PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 



343 



by merely changing the integration Hmits correspondingly — for instance, 

 in (5.11), from 0, i? to i?, oo ; in (5.13), from 0, L to L, ^ \ and so on. How- 

 ever, the first four formulas for Q*{K) so obtained would suffer .the disad- 

 vantage of each having an infinite limit of integration, rendering those 

 formulas unsatisfactory for numerical integration purposes. This difficulty 

 can be avoided by making the substitution R = \/r in each of those formulas 

 for Q*{R). The resulting formulas are the following five, corresponding to 

 (5.11) to (5.15) respectively :24 



()*(i?) 



Vi 



Q*{R) = 



Q*(R) 



VT 



2 rj_ 



_ /i2 Jo X^ 



b' Jo 



X2 



b/}C 



1 - F 

 b/\ 



d\, (5.16) 



1 



]jx, 



X' 



Vl - b^ [ 



exp 



exp 



dX, 



dX, 



expi—L) 



' Io{b log X) dX 



a 



(5.17) 



(5.18) 



(5.19) 



(5.20) 



As a check on (5.16), it is obtainable from (4.2) by integrating the latter 

 as to r. 



For purposes of evaluation by numerical integration, formula^ (5.11) 

 to (5.15) inclusive may evidently differ greatly as regards the amount of 

 labor involved and the nurrerical precision practically attainable. In 

 each of these formulas except (5.14) the integrand contains only one param- 

 eter, b, while the integration range involves either R or L = R-/{\ — b-). 

 In (5.14) the integrand contains two independent parameters, b and L, 

 while the integration range is a mere constant, 0-to-l. Similar statements 

 apply to formulas (5.16) to (5.20) inclusive. 



A partial check on any formula for Q(R) can be applied by setting R = <x> ^ 

 since Q(°o) should be equal to unity (representing certainty). If, for 

 instance, this procedure is applied to formula (5.13), the right side is found 

 to reduce to unity by aid of the known relation" 



exp (-^X) JoiBX) dX = 



} 



Jo 



1 



(5.21) 



together with Io{BX) — jQ(iBX). 



An integral formula of the secondary type for Q*(R) = Q*{R;b) can be 

 obtained by substituting (2.20) into the last integral in (5.8), utilizing (2.25), 



» Ref. 1, p. 384, Eq. (1); Ref. 2, p. 65, Eq. (2); Ref. 4, p. 58, Eq. (4.5). 



