PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 



335 



The cases b — Q and b = \ will be dealt with first, and then the general 

 case {b = b). 



For the case J = it is easily found by differentiating (3.20) that P{R;b) = 

 P{R; 0) is a maximum Sit R — 1/ \^2 = 0.7071 and hence that its maximum 

 value is \/2exp (—1/2) = 0.8578, agreeing with the curve for 6 = in 

 Fig. 3.1. 



For the case b = I, which is a limiting particular case, the maximum 

 value of P(R;b) — P(i?;l) apparently cannot be found driectly and simply, 

 as will be realized from the preceding discussion of this case. Near the 

 end of Appendix B, it is shown that the maximum value of P{R;\) occurs at 

 7? = (as would be expected) and is equal to 0.9376. This is the maximum 

 value of the part P(0;1 of P(R;1). The remaining part of P(R;l), namely 

 P'{R;1), whose formula is (3.21), is seen from direct inspection of that 

 formula to have a right-hand maximum value a.t R = 0+, whence this 

 m-aximum value is 2/v 2ir = 0.7979. 



For the general case when b has any fixed value within its possible positive 

 range (O^i^ 1), it is apparently not possible to obtain an explicit expression 

 (in closed form) either for the value of R at which P{R;b) has its maximum 

 value or for the maximum value of P(R;b); and hence it is not possible to 

 make explicit computations of these quantities for use in plotting curves of 

 them, versus b, of which they will evidently be functions. However, as 

 shown in Appendix B, these desired curves can be exactly computed, in an 

 indirect manner, by temporarily taking b as the dependent variable and 

 taking T, defined by (3.6), as an intermediate independent variable. For 

 let Re denote the critical value of R, that is, the value of R at which PiR;h) 

 has its maximum value; and let Tc denote the corresponding value of T, 

 whence, by (3.6), 



Tc= bRl/il-b'). 



(3.22) 



uj 0.8 



I 



UJ 



O 

 5 0.4 



gO.2 



»- 

 o 



z 



2 



0.1 



0.2 0.3 0.4 0.5 0.6 0.7 



PARAMETER, b 



0.8 0.9 1.0 



Fig. 3.3 — Functions relating to the maxima of the distribution function for the modulus. 



