2 3 8 



UNIVERSITY OF COLORADO STUDIES 



order. In this counting of the rational numbers all numbers of the 



ftti) 



form — where p and q are prime must be left out. According to this 



arrangement the one-to-one correspondence between the natural num- 

 bers and all rational numbers may be indicated by the two rows : 



i 2 3 4 5 6 7 8 9 io ii 12... 



iis 



+ 00 +1 o —i —oo —2 +2 — 3 +3 . . . 



2 2 2 



Fig. i 



3. Example II. Expansion of the function /(z) = -3- by 

 taylor's series and its analytic continuation. 



1. According to Taylor's series we have 



I(z)=Ka) + (z-a)na) + (l ^ 1 r(a) + . . ,+fezj2>(a)+. . . , (1) 

 where 



*^ n! C f( z ) dz fa \ 



/(a)= ^J c (^)^' (2) 



an integral taken over the boundary of a circle C having the point a as a 



