演講訊息 TALKS

The deficiency of Riemann Integral is that it cannot handle functions with singular points, which is why we need improper Riemann Integrals. The deficiency of Lebesgue Integral is that it cannot integrate highly oscillating functions with singular points because its primitive function needs to be absolutely continuous which means the total oscillations should be as small as we wish. In this talk, we will see how can easily extend the Riemann Integrals and Lebesgue Integrals by modifying Riemann Integration definition just a little,we will see some integrals that do not exist under Lebesgue sense but they exist under Henstock-Kurzweil (HK) sense, and we see how we compute such integrals. In addition, we discuss the relationship between the primitive functions of Lebesgue and HK integrable functions respectively. Under the HK integral, the Fundamental Theorem of Calculus holds in one dimension and Divergence Theorem holds in higher dimensions.