semiintegral

Semiintegral is not a standard, formally defined term in mathematics. In many texts, it appears as an informal label for concepts that lie between a primitive (an antiderivative) and a definite integral, or as shorthand for certain fractional or generalized integral operators. Because the term is not universally defined, its precise meaning is context dependent and should be clarified within the source.

One common context is fractional calculus. In this setting, a semi-integral (often referred to more precisely as a fractional integral) of order α in (0, 1) generalizes the ordinary integral. The Riemann–Liouville fractional integral of order α of a function f is defined by I^α f(x) = (1/Γ(α)) ∫_0^x (x − t)^{α−1} f(t) dt. When α = 1, this operator reduces to the ordinary definite integral. In texts that use the term semi-integral, this informal meaning may be implied, though the standard terminology is fractional integral.

Another usage can be informal or problem-specific, referring to an integral operator with a constrained domain, a weighted integral, or a mixed operation that is between differentiation and integration. In integral equations, phrases like “semi-integral equation” may appear to describe problems where the unknown function appears under an integral sign with special kernels, again depending on the author's definition.

Given the lack of a universal definition, readers should examine the precise definition provided in a given work when encountering the term semiintegral. Related concepts include indefinite integrals, definite integrals, fractional (or generalized) integrals, and integral equations.