Displaying similar documents to “Cauchy's residue theorem for a class of real valued functions”

The L r Henstock-Kurzweil integral

Paul M. Musial, Yoram Sagher (2004)

Studia Mathematica

Similarity:

We present a method of integration along the lines of the Henstock-Kurzweil integral. All L r -derivatives are integrable in this method.

Henstock-Kurzweil integral on BV sets

Jan Malý, Washek Frank Pfeffer (2016)

Mathematica Bohemica

Similarity:

The generalized Riemann integral of Pfeffer (1991) is defined on all bounded BV subsets of ℝ n , but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of σ -finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of BV sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect...

Continuity in the Alexiewicz norm

Erik Talvila (2006)

Mathematica Bohemica

Similarity:

If f is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of f is ∥ f ∥ = sup I | ∫ I f | where the supremum is taken over all intervals I ⊂ ℝ . Define the translation τ x by τ x f ( y ) = f ( y - x ) . Then ∥ τ x f - f ∥ tends to 0 as x tends to 0 , i.e., f is continuous in the Alexiewicz norm. For particular functions, ∥ τ x f - f ∥ can tend to 0 arbitrarily slowly. In general, ∥ τ x f - f ∥ ≥ osc f | x | as x → 0 , where osc f is the oscillation of f . It is shown that if F is a primitive of f then ∥ τ x F - F ∥ ≤ ∥ f ∥ | x | . An example shows that the function y ↦ τ x F ( y ) - F ( y ) need not be in L 1 . However, if...

The M α and C -integrals

Jae Myung Park, Hyung Won Ryu, Hoe Kyoung Lee, Deuk Ho Lee (2012)

Czechoslovak Mathematical Journal

Similarity:

In this paper, we define the M α -integral of real-valued functions defined on an interval [ a , b ] and investigate important properties of the M α -integral. In particular, we show that a function f : [ a , b ] → R is M α -integrable on [ a , b ] if and only if there exists an A C G α function F such that F ' = f almost everywhere on [ a , b ] . It can be seen easily that every McShane integrable function on [ a , b ] is M α -integrable and every M α -integrable function on [ a , b ] is Henstock integrable. In addition, we show that the M α -integral is equivalent to...

Boundary value problems for the Schrödinger equation involving the Henstock-Kurzweil integral

Salvador Sánchez-Perales, Francisco J. Mendoza-Torres (2020)

Czechoslovak Mathematical Journal

Similarity:

In the present paper, we investigate the existence of solutions to boundary value problems for the one-dimensional Schrödinger equation - y ' ' + q y = f , where q and f are Henstock-Kurzweil integrable functions on [ a , b ] . Results presented in this article are generalizations of the classical results for the Lebesgue integral.