FUNCTIONS IN THE TIIEOKY OF INTEGRAL EQUATIONS. 135 



the two functions being such that 



x 00 *'* (*)-** 00'*00 = -i. 



Writing 



-!$ *-$ 



\\r thus see that 



> . . ...... 



where ^ x (.s) is the solution of (5) denned hy the conditions A (0) = 1, 0' A (0) = h f , 

 <^0*) is the solution denned by <p A (ir) = 1, <j>\(ir) = H', and 8 (\) is the value of 



which is known to be independent of*. 



5. The result just stated may be employed to obtain an asymptotic formula for 

 KA (s, t), when X is negative and numerically large. For this purpose it will be 

 convenient to write X = p 3 , where p is supposed real and positive. If we denote by 



D the operator -j- , the equation (2) then becomes 



[D a -p a ] = -qu. 

 The complete primitive of this is 



u = Cj cosh ps + Ci sinh ps [D 3 /a 2 ]" 1 qu 



= r, cosh ps + c a sinh ps - {[D p]" 1 qu [D + p]" 1 qu} 



2p 



1 f* 



= c, cosh ps + Cg sinh ps -- I q^ sinh p (s s,) ds lt 



jo 



where c, and c a are constants, and q lt H, are what q, u become when s l is substituted 

 for s. 



Ifu = $i(s), the conditions A (0) = 1, 0^(0) = h' give 



Accordingly we have 



A (s) = cosh ps + sinh ps -- I 7,^ (i) sinh p (,)(/,. ... (7) 

 p pJo 



If we write 



^( g \ = AteL .......... ( 8 ) 



coshps 

 this equation becomes 



\ cosh ps, sinh p(ss t ) , / Q \ 



