PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 357 



h(T,)/hiT,) = 1 - l/2r, = 1. (C8) 



The first step toward computing the curves in Fig. 4.3 is to find approxi- 

 mately the Tc-range corresponding to the 6-range, O^b^l. This is done 

 in the course of the next four paragraphs. 



When b = 0, equation (C6) shows that Tc/b = 3/2 and hence that 

 Tc = 0; or, what is equivalent, b/Tc = 2/3 and hence l/Tc = oo (since 

 b^ 0). 



When 6 = 1, Tc = CO, as can be easily verified from equation (C4), 

 (C5) or (C6) by utilizing (C8). 



Thus, from the two preceding paragraphs, it is seen that, when b ranges 

 from to 1, b/Tc ranges from 2/3 to 0; Tc/b from 3/2 to cc ; and Tc from 



to 00. 



Since Tc = "^ when b = 1, the choosing of a set of finite values of Tc 

 will necessitate an approximate formula for computing Tc for values of 

 b nearly equal to 1 , which means for very large values of T. Such a formula 

 is easily obtainable from (C5) by utiUzing the approximation 1 — 1/2 Tc 

 in (C8), whereby it is found that, for large Tc, 



Tc = b/{l-b), (C9) b/Tc = l-^*. (CIO) 



As examples, these approximate formulas give: When b = 0.99, Tc ~ 99, 

 b/Tc = 0.01; when b = 0.9, Tc = 9, b/Tc = 0.1. It will be found that 

 even in the second example the results are pretty good approximations. 



The curves in Fig. 4.3 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 about 100 (the latter figure corresponding approximately 

 to b = 0.99). Second, for each such chosen Tc the right side of (C5) is 

 computed, thereby evaluating Tc/b and also 1/(1 — 6-)^^, these two quan- 

 tities being equal by (C2). Third, the corresponding value of b is found 

 by dividing Tc by Tc/b; less easily, it could be found by substituting Tc 

 into (C4). Fourth, from Tc/b the value of \/Tc/b is found, and thereby 

 the value of I/tc s/l — b^ and thence Tc . Finally, Max P{r;b) is computed 

 by inserting the critical values into any of the (equivalent) formulas for 

 P{r;b), namely (4.2), (4.3) or (4.4). 



APPENDIX D 



Some Simple General Considerations Regarding the Evaluation of 

 Cumulative Distribution Functions by Numerical Integration 



This appendix gives some simple general considerations and relations 

 that may sometimes facilitate and render more accurate the evaluation 

 of cumulative distribution functions by numerical integration. 



