66 BELL SYSTEM TECHNICAL JOURNAL 



Since the Z's in Theorem 3 need not be independent, the theorem 

 will continue to be valid when the Z's are any functions of any number 

 of other chance-variables Wi, • • • w^. 



The following six simple and useful equations, in which Z = X -\- iY 

 denotes any complex ^ chance-variable, are immediately obtainable 

 by means of Theorem 3. 



Z = X -\-iY, 



The following eight equations can be obtained by solving the fore- 

 going set of equations or by applying Theorem 3 to the appropriate 

 identities. 



X = ReZ = ReZ, (80) F = Im Z = Im Z, (81) 



IXY = Im Z\ (82) 



2X2 = |2I2 + ReZ2, (83) 



2F2 = |Z|2 - ReZ2, (84) 



_2 



2XF=ImZ, (85) 



2X = lZ|' + ReZ, (86) 



2f' = |Z1' - ReZ. (87) 

 Theorem 3 yields also the following two useful equations 



.2 



{z - zy =^ z-" - Z , (88) 



\Z - Z|2 = ]Zp - \Z\\ (89) 



The first can be obtained immediately by squaring Z — Z and then 

 applying Theorem 3; the second by expanding the product (Z — Z) 

 (Z — Z) and then applying Theorem 3 together with equation (75). 



When, instead of a single chance-variable Z, there are n chance- 

 variables Zi, • • • Z„, not restricted to being independent, equations 



