Formulation
A hyperfunction on the real line can be conceived of as the 'difference' between one holomorphic function on the upper half-plane and another on the lower half-plane. That is, a hyperfunction is specified by a pair (f, g), where f is a holomorphic function on the upper half-plane and g is a holomorphic function on the lower half-plane.
Informally, the hyperfunction is what the difference f − g would be at the real line itself. This difference is not affected by adding the same holomorphic function to both f and g, so if h is a holomorphic function on the whole complex plane, the hyperfunctions (f, g) and (f + h, g + h) are defined to be equivalent.
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