426 Dr Bromwich. Symbolical methods 



(viii) Interpretation of e-«^ and - e-^^, x being positive. 



We have proved that 



1 



Thus, on integrating with respect to x, we see that 



f dx 

 e-ea; = _ -— Q-x-iiKt _^ const. 



jy^TTKt) 



2 [^ 

 = — e^^^dv + const. 



V'^ -I 



where z = ^xl\^{Kt). 



Now when x^O, e-'^'^ ^ 1, and so 



2 r^ 2 r^ 



e-3a; ==1 ^ e-'^'dv = —J- e-'"'dv. 



yTT ' V ■"■ •' s 



We have now a verification of the work ; for as x tends to + oo , 

 e-3* tends to (since the real part of q is positive) ; and this property 

 is seen to hold for the integral just found. 



The value of [11 q) e""^ is easily found by observing that 



/I \ 1 occ2 So-^a;* bq^x^ 



1 



2 ! 4 ! 6 ! 



Thus - e-«« = 2 , /(-) e-^V^-^* _ ^ f e'^'-dv, 



q 'S/ \7T/ v^.'s 



where as before z = |a;/y'(/c^). 



This is a form suited for numerica] work ; but a more useful result 



is the simpler formula 



1 x f"^ .dv 



_e-<ix^ e-v- ^ 



q V'^ -I s v^ 



For large values of z this integral can be converted into an 



IT P — ^" 



asymptotic series of which the first term is ^ — -; ^ . 



(ix) The only point remaining is to establish the asymptotic 

 property for the series in powers of q, used in §§ 2, 3 above; this 

 series is derived from the function 



and suppose that as a matter of algebraic expansion we obtain 



/(g) = ^0+^l? + ^2?'+"-. . 



