621 



X X X 





{t)dt, 



which result, to begin with, is regular within the circle of con- 

 vergence ()') of D~^v. In order to find a majorant value of ]r?',„l we 

 reduce as follows 



X X 



\ a\„ \<C\t-x\ '" ,\v{t)\.\dt\< «'« M (a) f 



, dt 



0- 



or 



I a',n I < «'«+iiI/(«), 

 if t( =z [r! and iW («) be the maximum modulus of v on the circum- 

 ference of the circle (a). 



F'rom this it follows for values of a <^ r 



1 

 a'x ^ lii'i a'm I "* < « 



consequently 



a' {(() < (t, 



according to the general result (41^/) in N". 20, but here extended 

 to domains greater than (^ r), provided they are smaller than (?■). 

 Thus, in fact, the series P^ is complete in any domain (a) <^ (r) 

 the corresponding domain ((i') being at most equal to (2 a). The 

 series P^a, determined by (42), converges therefore within the circle 

 (^^ 6*), if the radius of convergence s of a is smaller than 2 r, and 

 otherwise within the circle (r). 



45 



Proceedings Royal Acad. Amsterdam, Vol. XX. 



