An Asymptotic Expansion of Definite Integrals at Two Points
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.
definite integralasymptotic expansionthe integral remainder of Taylor's formulaintegration by parts
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