PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 351 



by the general formula (7.3) together with the auxiliary general formulas 

 (7.4) and (7.5), beginning with the two latter. 



It will be convenient to dispose of (7.5) before dealing with (7.4), as (7.5) 

 turns out to be the less useful. For when P{R,d) given by (2.16) is sub- 

 stituted into (7.5), the indicated integration cannot be executed in general, 

 as (7.5) becomes (2.18), wherin the indicated integration can be executed 

 only for certain special values of the integration limit 6 — by means of the 

 special Bessel function formula (3.i). 



When PiR,d) given by (2.15), which is equivalent to (2.16) used above, 

 is substituted into (7.4), it is found that the indica^^ed integration can be 

 executed, giving the previously obtained formula (6.3) for F{d) = P{&',b). 



A 0-integral formula for Q{d) = Q{Q\h) can be obtained by substituting 

 P{e) = P{d;b) from (6.3) into the first integral in (7.3), giving 



Vi - 6- f' dd Vi - 62 r'" d<f> 



^^ ' ' 27r h \ - b cos 28 47r h 1 



b cos 



(7.6) 



This formula can also be obtained as a particular case of the more general 

 formulas (2.22) and (2.24) by setting i? = ^ in (2.22) or i? = in (2.24); 

 also by adding (2.22) to (2.24) and then utilizing (1.11). 



The integral in (7.6) is of well-known form, and the indicated integration 

 can be executed, yielding the following two equivalent formulas for Q{d\h): 



27r 



tan 



-1 1 cos 2^ - 6 n 

 ''' L i-6cos2d r 



In Q{d;b) it will evidently suffice to deal with values of 6 in the first quad- 

 rant, because of symmetry of the scatter diagram, and the resulting fact 

 that Q{n 90°) = n/i, where n = 1, 2, 3 or 4. 



In Q{6;b) it will suffice to deal with positive values of b, as (7.7) shows 

 that^i 



Q{e; -b) 



I-e i±M 



Fig. 7.1 gives curves of Q{d;b) = Q{6) computed from (7.7), as function 

 of d with b as parameter, for the ranges 0^0^90° and 0^6^ 1. 



Consideration of the scatter diagram of IF or of its equiprobability curves, 

 which are concentric similar ellipses, affords several partial checks on the 

 curves in Fig. 7.1 and on formula (7.7) from which they were plotted. 



^1 This relation can also be derived geometrically from the fact that the scatter dia- 

 gram for —b is obtainable by merely rotating that for b through 90°, as shown by (2.6), 

 or (2.7) and (2.8), or (2.11). 



