PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 359 



R , so that (D3) could be evaluated, it might be that (D4) would result 

 in greater accuracy; this would presumably be the case when p{R' <R ) 

 is nearly equal to unity. 

 Evidently an evaluation of 



P(R'>R') = P(R)dR (D5) 



directly by numerical integration would be less satisfactory than the evalua- 

 tion of p{R' <R ) in the preceding paragraph. For, due to the presence 

 of the infinite limit in the integral in (D5), the plot of P{R) would have to 

 be carried to a large enough value of R so that the integral from there to «^ 

 would be known to be negligible. This diflficulty can be avoided by start- 

 ing with the relation 



piR'>R') = 1 - piR'KR") (D6) 



and substituting therein the value of p{R' <R ) given by (D3) or (D4), 

 resulting respectively in the following two formulas: 



p(R'>R') = I - P(R)dR, (D7) 



P(R'>R') = p(r'<r') = / P{r)dr, (D8) 



the integrals in which are evidently suitable for evaluation by numerical 

 integration, none of the integration limits being infinite. If p{R'>R'^) 

 is small compared to unity, (D8) would presumably be more accurate 

 (percentagewise) than (D7). If the plot of P(R) is not sufficiently exten- 

 sive to include R , (D7) evidently could not be used; but, instead, (D8) 

 could be used if the plot of P{r) were sufficiently extensive to include r . 



References on Bessel Functions 



1. Watson, "Theory of Bessel Functions," 1st. Ed., 1922; or 2nd Ed., 1944. 



2. Gray, Mathews and MacRobert, "Bessel Functions," 2nd Ed., 1922. 



3. McLachlan, "Bessel Functions for Engineers," 1934. 



4. Bowman, "Introduction to Bessel Functions," 1938. 



5. Whittaker and Watson, "Modern Analysis," 2nd Ed., 1915. 



6. "British Association Mathematical Tables," Vol. VI: Bessel Functions, Part I, 1937. 



7. Anding, "Sechsstellige Tafeln der Bessel'schen Funktionen imaginaren Arguments," 



1911 (mentioned on p. 657 of Ref. 1). 



