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

There is no general product between hyperfunctions, but the product of a hyperfunction by a holomorphic function can be expressed as

A standard holomorphic function can also be defined as a hyperfunction,

The Heaviside step function and the delta function can be defined as the hyperfunctions

See also

Holomorphic Function

This entry contributed by Bryan Jacobs

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References

Isao, I. Applied Hyperfunction Theory. Amsterdam, Netherlands: Kluwer, 1992.Penrose, R. The Road to Reality: A Complete Guide to the Laws of the Universe. New York: Random House, 2006.

Referenced on Wolfram|Alpha

Hyperfunction

Cite this as:

Weisstein, Eric W., with contributions by Bryan Jacobs. "Hyperfunction." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Hyperfunction.html

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