A hyperfunction, discovered by Mikio Sato in 1958, is defined as a pair of holomorphic functions 
which are separated by a boundary 
. If 
is taken to be a segment on the real-line, then f is defined
on the open region 
below the boundary and 
is defined on the open region 
above the boundary. A hyperfunction 
defined on gamma is the "jump" across the boundary
from 
to 
.
This 
pair forms an equivalence class of pairs of holomorphic
functions 
,
where 
is a holomorphic function defined on the open region 
, comprised of both 
and 
.
Hyperfunctions can be shown to satisfy
 |  |  |
(1)
|
 |  |  |
(2)
|
There is no general product between hyperfunctions, but the product of a hyperfunction by a holomorphic function 
can be expressed as
 |
(3)
|
A standard holomorphic function 
can also be defined as a hyperfunction,
 |
(4)
|
The Heaviside step function 
and the delta function 
can be defined as the hyperfunctions
 |  |  |
(5)
|
 |  |  |
(6)
|
