PROPERTIES OF SINE WAVE PLUS NOISE 141 



The integration with respect to Ic2 and Is2 may be performed in a similar 

 manner. In this way we obtain a 12-fold integral. 



The integrations with respect to the /"s may be performed by using 



;i- dl e-'"f{z) dz = /(O) 



ZTT J—aa J—oo 



Ztt j-oo J-<» {_ az J«= 



(7.10) 



The result is the four-fold integral 



^-00 J-oo (Zi + 23) (Z2 + Z4) (7.11) 



exp [-(6o/2)(z? + zl + Z3 + zl) — g{ziZ2 + ZzZi) + iQ{zi + Z2)]. 



In the same way J2, J3, J4 may be reduced to the integrals obtained from 



(7.11) by replacing ZiZ2(g'' - g'^Z3Z4) by -gVizl, -g\lzl and 232,4(5" - 

 — g'^Zi22), respectively. When the /'s are combined in accordance with 

 (7.7) we obtain an integral which may be obtained from (7.11) by replacing 

 2iZ2(g" - ^'^2324) by 



g'[{ZiZ2 + Z324) + g'KZlZi - Z2ZzY (7.12) 



The terms zl + Z3 and Z2 + zl in the denominator may be represented as 

 infinite integrals. Interchanging the order of integration and expressing 



(7.12) in terms of partial derivatives of an exponential function leads to the 

 six-fold integral 



.(.) = (4^)-^P-f ^{-4+/%4j^^j_>2...7_;".. 



exp [- (^^0 + u){zl + 23')/2 - (^^0 + ^0(22 + z^)/2 ^^'^^^ 



— g{ziZ2 + Z3Z4) — q;(ziZ4 - Z2Z3) + iQ(zi + 22)] 



where the subscript a = indicates that a is to be set equal to zero after 

 the differentiations are performed. 

 When the four-fold integral in the z's is evaluated (7.13) becomes 



««= f.«f ..[-," 4 +/^£L 



^ exp [-Q\2b„ - 2g + u + v)/(2D)] (7.14) 



= [ du ( dviif- - gg")(2 -2F+ Q'/g) - «"QVg]«~7(4DS) 



JO JO 



