MATHEMATICAL AXALYSIS OF RANDOM NOISE 69 



Other integrals may be obtained by differentiation. Thus from 



[ dx [ dy e-''-'''-''"""'"' = y CSC cp (3.5-3) 



Jo Jo 



we obtain 



[ dx I dyxye-'"-'"-'''"''"'' = I esc' ^(1 - <p cot <p) (3.5-4) 



Jo •'0 



By using the same transformation we may obtain 



[ dx [ dy ye-''-"'-'"'" = ^ -\- (3.5-5) 



JO JO 4 1 + a 



Of course, we may expand part of the exponential in a power series and 

 integrate termwise but this leads to a series which has to be summed in each 

 particular case: 



r dx r dyx^y'^e-''-"'-'"'" 

 Jo JO 



If we take — 1 < R{m) < — |, —1 < R{m) < — ^, the series may be 

 summed when a = I. The result stated just below equation (3.8-9) is ob- 

 tained by continuing m and n analytically. 



The same methods will work when the limits are ± =o . We obtain, when 

 m and n are integers. 



-+00 -+Q0 



/ dx I dy x' y 



m —x^—y'^—lxy cos ip 



0, w + w odd 



r / ^ + w + 1 \ (3,5-6) 



^/ \ — n — m \ — cos ip\ 



F I —n, —m; ; \ , n -\- m even 



The hypergeometric function may also be written as 

 ^ ( n m \ — n — m . i \ 



^\-r—i' — 2 — '""7 



