• Title/Summary/Keyword: Euler constant

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INTEGRAL FORMULAS FOR EULER'S CONSTANT

  • JUNESANG CHOI;TAE YOUNG SEO
    • Communications of the Korean Mathematical Society
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    • v.13 no.3
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    • pp.683-689
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    • 1998
  • There have been developed many integral representations for Euler's constant some of which are recorded here. We are aiming at showing a (presumably) new integral form of Euler's constant and disproving another integral representation for this constant which were recently proposed by Jean Angelsio, Garches, France, in American Mathematical Monthly. By modifying the Angelsio's incorrectly proposed integral form of Euler's constant, we also provide an integral representation for Euler's constant.

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SERIES REPRESENTATIONS FOR THE EULER-MASCHERONI CONSTANT $\gamma$

  • Choi, June-Sang;Seo, Tae-Young
    • East Asian mathematical journal
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    • v.18 no.1
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    • pp.75-84
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    • 2002
  • The third important Euler-Mascheroni constant $\gamma$, like $\pi$ and e, is involved in representations, evaluations, and purely relationships among other mathematical constants and functions, in various ways. The main object of this note is to summarize some known series representaions for $\gamma$ with comments for their proofs, and to point out that one of those series representaions for $\gamma$ seems to be incorrectly recorded. A brief historical comment for $\gamma$ is also provided.

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Determination of Elastic Modulus by Time Average ESPI and Euler-Bernoulli Equation (Time Average ESPI와 Euler-Bernoulli 방정식에 의한 탄성계수 측정)

  • Kim, Koung-Suk;Lee, Hang-Seo;Kang, Young-June;Kang, Ki-Soo
    • Journal of the Korean Society for Precision Engineering
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    • v.24 no.7 s.196
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    • pp.69-74
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    • 2007
  • The paper proposes a new sonic resonance test for a elastic modulus measurement which is based on time-average electronic speckle pattern interferometry(TA-ESPI) and Euler-Bernoulli equation. Previous measurement technique of elastic constant has the limitation of application for thin film or polymer material because contact to specimen affects the result. TA-ESPI has been developed as a non-contact optical measurement technique which can visualize resonance vibration mode shapes with whole-field. The maximum vibration amplitude at each vibration mode shape is a clue to find the resonance frequencies. The dynamic elastic constant of test material can be easily estimated from Euler-Bernoulli equation using the measured resonance frequencies. The proposed technique is able to give high accurate elastic modulus of materials through a simple experiment set up and analysis.

THE E-EULER PROCESS FOR NONAUTONOMOUS SYSTEMS

  • Yu, Dong-Won
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.8 no.2
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    • pp.87-93
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    • 2004
  • The E-Euler process has been proposed for autonomous dynamical systems in [7]. In this paper, the E-Euler process is extended to nonautonomous dynamical systems. When a discrete function is bounded or gradually decreases to ${\epsilon}\;<<\;1$ as $n\;{\rightarrow}\;{\infty}$, it is shown that the relative error converges to a constant or decreases.

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Measurement of Dynamic Elastic Modulus of Foil Material by ESPI and Sonic Resonance Testing (ESPI와 음향공진법을 이용한 Foil 재료의 동적탄성계수 측정)

  • Lee H.S.;Kim K.S.;Kang K.S.;Ahmad Akhlaq
    • Proceedings of the Korean Society of Precision Engineering Conference
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    • 2005.10a
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    • pp.914-917
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    • 2005
  • The paper proposes a new sonic resonance test for a dynamic elastic constant measurement which is based on time-average electronic speckle pattern interferometry(TA-ESPI)and Euler-Bernoulli equation. Previous measurement technique of dynamic elastic constant has the limitation of application for thin film or polymer material because contact to specimen affects the result. TA-ESPI has been developed as a non-contact optical measurement technique which can visualize resonance vibration mode shapes with whole-field. The maximum vibration amplitude at each vibration mode shape is a clue to find the resonance frequencies. The dynamic elastic constant of test material can be easily estimated from Euler-Bernoulli equation using the measured resonance frequencies. The TA-ESPI dynamic elastic constant measurement technique is able to give high accurate elastic modulus of materials through a simple experiment and analysis.

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MATHEMATICAL CONSTANTS ASSOCIATED WITH THE MULTIPLE GAMMA FUNCTIONS

  • Jung, Myung-Ho;Cho, Young-Joon;Choi, June-Sang
    • East Asian mathematical journal
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    • v.21 no.1
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    • pp.77-103
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    • 2005
  • The theory of multiple Gamma functions was studied in about 1900 and has, recently, been revived in the study of determinants of Laplacians. There is a class of mathematical constants involved naturally in the multiple Gamma functions. Here we summarize those mathematical constants associated with the Gamma and multiple Gamma functions and will show how they are involved, if possible.

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GENERALIZED EULER PROCESS FOR SYSTEMS OF NONLINEAR DIFFERENTIAL EQUATIONS

  • Yu, Dong-Won
    • Journal of applied mathematics & informatics
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    • v.7 no.3
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    • pp.941-958
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    • 2000
  • Euler method is generalized to solve the system of nonlinear differential equations. The generalization is carried out by taking a special constant matrix S so that exp(tS) can be exactly computed. Such a matrix S is extracted from the Jacobian matrix of the given problem. Stability of the generalized Euler process is discussed. It is shown that the generalized Euler process is comparable to the fourth order Runge-Kutta method. We also exemplify that the important qualitative and geometric features of the underlying dynamical system can be recovered by the generalized Euler process.

A Second-Order Particle Tracking Method

  • Lee, Seok;Lie, Heung-Jae;Song, Kyu-Min;Lim, Chong-Jeanne
    • Ocean Science Journal
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    • v.40 no.4
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    • pp.201-208
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    • 2005
  • An accurate particle tracking method for a finite difference method model is developed using a constant acceleration method. Being assumed constant temporal and spatial gradients, the new method permits temporal-spatial variability of particle velocity. Test results in a solid rotating flow show that the new method has second-order accuracy. The performance of the new method is compared with that of other methods; the first-order Euler forward method, and the second-order Euler predictor-corrector method. The new method is the most efficient method among the three. It is more accurate and efficient than the other two.