MATHEMATICAL AX A IAS IS OF RANDOM^NOISE 115 



The 2^1 may be expressed in terms of the complete eUiptic functions E and 

 A' of modulus v^'~x~^''^. Thus 



.F,(-J,-J;l;>:)=*£-?('l-0^. 



.V / TT X 



.F.(^,i;l;l) = lK 



(3.10-28) 



and the higher terms may be computed from the recurrence relation 

 (3.10-29). The tirst term, ^ = 0, in (4.2-11) gives Idc when the noise is 

 absent. 



The mean square value of 1 1( is 



2 2 



la = %R' = -. [2.Ao + P' + Q'] (4.2-14) 



From this expression and our expression for Idc , the rms value of the low- 

 frequency current, If/ , excluding the d.c, may be computed. For example, 

 when the noise is small, 



+ ,,.(._.,,(_,, _.,;05)_ 



The term independent of xj/q gives the mean square low frequency current 

 in the absence of noise. As Q goes to zero (4.2-15) approaches the leading 

 term in (4.2-7), as it should. When P = Q our formula breaks down and 

 it appears that we need the asymptotic behavior of "" 



In view of the questionable nature of the derivation given in Section 3.10 

 of equations (4.2-9) and (4.2-11) it was thought that a numerical check on 

 their equivalence would be worth while. Accordingly, the values x = 4, 

 y = 3 were used in the second series of (4.2-9). It was found that the 

 largest term (about 130) in the summation occurred at ^ = 11. In all, 24 

 terms were taken. The result obtained was 



^ = 2.5502 



V2;/', 



« See \V. R. Bennett, B.S.TJ., Vol. 12 (1933), 228-243. 



^5 This mav be done bv the method given l)v W. B. Ford, Asymptotic Developments, 

 Univ. of Mich. Press (193'6), Chap. VI. 



