Any pair of equations giving the real part of a function as an integral of its imaginary part and the imaginary part as an integral of its real part. Dispersion relationships imply causality in physics. Let
 |
(1)
|
then
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
denotes the Cauchy principal value and 
and 
are Hilbert transforms
of each other. If the complex function is symmetric
such that 
,
then
 |  |  |
(4)
|
 |  |  |
(5)
|
