PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 



341 



small probability of exceeding a preassigned rather large value of R than to 

 deal with the corresponding rather large probability (nearly equal to 

 unity) of being less than the preassigned value of R. 



A 'double integral' for Q{R), in the form of two 'repeated integrals,' 

 can be written down directly by inspection of the p{ ) expression in 

 (5.1) or by specialization of (1.8); thus 



' / P{R,d) de clR = / P{R,d) dR dd. (5.3) 



Evidently these can be written formally as two 'single integrals,' 

 Q{R) = / P{R) dR = / P{e\ < R) dd, 



Jo Jn 



(5.4) 



by means of the distribution functions P(R) = P(R | ^i.) and P{e \ <R) 

 given by the formulas 



P{R) = [ P{R,e) dd, (5.5) P{d\<R) = [ P{R,d)dR. (5.6) 

 Jo Jo 



(5.5) is the same as (3.2). (5.6) is a special case of (1.6), and the left side 

 of (5.6) is a special case of P{p \ <a) detined by (1.13). 



Similarly, from (5.2), we arrive at the following formulas corresponding 

 to (5.3), (5.4), (5.5), and (5.6) respectively: 



dd, 



Q*{R) = ■ / PiR,d) dd dR = / P(R,d) dR 



J R \_Jo J •'oL'^'' 



^00 /.27r 



Q*(R) = p{R) dR = P{d\ > R) dd, 



J R Jo 



P{d\ > R) = f P{R,d) dR. 



J R 



P{R) = [ P{R,d) dd, 

 Jo 



(5.9) 



(5.7) 



(5.8) 



(5.10) 



The rest of this section deals with the case where W = i?(cos d -\- i sin 6) 

 is 'normal.'-^ Since this case depends on 6 as a parameter, Q{R) and Q*(R) 

 are here abbreviations for Q{R;b) and Q*{R;b) respectively. 



A natural and convenient way for deriving formulas for Q{R) is afforded 

 by the general formula (5.4) together with the auxiliary general formulas 

 (5.5) and (5.6), beginning with the two latter. 



For the 'normal' case, P{R,d) is given by (2.15). When this is sub- 

 stituted into (5.5) and (5.6), it is found that each of the indicated integra- 



23 For the 'normal' case, the cumulative distribution function was treated in a very 

 different manner in my 1933 paper and its unpublished Appendix A. That paper included 

 applications to two important practical problems, and its unpublished Appendix C treated 

 a third such problem. (The unpublished appendices, A, B and C, are mentioned in foot- 

 note 3 of the 1933 paper.) 



