MATHEMATICAL AXALYSIS OF RAX DOM XOISE 139 



double integral for ^(t). If there were some easy way to evaluate this in- 

 tegral then everything would be fine. Unfortunately, no simj^le method of 

 evaluation has yet been found. However, one method is available which is 

 closely related to the direct method used by Bennett. It is based on the 

 expansion 



gs{u, V, t) = Ja{P\/ifi + v^ -\- 2uv cos pr) 



00 



= Z en{-TJn{Pu)Jn{Pv) C05 npr (4.9-4) 



eo = 1, en — 2 for ii > 1 

 This expansion enables us to write the troublesome terms in (4.9-3) as 

 e~^^"Vo(P\/«2 + V' -{- 2uv cos pr) 



= 2^ 2^ { — ) en cos npT JniPu)Jn(Pv) 



n=0 k=0 kI 



The \drtue of this double sum is that it simplifies the integration. Thus, 

 putting it in (4.9-3) and setting 



•n+k r> 



hnk= — / F{iu)u'j„(Pu)e-^^'>""''du (4.9-6) 



Ztt J c 



gives 



00 oo ^ 



^(t-) = Z E t7 ^''rhlten COS 7lpT (4.9-7) 



n=0 k=0 k\ 



The correlation function ^oo(t) for the dc and periodic components of / 

 are obtained by letting t -^ 'x where xf/j — > 0. Only the terms for which 

 k = remain: 



00 



^«(t) = Z) en hlo COS npT (4.9-8) 



Comparing this with the known fact that the correlation function of 



yl + C cos {2Trfot - <p) • (2.2-2) 



is 



A^ + J cos 2irfoT (2.2-3) 



and remembering that eo is one while e„ is two for n > 1 shows that 

 Amplitude of dc component of / = //oo 



lip (4-9-9) 



Amplitude of ~- component of / = 2//„o 

 27r 



