Concerning Harnack’s theory of improper definite integrals

Abstract

In this paper I consicler the improper simple definite integrals of HARNACK (1883, 1884). In the introduction I wish to characterize somuewhat clearly the theories of the improper simple and multiple integrals recently given by JORDAN (1894) and STOLZ (1898, 1899), and in this introductory paragraph I summarize the contents of the whole introduction. These theories for the simple integrals have intimate relations with the HARNACK theory. The definition adopted for the multiple integrals is inore exacting than that for the simple ilntegrals. The miiultiple integrals converge or exist (as limits) only absolutely. For the simple integrals we have then two theories, on the one hanld, of the integrals with the milder definition, and, on the other hand, of the integrals with the stronger definition and so with a larger body of properties. The first class of integrals includes the second class of integrals. The HARNACK theory relates to the first and general class of integrals; this theory has not received systellmatic development; however, for the theory of the absolutely conivergent HARNACK integrals this is nlot true, and these initegrals conistitute the second and special class of integrals. I discuss both classes of simple integrals simultaneously and by uniform process; this is made possible by suitable determinations of the definitions; the absolute convergence of the integrals of the seconid class appears only at the conclusion, and hence it is desirable to introduce terms of discriminiation conlnoting the two definitions, the milder ancl the stronger; the terms chosen, narrow, broad, connote the geometric form of the definitions, and likewise the fact that the class of narrow integrals lhas a less extensive body of properties than the (included) class of broad integrals. There has been a tendency to do away with the non-absolutely convergent HARNACK integrals; I hope to show that this tendency rests uponi misconceptions.-The tlheory of DE LA VALL-fE POUSSIN (initiated in 1892) is in form distinct from the HARNACK theory and