Hyperfunction - Examples

Examples

• If f is any holomorphic function on the whole complex plane, then the restriction of f to the real axis is a hyperfunction, represented by either (f, 0) or (0, −f).

• The Heaviside step function can be represented as .

• The Dirac delta "function" is represented by . This is really a restatement of Cauchy's integral formula. To verify it one can calculate the integration of f just below the real line, and subtract integration of g just above the real line - both from left to right. Note that the hyperfunction can be non-trivial, even if the components are analytic continuation of the same function. Also this can be easily checked by differentiating the Heaviside function.

• If g is a continuous function (or more generally a distribution) on the real line with support contained in a bounded interval I, then g corresponds to the hyperfunction (f, −f), where f is a holomorphic function on the complement of I defined by

This function f jumps in value by g(x) when crossing the real axis at the point x. The formula for f follows from the previous example by writing g as the convolution of itself with the Dirac delta function.

• If f is any function that is holomorphic everywhere except for an essential singularity at 0 (for example, e1/z), then (f, −f) is a hyperfunction with support 0 that is not a distribution. If f has a pole of finite order at 0 then (f, −f) is a distribution, so when f has an essential singularity then (f,−f) looks like a "distribution of infinite order" at 0. (Note that distributions always have finite order at any point.)