CON FORM A L TRA NSFORMA TION 



127 



interchanging the order of integration, and integrating with respect to v 

 and j'o . Assuming 5 — /- > 0, the integral is then evaluated by closing the 

 [lath of integration by an infinite semicircle in the upper half plane and 

 rakulating the residues of the integrand at the poles ib, c + im, —c + i^l•. 



/« 1^ ii(»— r)— tc(r-(-«) 



« 7r(52 + x-')W 



Ab^inie^ 



dx 



= AdjjLin 

 + 



l' + ix+ cy][m' -\- {x - cy] 



-(6+ic)s+(«-ic)r 



LMm- + (c + i8y][m^ + {c - idY] 



— m»+(m— 2 »c)r 



(A2-6) 



m[8^ + (f + imYln^ + (2c + iniY] 



jir—(.ft+2ic)s 

 + 



m[52 + (c - i^iYlm'' + (2c - !>)']_ 



Substituting special values for the parameters gives the results required 

 in the text. Thus, 



J(fn, m, k,y;t,t) = g-2«>< /(^^ ,„^ ^, 7; 0, 0) 



J(m, m, k,y;—t, t) = e^'*' J{m, m, y^, 7; 0, 2/) 



/(w, w, ^, 7; -/, 0) - e-'*' J(m, ni, yfe, 7; 0, /) (A2-7) 



w s n n\ 2w(6 + 2m) 



J{m, m, c, 8; 0, 0) - — — — ,..,,. ' ,.., 

 (c2 + m2)[c2 + (w + 5)2] 



which hold irrespective of any relations between the parameters. The 

 derivation of the last result is simplified by setting a = c + im, a = c — im 

 and factoring the denominators in (A2-6) so as to obtain terms of the 

 form a ± id, a ± id. 



WTien 7' = w^ — k- considerable simplification is possible and we obtain 



1 m ~\ 



J{m, m, ^, 7;0, 0) = 



Jim, m, ^; 7, 0, /) = 



^2 1^^ ^2 _|_ ^2^ 



ye 



k"- 



e '' _ e "" (w cos kt — k sin kl) 



_ 7 vi^ -\r k^ 



(A2-8) 



If we put M = w — 1, w = w + 1, and set 6- == n- — 1 

 where k- = 1 + c^, (A2-6) yields, after some reduction, 



J{n - 1, w + l,c,5;0,0) = 

 + 



n — \ 



+ 



(« - 1)5 



(en + i5)2 2(« + 1)(1 - icy{n - ic) 



{n +J)5_ 

 2(w - 1)(1 - ic)2(7r+ ic) 



(A2-9) 



kXw2 - 1) 



+ 



nb[2hi + c') - k^(h'' - 1)] 

 K^n" - l)(w2 + c2) 



