• Title, Summary, Keyword: integral formula

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MODIFIED CONDITIONAL YEH-WIENER INTEGRAL WITH VECTOR-VALUED CONDITIONING FUNCTION

  • Chang, Joo-Sup
    • Journal of the Korean Mathematical Society
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    • v.38 no.1
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    • pp.49-59
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    • 2001
  • In this paper we introduce the modified conditional Yeh-Wiener integral. To do so, we first treat the modified Yeh-Wiener integral. And then we obtain the simple formula for the modified conditional Yeh-Wiener integral and valuate the modified conditional Yeh-Wiener integral for certain functional using the simple formula obtained. Here we consider the functional using the simple formula obtained. Here we consider the functional on a set of continuous functions which are defined on various regions, for example, triangular, parabolic and circular regions.

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RELATIONSHIP BETWEEN THE WIENER INTEGRAL AND THE ANALYTIC FEYNMAN INTEGRAL OF CYLINDER FUNCTION

  • Kim, Byoung Soo
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.2
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    • pp.249-260
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    • 2014
  • Cameron and Storvick discovered a change of scale formula for Wiener integral of functionals in a Banach algebra $\mathcal{S}$ on classical Wiener space. We express the analytic Feynman integral of cylinder function as a limit of Wiener integrals. Moreover we obtain the same change of scale formula as Cameron and Storvick's result for Wiener integral of cylinder function. Our result cover a restricted version of the change of scale formula by Kim.

A DEFINITE INTEGRAL FORMULA

  • Choi, Junesang
    • East Asian mathematical journal
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    • v.29 no.5
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    • pp.545-550
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    • 2013
  • A remarkably large number of integral formulas have been investigated and developed. Certain large number of integral formulas are expressed as derivatives of some known functions. Here we choose to recall such a formula to present explicit expressions in terms of Gamma function, Psi function and Polygamma functions. Some simple interesting special cases of our main formulas are also considered. It is also pointed out that the same argument can establish explicit integral formulas for other those expressed in terms of derivatives of some known functions.

THE BARTLE INTEGRAL AND THE CONDITIONAL WIENER INTEGRAL ON C[0,t]

  • Ryu, Kun-Sik;Im, Man-Kyu
    • Communications of the Korean Mathematical Society
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    • v.19 no.4
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    • pp.643-660
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    • 2004
  • In this paper, we give a new formula between the conditional Wiener integral E(F|X), the conditional Wiener integral of F given X, and the integral with respect to a measure-valued measure, a kind of Bartle integral. Using this formula, we give some examples of evaluation of E(F|X).

On the Evaluation of a Vortex-Related Definite Trigonometric Integral

  • Lee, Dong-Kee
    • Journal of Ocean Engineering and Technology
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    • v.18 no.1
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    • pp.7-9
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    • 2004
  • Using the solution to th contour integral of the complex logarithmic function ${\oint}_cIn(z-z_{0})dz$, the following definite integral, derived from the formula to calculate the forces exerted to n circular cylinder by the discrete vortices shed from it, has been evaluated (equation omitted)

A NOTE ON THE MODIFIED CONDITIONAL YEH-WIENER INTEGRAL

  • Chang, Joo-Sup;Ahn, Joong-Hyun
    • Communications of the Korean Mathematical Society
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    • v.16 no.4
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    • pp.627-635
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    • 2001
  • In this paper, we first introduce the modified Yeh-Wiener integral and then consider the modified conditional Yeh-Wiener integral. Here we use the space of continuous functions on a different region which was discussed before. We also evaluate some modified conditional Yeh-Wiener integral with examples using the simple formula for the modified conditional Yeh-Wiener integral.

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A CHANGE OF SCALE FORMULA FOR CONDITIONAL WIENER INTEGRALS ON CLASSICAL WIENER SPACE

  • Yoo, Il;Chang, Kun-Soo;Cho, Dong-Hyun;Kim, Byoung-Soo;Song, Teuk-Seob
    • Journal of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.1025-1050
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    • 2007
  • Let $X_k(x)=({\int}^T_o{\alpha}_1(s)dx(s),...,{\int}^T_o{\alpha}_k(s)dx(s))\;and\;X_{\tau}(x)=(x(t_1),...,x(t_k))$ on the classical Wiener space, where ${{\alpha}_1,...,{\alpha}_k}$ is an orthonormal subset of $L_2$ [0, T] and ${\tau}:0 is a partition of [0, T]. In this paper, we establish a change of scale formula for conditional Wiener integrals $E[G_{\gamma}|X_k]$ of functions on classical Wiener space having the form $$G_{\gamma}(x)=F(x){\Psi}({\int}^T_ov_1(s)dx(s),...,{\int}^T_o\;v_{\gamma}(s)dx(s))$$, for $F{\in}S\;and\;{\Psi}={\psi}+{\phi}({\psi}{\in}L_p(\mathbb{R}^{\gamma}),\;{\phi}{\in}\hat{M}(\mathbb{R}^{\gamma}))$, which need not be bounded or continuous. Here S is a Banach algebra on classical Wiener space and $\hat{M}(\mathbb{R}^{\gamma})$ is the space of Fourier transforms of measures of bounded variation over $\mathbb{R}^{\gamma}$. As results of the formula, we derive a change of scale formula for the conditional Wiener integrals $E[G_{\gamma}|X_{\tau}]\;and\;E[F|X_{\tau}]$. Finally, we show that the analytic Feynman integral of F can be expressed as a limit of a change of scale transformation of the conditional Wiener integral of F using an inversion formula which changes the conditional Wiener integral of F to an ordinary Wiener integral of F, and then we obtain another type of change of scale formula for Wiener integrals of F.

CHANGE OF SCALE FORMULAS FOR CONDITIONAL WIENER INTEGRALS AS INTEGRAL TRANSFORMS OVER WIENER PATHS IN ABSTRACT WIENER SPACE

  • Cho, Dong-Hyun
    • Communications of the Korean Mathematical Society
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    • v.22 no.1
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    • pp.91-109
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    • 2007
  • In this paper, we derive a change of scale formula for conditional Wiener integrals, as integral transforms, of possibly unbounded functions over Wiener paths in abstract Wiener space. In fact, we derive the change of scale formula for the product of the functions in a Banach algebra which is equivalent to both the Fresnel class and the space of measures of bounded variation over a real separable Hilbert space, and the $L_p-type$cylinder functions over Wiener paths in abstract Wiener space. As an application of the result, we obtain a change of scale formula for the conditional analytic Fourier-Feynman transform of the product of the functions.

A New Approach for the Derivation of a Discrete Approximation Formula on Uniform Grid for Harmonic Functions

  • Kim, Philsu;Choi, Hyun Jung;Ahn, Soyoung
    • Kyungpook Mathematical Journal
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    • v.47 no.4
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    • pp.529-548
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    • 2007
  • The purpose of this article is to find a relation between the finite difference method and the boundary element method, and propose a new approach deriving a discrete approximation formula as like that of the finite difference method for harmonic functions. We develop a discrete approximation formula on a uniform grid based on the boundary integral formulations. We consider three different boundary integral formulations and derive one discrete approximation formula on the uniform grid for the harmonic function. We show that the proposed discrete approximation formula has the same computational molecules with that of the finite difference formula for the Laplace operator ${\nabla}^2$.

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