), p, 44. Finally, the most general theorem, viz., that the circle of convergence is a cut if lim (Xn+I - An) n+m = m, is due to Fabry, Ann. de I'&cole Norm. Sup., t. xiii (896), p. 367.

14 352 Singularities of Functions We have, by hypothesis, An+, - An+p-l> hg+p-~* Obviously hnsk > k + for k sufficiently large. We may therefore assume that this inequality holds for all k. We have, then, An+p - An+p-l> hpa- Hence, An+,- A,> h(l"+ 2"+.***+p"), A,+,> h(l"+ 2"+.***+p") > h?x"dx h --- pa+' ff+l We may choose a number r, 0 < T <, such that r(a+ )> + 6, 6> 0, provided s is small enough. Consequently The series E-- is therefore convergent. Hence, by Theo- A:, rem 2, the circle of convergence of the given series is a cut. This transition from Theorem to Theorem 3 by means of Theorem 2 is due to Faber.' Starting with Theorem he also proves by an analogous process the theorem of Fabry referred to in the footnote on p. 35. The latter theorem includes as particular cases the theorems of Hadamard and Borel. S. MANDELDROJT. Sitzungsb. der Math.-Phys. Classe der Municher Acad. (904), p. 62.

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