906 DR JOHN DOUGALL ON AN ANALYTICAJ. THEORY OF THE 



This again is equal to 



|fE(K)e-'''(^^'WK^ (first patli) + I fE(K)e''<(^-2'W/<, (first path) . . . • (4) 



= :|fE(K)e-''«(^-2'WK, (first path) + ^fE(/<)e-''<(^-^)(^»c, (second patli), . • (5) 



as we see by changing k into —k in the second integral of (4). 

 The result may be written 



/ {E(/c) cos k{z - z) - E„(K)}rfK = 1 \^{K)e- '■'<(^-^'>'('k, (first and second paths). . (6) 



In the second integral of (6), change the variable from k to jS, where ^ = ua, as in 

 Art. 5 after equations (2). If E(«:) thus becomes E^, we obtain 



I {E(K)cosK(2-2')-E„(K)}(?K = -^lE,3e « J/3, (tliird and fourth paths), . . (7) 



where the third and fourth paths in the jS-plane pass from — oo i to oo i along the 

 imaginary axis, except near the origin, where they follow a small semicircle to the 

 left and right of respectively. 



Applying the transformation (7) to the integrals (1), we obtain 



M, ^,\ ^ ^ /■ 1 ( A,, B,.,„ , I j^^-^-i- cos ,^ _ ^ry^^ j^.^,^^ ,^^^^ ^.^^^^,^j^ j^ 



Here, of course, D,„, Ai„j, Bi „„ G■^^„^ are to be expressed in terms of ji. [Cf. Art. 3 

 and Art. 5 after (2).) 



Let iS = ^ + ir)^ where ^ and >; are real. 



Consider a rectangle SABN formed by the lines ^=0, (tSN) ; ^ = N'7r+ + — , 



(AB); 2/ = N, (NB); y-=-N, (SA). 



It can easily be seen that sufficient conditions that each of the integrals in (8) 

 taken over the path SABN from S to N may tend to zero when N is increased 

 indefinitely are 2>2' and p<jii. Hence, by Cauchy's Theorem, 



each of the integrals (8) over the fourth path I „, 



= - 2iTi (sum of the residues of tlie integrand at the zeroes of D,,„ to the riglit of 0), | ' ^ ' 



the terms of this infinite series being so arranged that the moduli of the zeroes are in 

 non-descending order of magnitude. 



A series taken over these zeroes so arranged will always be denoted by 2- 



Again, each of the integrals (8) over the tliird path I 



= integral over the fourth path - 2iri (residue of the integrand at 0) 

 Hence, from (8), (9), (10), in a more precise notation, 



/^r, ' ^p. A e,„ >r, 1 i A, „. , B, ,„ , ) . /Jp -^(i^) cos , ,, 



- —^ ■ coefhcient of -^ in -— { '•'" '■"" } J,„^e a . miw - w ). 



(11) 



