-• - - r ^*T.^V-1i4^if^i«»w-«-__,_ __ 



a standard form of W-H equation, with 



1 

 K( 8 ) - i - ( B » - k 2)"T (8.24) 



Y s ° 



The factorization of K. into factors analytic and non-zero in R 

 and of algebraic behavior at infinity there is again obvious: 



K. (s) « (s + k ) 2 , K (s) - <s- k ) 

 + o — o 



and after division of (8.23) by K_(s) we again have the additive split 

 of the function l/(s + q) K (s) to make. We thus arrive at the 

 equation 



v s > n< 8 -°> + Y q M^ )( s + q) < 8 - 25) 



* (s,0) iv 



K (s) Y.(s + q |K_(s) K_(-q) 



— it 



■ an entire function E(s) . 



We anticipate that E(s) will be a polynomial, and consider the implica- 



N N 

 tions that the degree of the polynomial be N, E(s) - as + a x s +. ..+a . 



iv (s + k ) 2 1 „ 



*'(s,0) - — 2 r + (s + k ) 2 {a s"+...} 



+ Y K (-q)(s + q) o' o 



— L i. 

 ~ 0(s~ 2 ) + 0(a o S N+2 ) 



as s ■* " in R, . The first term vanishes algebraically at » and is 

 + 



therefore the half-range transform of a function with at most an integrable 

 singularity at x - 0+. In fact it follows from Watpon's lemma that the 



57 



;-^;g^W«,;^jH^ay^^tr^^ 1^-. ^,.-...^ sisaii.^^^AkifiMd 



