PROBABILITY FUNCTIONS FOR COMPLEX VARIATi: 32^ 



It will often be advantageous to express P^; 6 in terms of b and one or 

 the other of the auxiliary variables L and T defined by the equations 



^ = r^2' (3-5) ^ = ^^ = 1^2- (^-6) 



Formula (3.4) thereby becomes, respectively, 



P{R;b) = 2VLexp{-L)h{bL), (3.7) 



P(R;b) = 2 y^l exp[^j h{T). (3.8) 



Formula (3.8) will often be preferable to (3.7) because the argument of 

 the Bessel function in (3.8) is a single quantity, T. 



Because tables of /o(-V) are much less easily interpolated than tables of 

 Mo(X) defined by the equation 



Mo(X) = exp(-X)h(X), (3.9) 



extensive tables of wiiich have beeo published," it is natural, at least for 

 computational purposes, to write (3.4) in the form 



2R r -R' 1 



Vl - b^ 



Mo 



• bR' 

 1 -b' 



(3.10) 



For use in equation (3.16), it is convenient to define here a function 

 Mi(X) by the equation 



M,(X) = exp(-A')/i(X), (3.11) 



corresponding to (3.9) defining Mo{X). Mi(X) has the similar property 

 that it is much more easily interpolated than is Ii(X); and extensive tables 

 of Ml (A') are constituent parts of the tables in Ref. 1 and Ref. 6. 



The quantity bR-/{l — b') = T, which occurs in (3.4) and (3.8) as the 

 argument of /o( ), and in (3.10) as the argument of Mo{ ), evidently 

 ranges from to co when R ranges from to co and also when b ranges 

 from to 1. Formula (3.10) is suitable for computational purposes for all 

 values of the above-mentioned argument bR~/(l — b'^) = T not exceeding 

 the largest values of X in the above-cited tables in Ref. 1 and Ref. 6. For 

 larger values of the argument, and partiularly for dealing with the limiting 



i-* Ref. 1, Table II (p. 698-713), for X = to 16 by .02. Ref. 6, Table VIII (p. 272- 

 283), for A^ = 5 to 10 by .01, and 10 to 20 by 0.1. Each of these references conveniently 

 includes a table of exp(A^) whereby values of /o(A') can be readily and accurately evalu- 

 ated if desired. Values of /o(A') so obtained would enable formulas (3.4), (3.7) and (3.8) 

 of the present paper to be used with high accuracy without any difficult interpolations, 

 since the table of exp(A'') is easily interpolated by utilizing the identity exp(A'i -)- A'2) = 

 exp(Ai) exp(A^2). 



