MATHEMATICAL ANALYSIS OF RANDOM NOISE 67 



negative slope when it is known that / goes through zero at :Vi with positive 

 slope: 





dx,, r. , .,., „2.i/2,.2 --^-^'^[l + H cot-' i-H)] 



2ir \_-\ 



This is the same as (3.4-1). 



The expression (3.4-10) is the same as the probability of I going through 

 zero in dr when it is known that / goes through zero at the origin with posi- 

 tive slope. This second probability may be obtained from (3.4-1) by add- 

 ing the probability that / goes through dr with positive slope when it is 

 known to go through zero with positive slope. Thus we must add the ex- 

 pression containing the integral in which the integration in both rji and r?2 

 run from to cc . In terms of x and y this integral is 



/ X dx I dy ye 



Jo •'0 



l2_2,2_2(3/23/jif22)3-y 



This is equivalent to a change in the sign of M23 and hence of H. After 

 this addition we must consider 



1 + H cot"' (-FI) + 1 - H cot"' H 

 = 2-\- H [cot"' i-H) - cot"' H] 

 = 2 + H[t - 2 cot"' H] 

 = 2[l + H tan"' H] 

 and this leads to (3.4-10). 



3.5 Multiple Integrals 

 We wish to evaluate integrals of the form 



/ = f dxi f t/A;.2e"'i"'"'i"^"'^ (3.5-1) 



Jo Jo 



Our method of procedure is to first reduce the exponent to the sum of 

 squares by a suitable linear change of variable and then change to polar 

 coordinates. This method appears to work also for triple integrals of the 

 same sort, but when it is applied to a four-fold integral, the last integration 

 apparently cannot be put in closed form. 



The reduction of the exponent to the sum of squares is based upon the 

 transformation: If* 



xi = hyi + hiDixyi + hzD^iyz + • • • + h„D„,iyn 



::C2 = + //2A2J2 + ••• + KDn.iyn (3.5-2) 



.T„ = -f-0 +••• +0 + KDn.nyn 



* T. Fort, Am. :Math. Monthly, 43 (1936), pp. 477-481. See also Scott and Mathews, 

 Theory of Determinants, Cambridge (1904), Prob. 63, p. 276. 



