For a wave, a dispersion relation is an equation relating the parameters 
and 
in a sinusoidal factor 
. It is often written 
. The ratio 
gives the speed of a fixed phase, while the derivative 
gives the speed of a slowly
varying envelope (Whitham 1974).
In complex analysis, dispersion relations are pairs of equations giving the real part of a function as an integral of its imaginary part and the imaginary part as an integral of the real part. Dispersion relationships imply causality in physics. Let
 |
(1)
|
then
 |  |  |
(2)
|
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(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)
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