Mathematics. - - "■Some Considerations on Complete Transmutation." 

 (Fifth Communication.) By Dr. H. B. A. Bockwlnkel. (Communi- 

 cated by Prof. L. E. J. Brouwer.) 



(Communicated in the meeting of January 27, 1917). 



23. If we tal^e, as we did in N°. 19 (preceding communication) 

 the development of T yvu) as a representation of the compound 

 transmutation T^ = {Tv) applied to u, of which transmutation the 

 first component is the operation of multiplication, and the second 

 an arbitrary one, that development is a special case of a more 

 general one that represents the resultant T of two transmutations 

 T^ and T^, which are both arbitrary. This development is also 

 mentioned by Bourlet, but merely in a formal way, without any 

 indication as to eventual domains of validity. As Booklet frequently 

 uses the formula, it is of importance to be on clear ground with 

 respect to it. 



Previously we observe something about the functions a„i(.r) of a 

 series P, representing a transmutation which is complete in certain 

 circular domains. If («) is such a domain of completeness, we saw 

 before (N". 4, 1^*^ communication) that this implies that the quantity 



1 

 ax — lim\am{x)\'^ (6) 



is limited in («) and consequently has a finite upper limit, which 

 we indicated by a («) or simply by a. We shall now consider the 

 majorant functions a„, {x) of «,„ {x) (cf. a note in N°. 7, 2"^^ com- 

 munication). Have they also the property that a limit exists as 

 indicated in (6)? 



According to a well-known formula from the theory of functions 

 which, moreover, we have already frequently used, we have in a 

 domain (a') <^ («) ^) 



a,n{x, + « )< — , 



a — a 



in which x^ is the point on the circumference of («), where am {x) 



1) The magnitude a»^ {Xq + a') is according to what we understand by a majorant 

 function, real and positive. 



63 

 Proceedings Royal Acad. Amsterdam. Vol. XX. 



