903 



d.^• öz óf O.c' öz 



In special cases this result may be further simplified and collected 

 into a closed expression; as stated already, the well-known analytical 

 reductions always hold. Of course, if (he result is to be used, we 

 have always to develop it again into a power-series according to z. 



We finally observe that in all that precedes we may replace the 

 three numbers «, q' q, respectively by a, /?, y, if F^ remains con- 

 tinuous if (y) and (/?) are taken for its associated numerical fields, 

 and P, Jf fli® same be done with («) and (y). This is the commonest case. 



25. To elucidate the general theory we shall give a few examples. 

 We first take T, = T, = D-\ so that r= Z>-2 and further assume 

 as before the origin as centre of our domains, consequently 

 Xo = 0. The transmutation D—^, as defined in N°. 16, is normal; 

 the corresponding series is complete in any circle (ï), with cor- 

 responding domain (2 5). For the three numbers a, y, ^, we can 

 take here «, 2«, 4«. The premise 3. is also satisfied ; for D~^ is 

 not only normal, if (i) and (2 >) are considered as associated 

 numerical fields, but also, if (ï) and (^) are taken as such. 



We can consequently apply our theorem, and infer from it that 

 the transmutation T^ D—- is normal with N.F.O. («) and N.F.F. 

 (4 a). This is correct, for of the simple transmutation D~^ it is 

 known that with preservation of (r<) as N.F.O., («) may as well be 

 taken as N.F.F. It remains still to verify point 2. of the theorem, 

 i.e. to prove that the series P corresponding to Z)~'^ is complete 

 in («), with a corresponding domain that is at most equal to (4 «). 

 To this purpose we shall apply Bourlet's formula. If we put 

 f^(x, 2)=f^{d', z) = fp {x, z), we have, as it appears from the series 

 belonging to D'^ (see among others N". 16), 



ff{x,z) z= \?n 

 



From this it follows 



( - i)'«.i;'«+'^'« 1— g- 



(m+1)/ 



=:: ( zY—^e—'' 



so that according to (59) we have. 





1 

 Further 



