MATHEMATICAL ANALYSIS OF RANDOM NOISE 305 



and the multiple integrals occurring in the expression for P{i) may be written 

 in terms of powers of 



q ^ [ p{x)e-°'dx (1.5-8) 



Jo 



Thus 



and since 



we have 



2cv/2(0 = va? + 2a V ^ 



1 - q 



1{1) = va \ F(l) dl = va/a 



J— 00 



Equations (1.5-8) and (1.5-9) give us an extension of Campbell's theorem 

 subject to the restrictions discussed in connection with equations (1.5-5) 

 and (1.5-7). Other generalizations have been made but we shall leave the 

 subject here. The reader may find it interesting to verify that (1.5-9) 

 gives the correct answer when p{x) is given by (1.5-6), and also to investi- 

 gate the case when the events are spaced equally. 



1.6 Approach of Distribution of / to a Normal Law 



In section 1.5 we saw that the probability density P{I) of the noise current 

 I may be expressed formally as 



-P(^) + T- [ exp\ -ilii + £ (iuYK/nl \du (1.6-1) 



ZTT J-oo L "=1 J 



where X„ is the nth semi-invariant given by (1.5-2). By setting 



X2 = 0" 



X = ^-^:^' = ^-^ (1.6-2) 



a a 



^See E. N. Rowland, Proc. Camb. Phil. Soc. 32 (1936), 580-507. He extends the 

 theorem to the case where there are two functions instead of a single one, which we here 

 denote by /(/). According to a review in the Zentralblatt fiir Math., 19, p. 224, Khint- 

 chine in the Bull. Acad. Sci. URSS, ssr. Math. Nr. 3 (1938), 313-322, has continued and 

 made precise the earlier work of Rowland. 



