212 



Proceedings of the Royal Irish Academy. 



the former holds when - 7r/2 < arg. y < 57r/2 (exclusive), the latter when 

 + 7r/2 < arg. y < 7ir/'2 (exclusive), and, accordingly, both . are valid when 

 4- 7r/2 < arg. y < ottI'I (exclusive). Thus, for arguments within the latter 

 limits, we obtain, on combining (18), (19) 



' K„{y)^{7rl2y)i\t-y + 2-Uozn7r.ey\. (20) 



On deforming the path of A, then, so that it is everywhere at a great 



distance from the origin, we have, for the portion which lies to the right of 



the axis of imaginaries, 



K„{:l\p) = {7rl2i\p)U-'^<', (21) 



for that which lies to the left, 



Kn{i\p) = {Trl2iXp)i {e-'^p + 2i cos mr . e'^'^}, (22) 



and throughout the range 



Kn (- iXr) = {7r/-2iXr)i e^. (23) 



Consider, then, the value obtainable for the left-hand member of (16) 



by substituting these approximate values, postponing for the moment the 

 discussion of the error, if anv. It is 



TrLt. 



dX 



gikr(^^-ikp ^ 2i cos mr . e'^p) (pj-r)i(p{p) dp 



. ki 



dX e'^'- e - '^p {plr)h (p) dp 



(24) 



The portion of this wliich is obtained by combining tlie second term with 

 the first part of the first is 77-/2 . (/j(r - j), by Fourier's integral theorem. 

 The remaining portion, on changing the order of integration, may be expressed 



as a multiple of 



Lt r '^ f - ^ ('■ + p) - e - '■'^ l'' + p) 



{p!r)^<p(p)dp. (25) 



r + 



It is shown in Dirichlet's proof of Fourier's theorem that, when h 

 increases indefinitely, the limit of the portion involving sin h (r + p) is zero, 

 and evidently the same argument holds for that involving the cosine. The 

 limit of the portion involving k is evidently zero also. Thus (25) vanishes, 

 and (24) is thus equal to 7r72.</)(r - c). 



Consider, next, the difference between (24) and the left-hand member 

 of (16). From the nature of the asymptotic equations its modulus is 

 evidently not greater than 



Lt. 



9'" 



\X-'dX 



\(Ap-' + Br-') e'M'--p} (p//-)4 (/)(p) dp 



rki 



+ 

 c J 



-h 



\X-'dX\ (Cp-' + Br-') e'Mr^p) (plr)h ^{p) dp\ 



(26) 



