• Title/Summary/Keyword: Improper integral

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The Understanding of Improper Integration - A Case Study

  • Camacho Matias;Gonzalez-Martin Alejandro S.
    • Research in Mathematical Education
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    • v.10 no.2
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    • pp.135-150
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    • 2006
  • Although improper integrals constitute a concept of great utility for Mathematics students, it appears that students are unable to assimilate this concept within the wider system of concepts they learn in their first year of Mathematics studies. In this paper we describe a competence model used in a study about the kind of understanding students possess about improper integral calculus when two registers of representation come into play. Competence will be considered as the coherent articulation of different semiotic registers. After analysing the results of a questionnaire, six students were selected to be interviewed on the basis of their overall results and the significance of their answers. For the interview, five original questions from the questionnaire were used together with a new question. In this article we will analyse, from our theoretical point of view, the work carried out by one student who was interviewed to show how our competence model works and we will discuss this formal competence model used.

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NOTE ON CONVERGENCE OF EULER'S GAMMA FUNCTION

  • Choi, Junesang
    • Honam Mathematical Journal
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    • v.35 no.1
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    • pp.101-107
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    • 2013
  • The Gamma function ${\Gamma}$ which was first introduced b Euler in 1730 has played a very important role in many branches of mathematics, especially, in the theory of special functions, and has been introduced in most of calculus textbooks. In this note, our major aim is to explain the convergence of the Euler's Gamma function expressed as an improper integral by using some elementary properties and a fundamental axiom holding on the set of real numbers $\mathbb{R}$, in a detailed and instructive manner. A brief history and origin of the Gamma function is also considered.

DEGENERATE VOLTERRA EQUATIONS IN BANACH SPACES

  • Favini, Angelo;Tanabe, Hiroki
    • Journal of the Korean Mathematical Society
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    • v.37 no.6
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    • pp.915-927
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    • 2000
  • This paper is concerned with degenerate Volterra equations Mu(t) + ∫(sub)0(sup)t k(t-s) Lu(s)ds = f(t) in Banach spaces both in the hyperbolic case, and the parabolic one. The key assumption is played by the representation of the underlying space X as a direct sum X = N(T) + R(T), where T is the bounded linear operator T = ML(sup)-1. Hyperbolicity means that the part T of T in R(T) is an abstract potential operator, i.e., -T(sup)-1 generates a C(sub)0-semigroup, and parabolicity means that -T(sup)-1 generates an analytic semigroup. A maximal regularity result is obtained for parabolic equations. We will also investigate the cases where the kernel k($.$) is degenerated or singular at t=0 using the results of Pruss[8] on analytic resolvents. Finally, we consider the case where $\lambda$ is a pole for ($\lambda$L + M)(sup)-1.

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EVALUATIONS OF THE IMPROPER INTEGRALS ${\int}_0^{\infty}$[sin$^{2m}({\alpha}x)]/(x^{2n})dx$ AND ${\int}_0^{\infty}$[sin$^{2m+1}({\alpha}x)]/(x^{2n+1})dx$

  • Qi, Feng;Luo, Qiu-Ming;Guo, Bai-Ni
    • The Pure and Applied Mathematics
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    • v.11 no.3
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    • pp.189-196
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    • 2004
  • In this article, using the L'Hospital rule, mathematical induction, the trigonometric power formulae and integration by parts, some integral formulae for the improper integrals ${\int}_0^{\infty}$[sin$^{2m}({\alpha}x)]/(x^{2n})dx$ AND ${\int}_0^{\infty}$[sin$^{2m+1}({\alpha}x)]/(x^{2n+1})dx$ are established, where m $\geq$ n are all positive integers and $\alpha$$\neq$ 0.

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