939 



B. The maximum value a [n) of nx for the lühole domain («) is 

 equal to that for the circumference. 

 That the series 



00 ^ V 

 Pv ^Z \ ,„ — 



Ü 



converges for all functions belonging to the circle with radius 

 a -\- a {(() and that this circle is the minimum circle for which this 

 property holds, therefore follows only from the hypothesis B. 

 This latter however is not sufficient to derive from it the uniform 

 convergence of the above series. But this at once follows if the 

 hypothesis A is added to B; it is not necessary to explain this 

 further since it may easily be derived from the proof given in N°. 4. 

 The uniformity of the convergence is, however, of interest since in 

 this case we may be certain that the transmuted of a function which 

 is regular in (/?) is itself a regular function, the domain of regularity 

 being at least («); in other words the transmutation in question is 

 then always a regular one for the F. F. of functions belonging to 

 (^) and the N.F.O. («). For this reason we shall retain the hypo- 

 thesis A for the extended theorem of Bourlet, treated in N". 4, 

 and thus substitute for the uniformity-supposition of that paragi^aph 

 the two suppositions A and B, ivhich are independent of one another. 

 That the uniformity-supposition of N°. 4 is narrower than the 

 two suppositions A and B together is proved by the following 

 example in which both A and B are satisfied, but not the former 

 supposition. Let 



/log 111 \ 

 a,n (^) = .r" — 1 , n = 2^\ï^) 



SO that n depends on m in such a way as to be equal to the 

 highest power of 2 that is contained in the number in ; thus n 

 passes through all integral powers of 2, but after every change of 

 its value it remains constant for a certain number of 7?i-values. For 

 « <^ 1 , a («) ^1, for a^l, a («) = o-, since n is never greater than 

 m but is equal to m for an infinite number of 7/2-values. The quantity 

 A is therefore, as the quantity R (see above) infinite, so that 

 the series having the above quantities a,^ {x) as its coefficients is 

 complete in any domain («), however large. If we imagine <^< ^ i 

 the circle of radius unity lies wholly in the domain («) ; at points 

 on the circumference of this circle having as their argument 



