PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 347 



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

 is 'normal.' Since this case depends on 6 as a parameter, P{d) is here an 

 abbreviation for P{B\b). 



A formula for P{d;b) = P{d) can be obtained by substituting P{R,d) 

 from (2.15) into (6.2) and executing the indicated integration, which can 

 be easily accomplished. The resulting formula is found to be 



2x(l — bcosld) 



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

 more general formulas (2.19) and (2.20) by setting R = co m (2.19) or 

 7? = in (2.20); also by adding (2.19) to (2.20) and then utilizing (1.10). 



In P{d) = P{d;b) it will evidently suffice to deal with values of 6 in the 

 first quadrant, because of symmetry of the scatter diagram. 



In P{d;b) it will suffice to deal with only positive values of b, as (6.3) 

 shows that changing b to —b has the same effect as changing 26 toir±26, 

 or 6 to 7r/2±0; that is, P{e;-b) = P{j/2±d;b). 



Fig. 6.1 gives curves of P{6;b), computed from (6.3), as function of 6 

 with b as parameter, for the ranges 0^^^90° and Q^b^l. 



The curves in Fig. 6.1 indicate that P{6;b) is a maximum at = 0° and 

 a minimum at 9 = 90°. These indications are verified by formula (6.3), 

 as this formula shows that: 



Max P{d;b) = P{0°;b) = ^ \/ H^ , (6.4) 



Thence 



Min P{e;b) = P{90°;b) = i- ^ j-qj] • (6-5) 



MmP{d;b)/MsixP(6;b) ^ (l-6)/(l + 6), (6.6) 



P{e;b)/MiixP{e;b) = P{d;b)/PiO°;b) = {l-b)/{l-b cos2d). (6.7) 



The curves in Fig. 6.1 indicate also that P{d;b) is independent of d in 

 the case b = 0. This is verified by formula (6.3), as this formula shows that 



P{6;0) = l/27r. (6.8) 



Thence (6.3) can be written 



P{d;b)/P{e;0) = (Vn^y2)/(l-6cos2^). (6.9) 



3" Beginning here, 6 will usually be expressed in degrees instead of radians, for prac- 

 tical convenience. 



