The Research On Several Questions Of E.Borel's Function Theory

E.Borel is an influential mathematican to the developments of the mathematics in twentieth century. In his work on the mathematics, there are theory of numbers, function theory, probability and applications to mechanics. The purpose of this paper is to probe and to analyze his mathematic work on function approximation, the summation of divergent series, function singularities, measure theory and analytic continuation. On the basis of reading original sources and of investigating previous researches on these subjects, the major contributions made in the paper are as follows.It examines the thought on the Borel's interpolation and concludes that his idea is relative to the 'zero theory of entire functions'. His interpolation is valuable. It discussed the important influence on M.Potron, J.W.Young and L.Kantorovitch.The theory of summable divergent series is an important problem from the end of 19^th to the beginning of 20^th. Mathematican obtained many methods of summability by certain techniques. Cesaro's method of arithmetic means and Borel's methods of the exponent and the integration are remarkable. It analyzes his thought and influence on analytic continuation and differential equations.This paper analyzes Borel's method of associated series which studies the function singularities. 'Hadamard theorem of the multiplication of function singularities' and 'Taylor's series generally considers its convergent circle as the cut secant' are studied. This paper discusses his influence on other mathematiciansThe background of Borel's idea on measure theory is studied. His source is from analytic continuation. In the point of view of the zero measure set, he established integral theory by constructive method. His idea influences F.Riesz and A.haar.It discusses the his work on these concepts of the function analysis and monogenicity.lt analyzes the development of his idea on these concepts and influence on J.Wolff, T.Carleman, A.Denjoy.