halfintegraal

Halfintegraal is a theoretical construct used in mathematics and related disciplines to denote a half-order integral operator. It is the half-order analogue of the standard integral and is often described as the fractional integral of order 1/2. The term is not universally standardized and appears mainly in didactic or speculative contexts; the more common terminology in literature is the fractional integral of order 1/2.

Formal definition: For a suitable function f defined on an interval [0, T], the halfintegraal I^{1/2} f is defined by I^{1/2} f(t) = 1/Γ(1/2) ∫_0^t (t − s)^{-1/2} f(s) ds. Since Γ(1/2) = √π, this can be written as I^{1/2} f(t) = (1/√π) ∫_0^t (t − s)^{-1/2} f(s) ds. This operator is the half-order counterpart to the standard integral in the sense of Riemann–Liouville fractional calculus.

Examples and properties: If f(s) = 1, then I^{1/2} 1(t) = (2/√π) √t. For f(s) = s^β with β > −1, I^{1/2} s^β = Γ(β+1)/Γ(β+3/2) t^{β+1/2}. The halfintegraal is linear, and under suitable conditions I^{1/2} I^{1/2} f = I^1 f, reflecting a semigroup property. It is related to half-derivatives via inverse relationships in fractional calculus.

Context and usage: The halfintegraal can model memory effects in materials or diffusion processes with intermediate memory. In practice, many discussions use the general fractional integral I^α and the term halfintegraal serves as a pedagogical or fictional label rather than a standardized designation. See also fractional calculus, Riemann–Liouville integral, Caputo derivative.