PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 333 



and includes therein curves for a considerable number of additional values 

 of b between 0.9 and 1 so chosen as to show clearly how, with b increasing 

 toward 1, the curves approach the curve for 5 = 1 as a limiting particular 

 curve; or, conversely, how the curve iov b — 1 constitutes a limiting par- 

 ticular curve — which, incidentally, will be found to be a natural and con- 

 venient reference curve. This curve, iov b = 1, will be considered more 

 fully a little further on, because it is a limiting particular curve and be- 

 cause of its resulting peculiarity at i? = 0, the curve iov b = 1 having at 

 R = a. projection, or spur, situated in the P{R;b) axis and extending from 

 0.7979 to 0.9376 therein (as shown a little further on). 



The formulas and curves iov b = and b = 1, being of especial interest 

 and importance, will be considered before the remaining curves of the set. 



For the case b = 0, formula (3.4) evidently reduces immediately to 



F{R;0) = 2Rexp(-R^). (3.20) 



This case, 6 = 0, is that degenerate particular case in which the equiprob- 

 ability curves in the scatter diagram of the complex variate, instead of 

 being ellipses (concentric), are merely circles, as noted in my 1933 paper, 

 near the bottom of p. 44 thereof (p. 10 of reprint). 



For the case b = 1, the formula for the entire curve of P{R; b) = P(R;1), 

 except only the part at R = 0, can be obtained by merely setting b = I 

 in^^ (3.12) as this, on account of (3.15), thereby reduces immediately to 



2_ 

 V2^ 



P'iR;\) denoting the value of P{R;b) when b = 1 but i? 5^ 0, the restriction 

 i? 5^ being necessary because the quantity R~/(l — b^) in (3.12) — and in 

 (3.4) — does not have a definite value when b — 1 if i? = 0. Thus, in Figs. 

 3.1 and 3.2, the curve of P'(R;\) is that part of the curve iov b = 1 which 

 does not include any point in the P{R; b) axis (where R — 0) but extends 

 rightward from that axis toward R = -f 00. The curve of P'{R;l) is the 

 'effective' part of the curve of P{R;l), in the sense that the area under the 

 former is equal to that under the latter, since the part of the curve of 

 P{R;l) at R = can have no area under it. 



P(0;1) denoting (by convention) the value, or values, of P{R;b) when 

 R — and b — 1, that is, the value, or values, of P{R'S) when R = 0, it 

 is seen, from consideration of the curves of P{R;b) in Figs. 3.1 and 3.2 when 

 b approaches 1 and ultimately becomes equal to 1, that the curve of P(0;1) 

 consists of all points in the vertical straight line segment extending upward 

 in the PiR;b) axis, from the origin to a height 0.9376 [= Max P(i?;l)],20 



'^ Use of (3.12) instead of (3.4), which is transformable into (3.12), avoids the indefinite 

 expression « .0.^ which would result directly from setting 6 = 1 in (3.4). 



^^ As shown near the end of Appendix B, MaxP(^;l) is situated at /? = and is 

 equal to 0.9376. 



^'(^; 1) = r7^exp|^-f]> (R ^ 0)> (3.21) 



