PROPERTIES OF CLASSES OF FUNCTIONS 



63 



+ - 



,;j^iMn'. ^luiii. goi, ^ [alV^»^%jun = 0]. 



,-^,u,ui. ^iwii. goff ^ 9[)f{. 



sj5ir.iii. ^ii:.ni. 2,^,^ ^ [allM^°-''Mii = 0]. 



,^;ii.ni. ^ii.iii. 93j^ ^ [all M*"'' 9 Mil = 0]. 



48p. L-C-D-A, 10 examples (§23alli2), on ^s"- (§44.1), 10 on 



(1) + + + +; + - :^;^"^'"; Al'^'"; 



a« ^ [all /x 9 (r . 3 . (£ Am ,v)™"""'*™') • (m?"") • 



(ajV'j ^i^ •^- M2i = -Ms;)].'" 



(2) - + + - 



(3) - + + + 



(4) + - + - 



(5) - - + - 



(6) + + - - 



(7) - + - - 



(8) + 



(9) 



(10) + + + - 



<^^[ij]^[iji] {i = l,2, . . ., n, ...;f^-i, -i+l, . . ., 0, 1, 2, . . .,i); 



Each stage m of the development A consists of the classes : 



spmu ^ ij (i = 1, 2, . . ., to; j = - i, - i + 1, . . ., 0, 1,2,.. ., i) ; 



0? = Tall M 3 (3 (v • {q.-Im • !&.! J » (L«. = I^^'- = 0) • 



[i^^,.J .:d. Mu = a. cos 2^ + 6, sin ^^j jj . 

 Proof of W.— 



3{fxn] ''"'" 9 ({i ^ n .f) . 3 . M«o = i)\^^> ^^ ••?'') • ^ • i^"i = i ''^^ 2l)- 



9J? 



3/ir"3 



LMn = e (i^; mo) 



where ,-y = ^ (y ) . 



Proof of 93J^'.— Given a function /n in the definition of which 



* Cf. § 48^-1; § 48/«l. The proof of 9JIA'i2» in this example follows in the same way as 

 proof of § 48/tl follows from § 4SA'l and § 48il. 



