Halfintegral

Halfintegral, also called a half-order integral, is a fractional calculus operator of order 1/2 that generalizes the concept of repeated integration to non-integer orders. It is most commonly defined within the Riemann–Liouville framework as the half-order integral of a function f on an interval [a, t].

Definition: The Riemann–Liouville half-integral of f, denoted I^{1/2} f, is

I^{1/2} f(t) = (1/Γ(1/2)) ∫_a^t (t − τ)^{−1/2} f(τ) dτ,

which equals (1/√π) ∫_a^t (t − τ)^{−1/2} f(τ) dτ for a < t. This operator is linear and extends to general orders α > 0 as I^α, with I^1 reducing to the ordinary integral.

Key properties: Composition of fractional integrals obeys I^α I^β = I^{α+β}. The Laplace transform of a half-integral satisfies L{I^{1/2} f}(s) = s^{−1/2} F(s), where F(s) is the Laplace transform of f. The half-integral is related to the half-derivative D^{1/2} via D^{1/2} I^{1/2} f = f in suitable function spaces. For a constant function f(t) = 1, I^{1/2} 1(t) = (2/√π) t^{1/2}.

Applications: The halfintegral appears in solutions to fractional differential equations, models of processes with memory or anomalous diffusion, viscoelasticity, and signal processing. It is one component of the broader field of fractional calculus, which includes various definitions of fractional integrals and derivatives, such as the Caputo derivative.