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BELL SYSTEM TECHNICAL JOURNAL 



tions can be executed, giving the two previously obtained formulas (3.4) 

 and (2.19) for P(i?) = P(R;b) and P{d\ <R) respectively. When these 

 are substituted into (5.4), there result two types of single-integral formulas 

 for Q{R): A prirrary type, involving an indicated integration as to R; and 

 a secondary tyj^e, involving an indicated integration as to 6. Formulas 

 of these two types for Q{R) will now be derived. 



An integral formula of the primary type for Q{R) = Q{R;b) can be ob- 

 tained by substituting P(R) = P(.R',b) from (3.4) into the first integral in 

 (5.4), giving 



Q{R) = 2 [ 

 Jo 



X 



Vl - b- 



exp 



d\. (5.11) 



This can also be obtained as a particular case of the more general formula 

 (2.21) by setting d = 2ir in the upper limit of integration and then apply- 

 ing {i.2,). 



In (5.11), X is used instead of R as the integration variable in order to 

 avoid any possible confusion wdth R as an integration limit. Thus the 

 integrand is a function of X with 6 as a parameter. Evidently Q{R;b) — 

 Q(R;—b). Formula (5.11) is evidently suitable for evaluation of ()(i?) by 

 numerical integration.-^ 



By suitably changing the variable in (5.11), we arrive at the following 

 various additional formulas, which, though equivalent to (5.11), are very 

 different as regards the integrand and the limits of integration. As previ- 

 ously, L denotes R-/{\ — b-). 



Q{R) 



1 



Vl 



K2 Jo 



exp 



■X 



1 



b' 



dX, 



Q{R) = Vl - b^ I exp(-X) h{b\) dX, 

 Jq 



Q(R) = LVi - b'~ I exp(-LX) h{bLX) dX, 

 Jo 



J PYn(—l 



(5.12) 

 (5.13) 

 (5.14) 

 (5.15) 



Q{R) = Vl - ^'M h{b log X) r/X. 



Jexp{-L) 



These four additional formulas are of some theoretical interest, but ap- 

 parently they are less suitable than (5.11) for numerical integration with 

 respect to R. A formula differing slightly from (5.11) could evidently be 

 obtained by taking X/-\/l — 6^ as a new variable, and hence R/y/l — b^ 

 as the upper limit of integration. 



Corresponding formulas for Q*(R) = Q*{R;b) can of course be obtained 

 from the preceding formulas (5.11) to (5.15) inclusive for Q{R) = Q{R;b) 



^* In this connection, Appendix D may be of interest. 



