E 



904 



( — z)n d"(p{a; z) 



n! 



Ö2" 



and thus 



f{x,z) 



= (f (.r, z — z) — (f{.T,z), 



l—{\-\-xz)e-^^ 



z 

 If we develop this according to powers of z, we arrive at 



{m-\-2)mf 







SO that to D~"ii corresponds the series 



V^ (_.,)«+2,,(,«) 



^ (m + 2)7n/ 



This series, which according to the theorem is known to converge 

 uniformly in the domain («), for functions t)elonging to (4 a), does 

 so already for functions belonging to (2 a), as may at once be 

 vei-ified by means of the often used raajorant value for !«('"'. 



26. As a second example we take 7'i = 7', =3>S^aj, consequently 

 T=:S\.>(x), in which Sr„(x) is the operation of substitution, with ty(.i;) 

 or (o as substitution-function, which we defined in N". 17 and 

 recognized there as a normal one. We consider, as we did there, 

 a neighbourhood of the origin, and have seen then that as a pair 

 of associated tields may serve: any pair of tields of which the N.F. 

 is a circle (ï) smaller than the circle of convergence (.4) of io{.x), 

 and tlie F. F. consists of functions belonging to the circle (o), if<Jis 

 the maximum modulus of a}{x) in the domain (|). We found further 

 that the series belonging to S is complete in any domain (i), as 

 meant here, with as its corresponding domain a circle ir[), the radius 

 of which is at least equal to the number o mentioned; from which 

 we can at once derive that premise 3 of our theorem is also 

 satisfied. Of the three-numbers mentioned in premise 2, « must be 

 chosen small enough to ascertain that the number y, which corresponds 

 to « for the series I\ {= P^) answering to S, is less than the radius 

 of convergence A of «>. It may of course occur that this is impossible, 

 viz. if a number y greater than A already corresponds to « = 0. 

 We shall therefore consider a case in which this last circumstance 

 does not occur; premise 2 is then satisfied and we have 



y = a + lw{.^•„i) — A'„i I, (60) 



/?= y -f |a>(.i;',„) — A'',„ |, (61) 



if x,n is the point on the circumference of («\ and .r„/ that on the 



