Algebraic analysis

Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of functions such as hyperfunctions and microfunctions. As a research programme, it was started by Mikio Sato in 1959.
Microfunction.
Let "M" be a real-analytic manifold of dimension "n", and let "X" be its complexification. The sheaf of microlocal functions on "M" is given as
where
A microfunction can be used to define a Sato's hyperfunction. By definition, the sheaf of Sato's hyperfunctions on "M" is the restriction of the sheaf of microfunctions to "M", in parallel to the fact the sheaf of real-analytic functions on "M" is the restriction of the sheaf of holomorphic functions on "X" to "M".