PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 337 



the result by 1/r, in accordance with (4.1). Evidently P{r; — b) 

 = P(r;b). 

 By means of (4.1), formulas (3.7) and (3.8) give, respectively, 



P{r;b) = 2(l-b^')L"'exp{-L)Io{bL), (4.3) 



P(r;b) = 2(1 - b') l^lj exp|^^j /o(T), (4.4) 



wherein L and T are defined by (3.5) and (3.6) respectively, but will now 

 be written in the equivalent forms 



i = (T^- (4.5) r = Si=_^_A_, (4.6) 



which are evidently more suitable for the present section. 



A few particular cases that are especially important will be dealt with 

 in the following brief paragraph, ending with equation (4.8). 



For the two extreme values of r, namely and oc , P{r;b) is zero for all 

 values of b in the b- range (0^6^ 1). 



When b = 0, 



When b = I, 



P{r-b) = P{r;0) = ^^expf-ij. (4.7) 



^f ] 



Pir;b) = P{r;\) = ^^ ;;;, exp| ~, \. (4.8) 



Fig. 4.1 gives curves of P(r;b), with the variable r ranging continuously 

 from to 1.4 and the parameter b ranging by steps from to 1; however, 

 in the r-range where r is less than about 0.6, alternate curves had to be 

 omitted to avoid undue crowding. Fig. 4.2 gives an enlargement of the 

 section betwen r = 0.2 and r = 0.5, and includes therein the curves that 

 had to be omitted from Fig. 4.1. 



In Fig. 4.1 it will be noted that with the scale there used for P(r;b) the 

 values of P(r;b) are too small to be even detectable for values of r less 

 than about 0.25. Even in the enlargement supplied by Fig. 4.2, the values 

 of P{r;b) are not detectable for r less than about 0.2. 



The curves of P{r;b) in Figs. 4.1 and 4.2 would have had to be computed 

 from the lengthy formula (4.2) — or its equivalents — except for the fact 

 that curves of P{R;b) had already been computed in the preceding section 

 of the paper. The last circumstance enabled the P{r;b) curves to be 

 obtained from the P{R;b) curves by means of the very simple relation (4.1). 



It will be observed that each curve of P{r;b) [Fig. 4.1] has a maximum 



