Related papers: Redefining the integral
In this paper, a new calculus on sequences is defined. Also, the $\lambda$-derivative and the $\lambda$-integration are investigated. The fundamental theorem of $\lambda$-calculus is included. A suitable function basis for the…
We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…
It is noted that the standard definition of the fractional Laplacian leads to a hyper-singular convolution integral and is also obscure about how to implement the boundary conditions. This purpose of this note is to introduce a new…
In the present paper two certain subclasses of the starlike functions associated with the vertical strip are considered. The main aim of this paper is to investigate some basic properties of these classes such as, subordination relations,…
We consider several possible approaches to evaluating an integral involving the digamma function and a related logarithmic series.
It is known in the case of the Stieltjes transform that evaluating the integral by expanding the kernel of transformation followed by term by term integration leads to an infinite series of divergent integrals. Moreover, it is known that…
General prescriptions of differential renormalization are presented. It is shown that renormalization group functions are straightforwardly expressed through some constants that naturally arise within this approach. The status of the action…
Although Feynman integrals in general cannot be expressed as well-studied special functions, they can be calculated systematically and efficiently using the \texttt{AMFlow} method in combination with differential equations in the kinematic…
We obtain new inversion formulas for the Funk type transforms of two kinds associated to spherical sections by hyperplanes passing through a common point $A$ which lies inside the n-dimensional unit sphere or on the sphere itself.…
A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we…
Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.
Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial…
This paper generalizes beta divergence beyond its classical form associated with power variance functions of Tweedie models. Generalized form is represented by a compact definite integral as a function of variance function of the…
n this article, firstly, we introduce the notion of star modules with respect to a balanced pair and obtain some properties. We mainly give the relationship between n-X star modules and n-X tilting modules [9], and a new characterization of…
In this manuscript, we generalize F-calculus to apply it on fractal Tartan spaces. The generalized standard F-calculus is used to obtain the integral and derivative of the functions on the fractal Tartan with different dimensions. The…
Although being powerful, the differential transform method yet suffers from a drawback which is how to compute the differential transform of nonlinear non-autonomous functions that can limit its applicability. In order to overcome this…
This paper introduces a new method for redefining the Roman factorial using universally applicable functions that are not expressed in closed form. We present a set of foundational functions, similar to Boolean operations, to simplify the…
This paper provides an approach to establishing the calculus method from the concept of mean, i.e., average. This approach is from a statistics perspective and can help calculus learners understand calculus ideas and analyze a function…
Subdifferentials (in the sense of convex analysis) of matrix-valued functions defined on $\mathbb{R}^d$ that are convex with respect to the L\"{o}wner partial order can have a complicated structure and might be very difficult to compute…
A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order $\alpha, 0 < \alpha \leq 1$, called $F^\alpha$-integral, is defined, which is suitable to integrate functions with fractal…