A hyperfunction, discovered by Mikio Sato in 1958, is defined as a pair of holomorphic functions (f, g) which are separated by a boundary γ. If γ is taken to be a segment on the real-line, then f is defined on the open region R^- below the boundary and g is defined on the open region R^+ above the boundary. A hyperfunction (f, g) defined on gamma is the "jump" across the boundary from f to g. This (f, g) pair forms an equivalence class of pairs of holomorphic functions (f + h, g + h), where h is a holomorphic function defined on the open region R, comprised of both R^- and R^+.

holomorphic function