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General information
The Henstock-Kurzweil integral was introduced in 1957-62 by Ralph Henstock and independently by Jaroslav Kurzweil. The H-K integral gives a Riemannian definition for the Perron and restricted Denjoy integrals. Thus the H-K integral encompasses the integrals of Newton, Riemann e Lebesgue as well as its improper integrals.
Also known as the Generalized Riemann Integral or the Complete Riemann Integral, the H-K integral has good convergence and functional analytic properties. Moreover, its definition can be easily extended to functions defined on unbounded intervals and Banach space-valued.
The main feature of the non-absolute H-K integral is the fact that functions with not anly many discontinuities, but also highly oscillating can be integrable in Henstiock-Kurzweil sense. For instance, the function f defined on the compact interval [0,1] of the real line and given by f(t)=F'(t), where F(t)=t^2sen(1/t^2) on ]0,1] and F(0)=0 is H-K integrable (but not Lebesgue nor Riemann integrable).
With a slight modification in its
definition,
the H-K integral gives way to a more restricted integral known as the
McShane
integral which coincides with the Lebesgue integral. Thus, we have a
constructive
definition for the Lebesgue integral by means of Riemann sums. This
fact
enables one to apply the techniques and tools of the Henstock-Kurzweil
Integration
Theory to treat problems which may involve Lebesgue integrable
functions
(with many discontinuities, for instance). In this manner, this theory
has
been shown to be useful in the more modern treatment of integral and
differential
equations centered in weak solutions (almost everywhere differentiable,
for
instante).
Some textbooks in this area
Some
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