328 BELL S YSTEM TECH NIC A L JOURNA L 



range can be reduced to 0-to-7r/2 provided the resulting integral is mul- 

 tiplied by q; that is, 



/«5(7r/2) ^jr/2 



/ F{e)(W = q / F{e)dd, (2.25) 



Jo •'0 



because of symmetry of the scatter diagram. 



3. The Distribution Function for the Modulus 



The distribution function P{R | dv2) = F{R) for the modulus R of any 

 complex variate IT = R(cos 6 + / sin 0) is defined by equation (1.10) on 

 setting p = R, a = 9, ffi = 6] — and (r2 — 62 — 2ir; thus 



P{R)dR = p(R<R'<R+dR, (xe'KlTv). (3.1) 



An integral formula for F(R) is immediately obtainable from (1.6) by 

 setting p = R, o — 6, (Ti = ai = 61 = and 04 = a^ ~ S2 = 2x; thus 



F{R) = [ F{R,d) do. (3.2) 



Jo 



The rest of this section deals with the case where \V = R(cos 6 + / sin 6) 

 is 'normal.' Since this case depends on i as a parameter, F(R) is here an 

 abbreviation for F{R;h). A formula for F{R;b) can be obtained by sub- 

 stituting F{R, 6) from (2.15) into (3.2) and executing the indicated integra- 

 tion by means of the known Bessel function formula 



i: 



exp(r} cos \f/) dip = 7r/o(r/), (3.3) 



/o( ) being the so-called 'modified Bessel function of the first kind,' of 

 order zero.^'- The resulting formula is found to be^^ 



2R 



.1 - d^Ti 



bR^ 



- b' 



(3.4) 



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

 (2.18) by setting 6 — 2t in the upper limit of integration and then apply- 

 ing (3.3). 



In F(R;b) it will suffice to deal with positive values of b, that is, with 

 U^6^1, as (3.4) shows that F(R; -b) = F{R;b). 



12 It may be recalled that /o(c) = /o(/-), and in general that /„(;) = i-"Jn{i~). 



In the list of references on Bessel functions, on the last page of this paper, the 'modified 

 Bessel function' is treated in Ref. 2, p. 20; Ref. 3, p. 102; Ref. 4, p. 41; Ref. 1, p. 77. 



Regarding formula {3.3), see Ref. 1, p. 181, Eq. (4), i. = 0; Ref. 1, p. 19, Eq. (9), fourth 

 expression, p = 0; Ref. 2, p. 46, Eq. (10), n = 0; Ref. 3, p. 164, Eq. 103, n = 0. 



^' This formula was given in its cumulative forms, / P{R; b)dR, as fornuilas (Sl-.A) 



and (53-A) of the unpublished .\ppendix A to my 1933 paper. 



