MATHEMATICAL AX A FAS IS OF RAX DOM XOISE 95 



We start by setting t — t — u which transforms the integral for A{t) into 



A{t) = I I\t - u)e~"''du (3.9-23) 



In order to obtain the correlation function ^(t) for A(t) we multiply A{t) 

 by A{t + t) and average over all the possible currents 



^(t) = A{t)A{t + r) 



= f e~"" du [ e~"" dv ave. l\t - u)l\t + t - v) 

 Jo Jo 



Just as in (3.9-4) the average in the integrand is the correlation function of 

 /■(/), the argument being t -}- t — v — t -\- u = t -\- u — v. From (3.9-7) 

 it is seen that this is 



ypl + 2\P~{t + u — v) 

 where \1/{t) is the correlation function of /(/). Hence 



^(t) =tl + 2 I du I ^z; e-""-"V'(r + u- i) (3.9-24) 

 a- Jo Jo 



From the integral (3.9-23) for A{t) it is seen that the average value of 

 ^(0 is 



A = ^- = ^" (3.9-25) 



where we have used 



h = 4^(0) = [ w(f) df = p 

 Jq 



Jo 



Using this result again, only this time applying it to A{t), gives 



.42(7) = ^(0) 



r" r (3.9-26) 



= A +2 du dv e-""~"V'(« - v) 

 Jo Jo 



The double integrals may be transformed by means of the change of 

 variable « + z' = x, u ~ v = y. Then (3.9-24) becomes 



^(r) = A' +\ [ dy f dx + I dy f dx e~"' yp\T + y) 



\_Jo Jy •'-00 J—y J 



(3.9-27) 



= ^" + i [ e-"'[4^\r +y) + rl^'ir - y)] dy 

 a Jq 



