Thesis--Université de Paris, 1902. / Bibliographical foot-notes.

Thesis (doctoral)--Georg-August-Universität zu Göttingen, 1914. / Cover title. Vita. Includes bibliographical references.

No description available.

No description available.

Inaug.-diss.--Erlangen. Lebenslaluf. / In both equations in the title, n̳ is superscript.

Inaug.-diss.--Erlangen. Lebenslaluf. / In both equations in the title, n̳ is superscript.

Thesis--Faculté des sciences de Paris, 1898.

In the definition of the Perron integral of a function f (x) over a closed interval [a, b] a major function M(x) and a minor function m(x) are required to satisfy the conditions (i) M(x) and m(x) are continuous on [a, b] and M(a) = m(a) = 0 ; (ii) - ∞ ≠ Ḏ M(x) ≥ f (x) ≥ D [overscored] m(x) ≠ + ∞. It is shown that without restricting the generality of the integral one may impose the additional condition (iii) M(x) and m(x) are differentiable on [a, b]. / Science, Faculty of / Mathematics, Department of / Graduate

No abstract included. / Science, Faculty of / Mathematics, Department of / Graduate

The purpose of this paper is to examine properties of the mean integral. The mean integral is compared with the regular integral. If [a;b] is an interval, f is quasicontinuous on [a;b] and g has bounded variation on [a;b], then the man integral of f with respect to g exists on [a;b]. The following theorem is proved. If [a*;b*] and [a;b] each is an interval and h is a function from [a*;b*] into R, then the following two statements are equivalent: 1) If f is a function from [a;b] into [a*;b*], gi is a function from [a;b] into R with bounded variation and (m)∫^b_afdg exists then (m)∫^b_ah(f)dg exists. 2) h is continuous.