70 



BELL SYSTEM TECH MCA L JOURNAL 



By transformations of this we are led to the following expression for the 

 integral 



0, n + m odd, 



(sin ^)"-H»+i V 2 ' 



^"1 2 \ , ^, 



— - , - ; cos (^ I , w, « both even, 



r{i + ^ir 



(•+!) 



cos ^F 



e 



2 ' 2 

 — w \ — n 



3 2 \ 



2 ; cos ^\ , 



(sin ^)»+">+i "" "" \ 2 ' 2 



w?, n odd 



As was mentioned earlier, the method used to evaluate the double inte- 

 grals may also be applied to similar triple integrals. Here we state two 

 results obtained in this way. 



«Q0 rtOO «GO 



/ dx I dy dz exp [—x^ — y — z — Icxy — Ihzx — 'layz\ 

 Jo Jo Jo 



*00 /.GO ^QO 



/ dx I dy dz yz exp [—x — y — z' — Icxy — Ihzx — layz] 

 Jo Jo Jo 



\/Tr[l+ a -b-c (^ - be 1 ,2 c '7\ 



= m L l + a - ^^ (« + /5 + T - -) J (3.5-7) 



where and 7 are obtained by cyclic permutation of a, b, c from 



a-cb ^ . _: r D^ Y 



(1 - c2)i/2(l - ^,2)1/2 ^'"^ L(l - c')(l - b^)j 



a = cos 



_i a — ic 



= cot ^1/2 



where a, (3, 7 all lie in the range 0, t and where 



D, = 



= 1 + 2 abc —a — b — c" 



For reference we state the integrals which arise from the definition of the 

 normal distribution given in section (2.9) 



dxi • • • I dxn exp — X) «rs .^V •^*« = I — \\ 



/+00 /.+00 r « "IF" 'V-l- A 



dxi • • • I dXnXtXu exp | — 2Z arsXrX^ = I — 1^ ^-~ 



(3.5-8) 



