Legendre级数的积分定理

AN INTEGRAL THEOREM FOR LEGENDRE SERIES

本文对P.Heywood研究的广义积分:integral from 0 to 1 (f(x)/(1-x)~W dx)进行了探讨。在莫叶、陈留琨、霍守诚、蒋润勃等人的研究基础上,将结果推广到:W=4,或4<W<5。 主要内容如下: 定理:设a_#终归为正,且: sum from n=0 to ∞ (a_n)=sum from n=0 to ∞ ((n+1)a_n) =sum from n=0 to ∞ (n(n-1)(n+1)(n+2)an) =sum from n=0 to ∞ (n(n-1)(n-2)(n+1)(n+2)(n+3)an) =0 f(x)=sum from n=0 to ∞ (a_np_n(x)) 这里:P_π(x)为Legendre多项式,则按照: w=4,或4<w<5,积分: I(?)=intergal form 0 to 1(f(x)/(1-x)~w dx) 存在的充要条件是: 级数∑n^(?)a_nlogn,或级数∑n^2(w^(-1))a_n收敛。

This paper investigates the improper integral studied by P. Heywood and extends the result to {W: 4≤W∠5Based on researches of Mo Ye, Chen Liukun, Huo Shouchen. JiangRunbo, and others. The chief result is as follows.Theorem Suppose a is ultimately positive satisfying the following conditions. Where {Pn(x)} is a sequence of Legendre Polynomials. Then theimproper integral I(W) = d x is convergent for 4≤W<5 ifand only if either Σn6 α n logn or Σn2 αn is convergent.