FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS 

 It follows that, for unequal values of s and t, 



lini pF A (M) (/>.*> = - 



(> 



Applying the theorem of I, 1, we deduce that, as p tends to oo, 



139 



I.C., 



tends to zero, for all values of s in (0, TT). 



In what follows we shall express the result just obtained by the notation 



-\ f K, (.s, <)/() dt ^^ -X f F A (s, t)f(t) dt, 



. JO 



the symbol between the left- and right-band members indicating that their difference 

 tends to zero, as X tends to oo. 

 9. Let us now consider 



The value of this is evidently 



p cosh p (n-s) (' h 



sinh 37T Jo smh 



sinh /37T 

 Now, whatever value .s- may have in the closed interval (0, IT), 



r,, (ir _,)/() j,. . (14) 

 J. 



p cosh p(*-s) ['.-,*,)*. g ^egdl.(^i) C|/(t)|rff : . . (15) 

 J mh 37T 



hence we see that 



sinh pn Jo 

 converges to zero, as X tends to oo. It follows that 



p C08h P (*-) (' cosh pt/(t)dt ^ p T 1 '. p . (7r - s) (V/W*. 



sinh PTT Jo 2 sinh pw Jo 



Since a corresponding result applies to the second member of (14), we finally 

 obtain 



2 sinh 



T 2 



