Mathematics. — "On the evaluation of ?(27i-f-l)". By Prof. J. 

 C. Kluyvkh. 



(Communicated in the meeting of September 28, 1918). 



By means of a characteristic and very general metliod Markoff ') 



00 00 



transformed the very slowly convergent series 2n~'^ and 2n~^ 'mio 



11 



other series, that converge more rapidly, and J. G. van der Oorput ') 

 described a special method of transformation applicable to the series 



00 



^ji-ah-^x] f-Q,. larger values of h. 1 propose to deal anew^ with the 

 1 



transformation of these series and to add a few residts to those 

 previously obtained. In order to appreciate the increase of convergence 

 resulting from the transformation, 1 will consider d'At-kmbert's ratio 

 for the transformed expansion, which I will call its index. For the 

 series given by Markoff the index is -^, and I will show that a 

 lower index can be attained. 



In the first place 1 base my deductions on the properties of the 

 function 



n=oo 

 (fk{z) = > — , 



n=l 



where k denotes a positive integer. In order to uniformise rpkiz), it 

 is convenient to regard the right line (-f-1 , -|- Q*^) ^s a cut in the 

 complex 2-plane, and with this convention we may enunciate the 

 following properties of (p^^z) in substance deduced by Abel: 



qp. {z) + <P. (1—^) = - log ^ log (1 -^) + ? (2), 

 y^. (^) -f- ^. (-^ = - è log' z Ar ni log 2 + 2 S (2) 



y. {z) - r^ ( -^J = è log' ^ - log z log (1-^) + S (2) 



Obviously in these formulae we have to take real values for log^ 

 and for log (1 — z), when z itself is real and between and 1. 



') Gompies rendus, t. 109, p. 934. 

 ») These Proc. XIX, p. 489. 



