广义Taylor中值定理“中间点函数”的性质  

Properties of ″Intermediate Point Function″ for Generalized Taylor Mean Value Theorem

作　　者：赵美娜[1] 张树义[1] 郑晓迪 

机构地区：[1]渤海大学数理学院,辽宁锦州121013 [2]锦州师范高等专科学校,辽宁锦州121001

出　　处：《南通大学学报（自然科学版）》2016年第3期80-85,89,共7页Journal of Nantong University(Natural Science Edition)　

摘　　要：在一定条件下,研究了广义Taylor中值定理"中间点函数"的可微性.设I是R上一区间,a∈I是区间I的左端点,函数f,g∶I→R满足条件:(i)在区间I上有n阶连续导数且g^(n)(x)≠0,(ii)存在实数α>0,使limx→a^+(f^(n)(x)-f^(n)(a)/(x-a)~α=A,limx→a^+(g^(n)(x)-g^(n)(a))/(x-a)~α=B,(iii)f^(n)(a)B≠Ag^(n)(a),其中A,B是常数,则广义Taylor中值定理"中间点函数"c(x)在点a可微且c^(1)=(n!Γ(α + 1)/Γ(n+α + 1))^(1/α).该结果丰富了数学分析中值定理理论.In this paper, the differentiable behavior of the intermediate point function in generalized Taylor mean value theorem under certain conditions was studied. Let I be an interval on R, a the left extremity of I and f, g ∶ I → R be two functions satisfying the conditions：（i）the functions f and g have n order continuous derivative on interval I and gn（x）≠0,（ii）there exists a real number α 0, such that limx→a＋（fn）（x）-fn（a）/（x-a）α=A,limx→a＋（gn（x）-gn（a））/（x-a）α=B,（iii）f^（n）（a）B≠Ag^（n）（a）,where A and B are constants. Then ″the intermediate point function″ c（x） is differentiable at a point for generalized Taylor mean value theorem and c1=（n!Γ（α ＋ 1）/Γ（n＋α ＋ 1））^（1/α）.. The results can contributeto the theory of mean value theorem in mathematical analysis.

关 键 词：广义Taylor中值定理 中间点函数 可微性 渐近性 

分 类 号：O177.91[理学—数学]