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Cauchy's residue theorem for a class of real valued functions

Branko Sarić (2010)

Czechoslovak Mathematical Journal

Let $[a, b]$ be an interval in $ℝ$ and let $F$ be a real valued function defined at the endpoints of $[a, b]$ and with a certain number of discontinuities within $[a, b]$ . Assuming $F$ to be differentiable on a set $[a, b] ∖ E$ to the derivative $f$ , where $E$ is a subset of $[a, b]$ at whose points $F$ can take values $\pm \infty$ or not be defined at all, we adopt the convention that $F$ and $f$ are equal to $0$ at all points of $E$ and show that $𝒦ℋ -vt \int_{a}^{b} f = F (b) - F (a)$ , where $𝒦ℋ -vt$ denotes the total value of the Kurzweil-Henstock integral. The paper ends with a few examples that illustrate...

Changes of variable which preserve almost everywhere approximate differentiability

James Foran (1981)

Colloquium Mathematicae

Characterizations of convex vector functions and optimization.

Cusano, Claudio, Fini, Matteo, La Torre, Davide (2004)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

Characterizations of sum form information measures on open domains.

B.R. Ebanks, P.K. Sahoo, P. Kannappan (1997)

Aequationes mathematicae

Conditions implying the summability of approximate derivatives

R. Fleissner, R. J. O'Malley (1979)

Colloquium Mathematicae

Continuity properties relative to the intermediate point in a mean value theorem.

Popa, Emil.C (2004)

General Mathematics

Continuous-, derivative-, and differentiable-restrictions of measurable functions

Jack Brown (1992)

Fundamenta Mathematicae

We review the known facts and establish some new results concerning continuous-restrictions, derivative-restrictions, and differentiable-restrictions of Lebesgue measurable, universally measurable, and Marczewski measurable functions, as well as functions which have the Baire properties in the wide and restricted senses. We also discuss some known examples and present a number of new examples to show that the theorems are sharp.

Correction to: An asymptotic form of Cauchy's functional equation; Volume 24, 2/3 (1982). (Short Communication).

A.J. Parsloe (1982)

Aequationes mathematicae

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