定积分在两点展开的渐近公式

An Asymptotic Expansion of Definite Integrals at Two Points

对于具有m+n阶连续导数的被积函数的定积分,通过多次分部积分给出了在积分上限和积分下限两点处同时展开的定积分的渐近展开式,其余项类似于Taylor公式的积分型余项,这种展开式可看作是Taylor公式的一种推广.被积函数在积分上下限处的值有不同情形,展开式也会随之变化而具有多种形式.通过分析与举例也发现这种展开式的近似计算优于Taylor公式的近似计算,而且在某些积分不等式的证明中也体现了其快捷方便的优点.

For the definite integral of the integrand with a continuous derivative of order m+n,the asymptotic expansion of the definite integral at the two points of the upper and lower bounds of the integral is given by multiple integration by parts.The remaining terms are similar to the remainder of the integral form of Taylor′s formula,so this expansion can be regarded as a generalization of Taylor′s formula.Because of the different values of the integrand function at the upper and lower limits of the integral,the expansion will also change and have multiple forms.Through analysis and examples,it is found that the approximate calculation of this expansion formula is better than that of Taylor formula,and it is also fast and convenient in the proof of some integral inequalities.