PROPERTIES OF SINE WAVE PLUS NOISE 155 



entiation with respect to /, and do the same for /«(/) and its derivatives we 

 have, from Section 3.8 of Reference A, 



ll = Jl = bo, lj]=0 

 TJ[ = -Tjs = hu TJ'c = 'Tj's = 



./ -,// 



/7^=/y=^„, lX = (A2-2) 



When we deal with moments in which the arguments of the two variables 

 are separated by an interval r as in (see the last of equations (3.7-11) of 

 Reference A) 



Ic{t)Is{t -i-r) = h, 



it is convenient to denote the argument t by the subscript 1 and the argu- 

 ment / + r by 2. Then our example becomes 



IcJsi = h 

 We shall need the following moments of this type. 



I cll c2 — I sll s2 — g, I cll s2 — ~Ic2l si = h 



Icllc2 = Islls2 = ~IclIc2 — ~IslI s2 — g 



(A2-3) 



Iclls — I c2lsl — ~IclI s2 — ~Ic2l si — h 



rf -rf 



Icllc2 — Islls2 = —g'\ Iclls2 — —Ic2lsl = " /?'' 



It should be remembered that in these equations the primes on the /'s 

 denote differentiation with respect to t while the primes on g and h denote 

 differentiation with respect to r. 



APPENDIX III 

 Evaluation of a Multiple Integral 



Several multiple integrals encountered during the preparation of this 

 paper were initially evaluated by the following procedure. The integral 

 was first converted into a multiple series by expanding a portion of the inte- 

 grand and integrating termwise. It was found possible to sum these series 

 when one of the factorials in the denominator was represented as a contour 

 integral. This reduced the multiple integral to a contour integral and some- 

 times the latter could be evaluated. 



