• Title, Summary, Keyword: Improper integral

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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.

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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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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Gate Location Design of an Automobile Junction Box with Integral Hinges (복합힌지를 갖는 차량용 정션박스의 게이트 위치설계)

  • 김홍석
    • Transactions of Materials Processing
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    • v.12 no.2
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    • pp.134-140
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    • 2003
  • Polymers such as polypropylene or polyethylene offer a unique feature of producing an integral hinge, which can flex over a million times without causing a failure. With such advantage manufacturing, time and cost required at the assembly stage can be eliminated by injecting the whole part as one piece. However, due to increased fluidity resistance at hinges during molding, several defects such as short shot or premature hinge failure can occur with the improper selection of gate locations. Therefore, it is necessary to optimize flow balancer in injection molding of part with hinges before actually producing molds. In this paper, resin flow patterns depending on several gate positions were investigated by numerical analyses of a simple strip part with a hinge. As a result, we found that the properly determined gate location leads to better resin flow and shorter hesitation time. Finally, injection molding tryouts using a mold that was designed one of the proposed gate systems were conducted using polypropylene that contained 20% talc. The experiment showed that hinges without defects could be produced by using the designed gate location.

Root Cause Analysis on the Steam Turbine Blade Damage of the Combined Cycle Power Plant (복합화력발전소 증기터빈 동익 손상 원인분석)

  • Kang, M.S.;Kim, K.Y.;Yun, W.N.;Lee, W.K.
    • Journal of Power System Engineering
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    • v.12 no.4
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    • pp.57-63
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    • 2008
  • The last stage blade of the low pressure steam turbine remarkably affects turbine plant performance and availability Turbine manufacturers are continuously developing the low pressure last stage blades using the latest technology in order to achieve higher reliability and improved efficiency. They tend to lengthen the last stage blade and apply shrouds at the blades to enhance turbine efficiency. The long blades increase the blade tip circumferential speed and water droplet erosion at shroud is anticipated. Parts of integral shrouds of the last stage 40 inch blades were cracked and liberated recently in a combined cycle power plant. In order to analyze the root cause of the last stage blades shroud cracks, we investigated operational history, heat balance diagram, damaged blades shape, fractured surface of damaged blades, microstructure examination and design data, etc. Root causes were analyzed as the improper material and design of the blade. Notches induced by erosion and blade shroud were failed eventually by high cycle fatigue. This paper describes the root cause analysis and countermeasures for the steam turbine last stage blade shroud cracks of the combined cycle power plant.

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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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