PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 327 



where 



L= Ry{\-b-'). (2.17) 



In P{R,d) it will evidently suffice to deal with values of 6 in the first 

 quadrant, because of symmetry of the scatter diagram. 



The fact that P(R,6) depends on 6 as a parameter when W is 'norma]' 

 may be indicated explicitly by employing the fuller symbol P{R,d;b) 

 when desired; thus the former symbol is here an abbreviation for the latter. 



In P{R,d) = P(R, 6; b) it will suffice to deal with only positive values of 

 b, that is, with O^b^l (whereas the total possible range of b is — l^^^l). 

 For (2.15) shows that changing b to —b has the same effect as changing 2d 

 to 7r±2e, or d to T/2±d; that is, P{R,d; -b) ^ P(R, ir/2±d; b). 



Seven formulas which will find considerable use subsequently are obtain- 

 able from the integrals corresponding to equations (1.13) to (1.16), by setting 

 p = R and a = 6 or else p = 6 and c = R, whichever is appropriate, and 

 thereafter substituting for P{R,6) the expression given by (2.16), and 

 lastly executing the indicated integrations wherever they appear possible." 

 The resulting formulas are as follows: 



P(R \ < d) = y^ exp(-Z) / expibL cos 26) dd, (2.18) 



T Jo 



(2.19) 



P{e \ < R) = ^^ ~ ^' 1 - exp[-i:(l - b cos 2d)] 

 2ir I — b cos 20 



P(e \> R) = ^^ ~ ^' exp[-£(l - b cos 29)] 

 2t 1 — b cos 26 



(2.20) 

 dR (2.21) 



Q{< R, < 6) = - [ \ \/l exp(-L) [ exp{bL cos 26) dd 



TT Jo L "^O 



Vnili r" 1 - exp[-I(l - b cos 26)] 



~27~ io 1 - b cos 26 ^^' ^^-^^^ 



Q(> R, < 6) =- I VL exp(-L) j exp {bL cos 26) dd dR (2.23) 



Ztt Jo 1 — b cos 26 



Formulas (2.21) to (2.24) are obtainable also by substituting (2.18) to 

 (2.20) into the appropriate particular forms of (1.9). 

 When a ^-range of integration is 0-to-5(7r/2), where q = 1, 2, 3 or 4, this 



" Except that in (2.22) the part 1/(1 — b cos 26) is integrable, as found in Sec. 7, 

 equations (7.6) and (7.7). 



