PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 355 



On account of (B2), the right sides of (B5) and (B6) are equal not only to 

 Tc/b but also to i?c/a-6"). 



Since the utilization of formulas (B4), (B5) and (B6) for computing the 

 curves in Fig. 2).di will involve taking Tc as the independent variable and 

 assigning to it a set of chosen numerical values, the natural first step is to 

 find approximately the range of Tc corresponding to the 6-range, O^^^l, 

 in order to be able to choose only useful values of Tc. This step will be 

 taken in the next paragraph. 



Equation (B6) shows that Tc/b = 1/2 when 6 = 0, and hence that Tc ~ 

 when b = 0; and this last is verified by (B4). The other end-value of the 

 Tc-range, namely the value of Tc iox b = 1, cannot be found explicitly 

 and exactly. However, rough values of limits between which it must lie 

 can be found fairly easily as follows: To begin with, each of the equations 

 (B5) and (B6) shows that Tc^ b/2, for all values of b in O^b^l; in par- 

 ticular, Tc > 1/2 when b = I. An upper limit for Tc for any value of 

 b can be found from (B5) by utilizing the power series expressions for 

 Ii{Tc) and lo(Tc), whereby it is found that 



^ -H^, (B7) where H =^ I - %' < 1. (B8) 



Io{I c) ^ o 



On substituting (B7) into (B5) and then solving for Tc in terms of b and 

 H, it is found that 



Tc = b/(l + Vl - Hb'). (B9) 



On account of (B8), (B9) shows that 



Tc < b/{l + Vn^2), (BIO) 



whence, in particular, Tc<l when b = 1. By successive approximation 

 or otherwise, it can now be rather quickly found that, when b — 1, Tc = 

 0.79 (to two significant figures).^^ 



From the preceding paragraph, it is seen that, when b ranges from to 1, 

 Tc ranges from to about 0.79; Tc/b ranges from 0.5 to about 0.79; and, 

 on account of (B2), Re ranges from ^/O.S = 0.707 down to 0. 



The curves in Fig. 3.3 are constructed with the aid of the formulas and 

 methods of this appendix as follows: First, a set of values of Tc is chosen, 

 ranging from to 0.79 and slightly larger. Second, for each such chosen 

 Tc the right side of (B5) is computed, thereby evaluating Tc/b and also 

 Rc/{l — b^), these two quantities being equal by (B2). Third, the cor- 

 responding value of b is found by dividing Tc by Tc/b; less easily, it could 



^' Because of the special importance oi b = 1 in other connections, Tc for b = I was 

 later evaluated to four significant figures and found to be Tc = 0.7900; thence, by sub- 

 stituting this value of T into (3.8), along with b = 1, it was found that Max. P{R;l) 

 = 0.9376, which occurs at R = Re = 0,hy (B2). 



