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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Generating Functions for the Mean Value of a Function on a Sphere and Its Associated Ball in Rn | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 656329 14 pages doi 10.1155 2008 656329 Research Article Generating Functions for the Mean Value of a Function on a Sphere and Its Associated Ball in R Antonela Toma Department of Mathematics II University Politehnica of Bucharest Splaiul Independent 313 060042 Bucharest Romania Correspondence should be addressed to Antonela Toma antonela2222@yahoo.com Received 20 April 2008 Accepted 22 May 2008 Recommended by Patricia Wong We define two functions which determine the properties and the representation of the mean value of a function on a ball and on its associated sphere. Using these two functions we obtain Pizzetti s formula in R as well as a similar formula for the mean value of a function on the ball associated to the sphere. We also give the expressions of the remainders in these two formulas using the surface integral on a sphere. Copyright 2008 Antonela Toma. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction The mean function over values of a sphere and over its associated ball are very important in the study of some mathematical-physics problems as well as in the theory of a potential and in partial differential equations. A representation for the mean values of a function over a sphere in R was given by Pizzetti 1 using polyharmonic operators. In 2 volume II chapter IV Section 3 page 258 using the second Green s formula it is given the proof for Pizzetti s formula and then its generalization in R . In this paper we define two functions which determine the properties and the representations of the mean values of a function over a sphere and over its associated ball. These functions are called generating functions of these two averages and using these functions we obtain Pizzetti s formula in
