一个广义连续原积分及其应用
A Generalised Continuous Primitive Integral and Some of Its Applications
摘要：
利用Laplace导数定义了Perron型积分，即Laplace积分.证明了该积分包含Perron积分，并构造了一个Laplace可积而Perron不可积的函数的例子，说明了该积分包含Perron积分的正确性.建立了微积分基本定理、Hake定理、分部积分法、收敛定理、中值定理、Taylor定理的积分余式及其余式的估计等积分性质。证明了在Alexiewicz范数意义下，所有Laplace可积函数的空间是不完备的，并且稠密地包含所有多项式的集合。应用程序显示泊松积分，系统的广义常微分方程和高阶广义常微分方程。
英文摘要：
Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such as fundamental theorem of calculus, Hake's theorem, integration by parts, convergence theorems, mean value theorems, the integral remainder form of Taylor's theorem with an estimation of the remainder, are established. It turns out that concerning the Alexiewicz's norm, the space of all Laplace integrable functions is incomplete and contains the set of all polynomials densely. Applications are shown to Poisson integral, a system of generalised ordinary differential equations and higher-order generalised ordinary differential equation.
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