58 



pitcher: interrelations of 



toJP?'". Owing to the peculiaHnitiaHunctional value ch aract erizing 

 the functions of each component class of S)t, the closure of each com- 

 ponent class as to 9[)?'" insures the closure of 9)? as to 9JJ'" and therefore 

 the closure of SOf. 

 Proof of m''''^— 



m^ = 9JJ'". 



<P' 



Kum 



if has infinite number of zeros. 

 .-. »U + [all ^^"-"]. 



48h. ~L-C-DA, 10 examples (§23«lli2), on "$!' (§41.1), 8 on 

 ^"^ (§ 42.812), 9 on ^]5"^ (§ 44.5), 10 on ^V". 



(1) 

 (2) 

 (3) 

 (4) 

 (5) 

 (6) 

 (7) 

 (8) 

 (9) 



^"=; A^'=; m 



(0, 1/n), (1/n, 0) {n). 

 (0, l/2«), (l/2n, 0) {n). 

 (1/n, 0), (0, IM) (n). 

 (l/2n, 0), (0, l/2n) (n). 

 (1/n, 0), (0, 1/n) (n). 

 (1/n, 0), (0, 1/n) (n). 

 (1/n, 0), (0, 1/n) (n). 



^;^"=; Af/- 50? = (1/n, 0), (0, 1/n) (n). 

 ^;^"- A5'- 93? = (1/n, 0, 0), (0, 1/n, 0) (n). 

 iV"; Al"; 



2)? = [all M J ( 3 P. 9 p""" ^ 2J„ . 3 . P„ = 0)] 

 + [allM9(3p„ap"™ ^p, .a.Mp = 0)]. 



Proof of W^. — Consider sequence {ju„!, such that 



M«p = (p-- > n); M.p = 1 (P ^ n); ^.^ = 1 iv"'") 



and the functions ^o and 6 such that 



Mop = p(p^^™); MOP = 0(p"""); e, = l(p). 

 LMn = e (^;>"';mo). 



n 



For proof of W^'"- see § 48^10. 



48{. -LCD- A, 10 examples (§23alli2), on %"■ (§41.1), 8 on 

 ^"' (§ 42.812), 9 on ^"^ (§ 44.5), 10 on ^"'.— 



A A', ivj 2v,2, ; A'l, Aj, 



(1) + + + +;+ + : r'=; A^-gr)? = (0, 1), (1, 0), (- 1/2, 0). 



(2) _ + + +;+ + : sp"=; Ar- m - (0, 1/2), (-1/2, 0). 



