The absolute square of a complex number 
, also known as the squared norm, is defined as
 |
(1)
|
where 
denotes the complex conjugate of 
and 
is the complex modulus.
If the complex number is written 
, with 
and 
real, then the absolute square can be written
 |
(2)
|
If 
is a real number, then (1) simplifies
to
 |
(3)
|
An absolute square can be computed in terms of 
and 
using the Wolfram
Language command ComplexExpand[Abs[z]^2,
TargetFunctions -> 
Conjugate
].
An important identity involving the absolute square is given by
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
If 
,
then (6) becomes
 |  |  |
(7)
|
 |  |  |
(8)
|
If 
,
and 
,
then
 |
(9)
|
Finally,
 |  |  |
(10)
|
 |  | ![2[1+cos(phi_2-phi_1)]](/images/equations/AbsoluteSquare/Inline36.svg) |
(11)
|
 |  | ![4cos^2[1/2(phi_2-phi_1)].](/images/equations/AbsoluteSquare/Inline39.svg) |
(12)
|
