619 



only complete in any domain smaller than (r,) but in niost cases in 

 any domain smaller than (/•') ifr' is the radius of convergence of Tv. 

 If now tlie radius of convergence of the function u be greater 

 than r, the series will converge in a domain greater than {r^}. We 

 have in this case been able to find for the transmuted Tio of n?, 



by developing in the point lü = v according to powers of 73 ( - 



an expression representing this transmuted in a greater numerical 

 field than would have been the case if we had developed in the 

 point w=^l. Or we may also consider the thing in this maimer 

 that, with the preservation of the numerical field a ^ (rj, {he func- 

 tional field that at first consisted of functions belonging to a circle 

 (/?) ^ (r), has now been extended to some functions not belonging to (/?), 

 namely such as show the same kind of singularity in the domain 

 (|i) as a fixed function v (.r), their quotient by v {:c) being regular in 

 {^) ; functions of which we might say that they lie in a geometrical 

 vicinity of the functional point v (/;)• Taken thus we observe an 

 analogy in the present phenomenon with the one called analytical 

 continuation in the theory of functions (Cf. N". 8). 



21. We shall now elucidate the more general theorem of Taylor 

 by two examples. As transmutation we take in the first place, in 

 a vicinity of the origin, the substitution So,, in which (o is a function 

 of ii', which has .v = as ordinary point, the radius of convergence 

 for that point being denoted by A. We fully explained in N°. 17, 

 (3'"^ communication) that *§ is a normal transmutation, the numerical 

 field being a circle with radius <( <^ A, and the functional field con- 

 sisting of the functions belonging to the circle (o) concentric with 

 {a), the radius of which a is equal to the greatest modulus of to in 

 the domain («)• We further saw that to this transmutation belongs 

 a series P, which is complete in [a), with a corresponding domain 

 ((li), which is determined by the equation 



^ z= a -{- \ Iti (A',„ ) — Xm I , 



where Xm is the point on the circumference of («)> where j to — x\ 

 attains its maximum ; we pointed out that ^ is at least equal to o. 

 We can therefore apply the functional theorem of Taylor to a 

 function w = vu^ provided we suppose the radius of convergence r 

 of the fixed initial function v to be greater than ioj(0)j. The circle 

 (rj to which, for the 'series 1\ (r) corresponds is determined here 

 by the condition that in it the maximum modulus of to — x is equal 

 to r—r^, while the circle of .convergence (r') of S,,{v) satisfies the 

 condition that in it the maximum modulus of oj is equal to r. The 



