Comiwtation of the Functions Go(a'), Gri(.''), and ^ni:'-\/i)' 33 



' - - 1 1 71 - 1 . - 2 . 1 . 2 



and, if ii be any positive integer or zero, 



Kn{x) = YAn{x)-K{;r) (2), 



where 



An (a-) = ln{x)\ogx 



I « - 1 . 1 



Sr denoting l+^+l"^ + i, with the special case So = 0, and [•£ 



being log 2 



2. When a; i§ a real quantity, the function increases from zero 

 (or unity, when ti = 0) to an infinitely large quantity, as x passes from 

 zero to infinity, while ^nip'^ decreases numerically from infinity to 

 zero under the same circumstances. 



The values of the functions Ko(^;), Ki(.?) have been tabulated by the 

 present writer, and published in the ' Proceedings,' for values of x at 

 intervals of 0*1 from 0*1 to 12-0. The elements used in the calcula- 

 tion of the earlier half of these results are available for computing the 

 values of Ko(;«) and Ki(a') in some cases when is a complex quantity. 



If be a pure imaginary = z being a scalar, it is easily seen 

 that 



ln{x) = i«J.«(.~) (4), 



where Jft(4 is the ordinary Bessel's function of the first kind and 

 wth order. 



If also YJ/} denote Neumann's function of the wth order, and Qcn{z) 

 be a function defined by the relation 



G.(4 = E. J4^)-Y40) (5), 



it can be shown without much difficulty that 



K^o^) = ^«G4^)-^^•-+lJ„(^) (6). 



3. The numerical calculation of the functions Go(^') and Gi(.i;) can 

 be made to depend on that of 1^q{x) and Ki(a) for any values of x for 

 which the convergent series (1) and (3) are applicable. In doing 

 this it is necessary to calculate the elements of Jo(a;) and Ji(a'), and in- 

 cidentally to compute these functions. 



With the notation used in the writer's paper on the computation of 

 Ko(.^') and Ki(.');), it is easily seen that 



VOL. LXVL D 



