PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 345 



In the present paper, formulas (5.11) and (5.16) have been used for numer- 

 ical evaluation of QiR) = p{R'<R) and of Q*(R) = p{R'>R) by numerical 

 integration (employing 'Simpson's one-third rule'), aided by some of the 

 considerations set forth in Appendix D. However, only a moderate number 

 of values of these quantities have been thus evaluated — merely enough to 

 afford a fairly comprehensive check on Table I of my 1933 paper, by means 

 of a sample consisting of 60 values (about 26%) distributed in a somewhat 

 representative manner over that table. These new values of Q*{R) = 

 p{R'>R) = 1 — Q(R) are presented in Table 5.1 (at the end of this section) 

 in such a way as to facilitate comparison with the old values, namely those 

 in the 1933 paper. Thus, for any fixed value of R in Table 5.1, there are 

 two horizontal rows of computed values of Q*{R), the first row (top row) 

 coming from the 1933 paper, and the second row coming from the present 

 paper. The third row of each set of four rows gives the deviations of the 

 second row from the first row; and the fourth row expresses these deviations 

 as percentages of the values in the first row. 



In the first row of any set of four rows, any value represents Q*{R) = 

 pb{R'>R) obtained, in accordance with Eq. (22) of my 1933 paper, by 

 adding exp (— i?-) to pb^o{R'>R) given in Table I there. In the second 

 row of a set, any value represents Q*{R) = 1 — Q{R) as computed by for- 

 mula (5.11) or (5.16) of the present paper: more specifically, the values for 

 R = 0.2, 0.4, 0.6 and 0.8 were computed by (5.11); and the values for 

 R = \.6 and i? = 2 by (5.16), taking r = 1/1.6 = 0.625 and r - 1/2 = 0.5 

 respectively." 



In the 1933 paper, the values of Pb{R'>R) = Q*{R;b) for J = and for 

 b — I were omitted as being unnecessary there because their values could 

 be easily obtained from the simple exact formulas to which the general 

 formulas there reduced, ior b = and ^ = 1. Those reduced formulas 

 were the same as (5.23) and (5.24) of the present paper, except that (5.24) 

 gives Q(R;\) instead of giving Q*{R;\) = 1 - QiR;!). The values obtained 

 from these two formulas, exact to the number of significant figures here 

 retained, are given in Table 5.1 at the intersections of the first row of each 

 set of four rows with the columns 6 = and b = I. Therefore in these two 

 columns the deviations (in the third row of each set of four rows) are devia- 

 tions from exact values; the values in the second row of each set are, as 



use of such a formula for numerical computations, the expansion producing the con- 

 vergent series was carried far enough to insure that the remainder deiinite integral would 

 be relatively small, though usually not negligible; and then this remainder definite integral 

 was evaluated sufficiently accurately by numerical integration. 



2s In the work of numerical integration, ' Simpson's one-third rule' was employed for 

 R = 0.2, 0.4, 0.6, 0.8 and 2. For R = 1.6, so that r = 1/1.6 = 0.625, 'Simpson's one- 

 third rule' was employed up to r = 0.620, and the ' trapezoidal rule' from r = 0.620 to 

 r = 0.625. 



