摘要
复合函数的高阶微商公式在涉及到高阶偏微商的偏微分方程中极有帮助,而国内外对此专门研究的文献要么极少,要么就是借用泛函分析等较深的方法。文章在已有的单变数复合函数高阶微商公式(即Bruno公式)的基础上,利用母函数、Bell多项式这两个组合学工具和多元函数Taylor公式这一个分析工具分别从两个方面将复合函数高阶微商公式推广到多元复合函数的一般情形,得到了在形式上更有条理而且在结论上更一般的复合函数高阶微商表达式。
The formula for higher - order derivatives of composite functions is rather helpful in the study of partial differential equations involving higher - order partial differentials while in the domestic and foreign literatures, it is either rarely studied or studied using some complicated methods like functional analysis. Here,the formula for higher -order derivatives of composite functions is extended to multivariate composite functions from two aspects, which base on the existing formula for higher - order derivatives of composite functions which is called Bruno formula. By utilizing the three important tools consisting of the generating functions, Bell polynomials and multivariate Taylor formulas,the more methodic and more general expressions of Bruno formula are finally deduced.
出处
《忻州师范学院学报》
2017年第2期8-13,共6页
Journal of Xinzhou Teachers University
关键词
复合函数高阶微商
母函数
BELL多项式
TAYLOR公式
higher - order derivative of composite function
generating function
Bell polynomial
Taylor formula
