The Improper Integral

The stationary output of a stable linear dynamic system to white noise input is considered. It is shown that the spectral moments of the output can be determined as the solution of a set of linear equations. The corresponding solutions are utilized for determining an improper integral which can be quite useful in random vibration analyses of linear dynamic systems.

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Aumann Fuzzy Improper Integral and Its Application to Solve Fuzzy Integro-Differential Equations by Laplace Transform Method

Advances in Fuzzy Systems ◽

10.1155/2018/9730502 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Elhassan Eljaoui ◽

Said Melliani ◽

L. Saadia Chadli

Keyword(s):

Differential Equations ◽

Laplace Transform ◽

Convolution Type ◽

Convolution Product ◽

Fuzzy Mapping ◽

Laplace Transform Method ◽

Transform Method ◽

Improper Integral ◽

Convolution Formula

We introduce the Aumann fuzzy improper integral to define the convolution product of a fuzzy mapping and a crisp function in this paper. The Laplace convolution formula is proved in this case and used to solve fuzzy integro-differential equations with kernel of convolution type. Then, we report and correct an error in the article by Salahshour et al. dealing with the same topic.

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Power Series.Orthogonal Operational Expansions

10.14293/s2199-1006.1.sor-.ppk4cqc.v1 ◽

2020 ◽

Author(s):

Pablo Enrique Aballe Vazquez

Keyword(s):

Hilbert Space ◽

Taylor Series ◽

Laplace Transforms ◽

Unique Function ◽

Differential Transformation ◽

Improper Integral ◽

Analytical Functions ◽

Positive Semiaxis ◽

Function Definition ◽

Definition Of

Formulation of the classic Taylor series as an orthogonal concept based on identifying the expansions coefficients as differential transformation applied to a unique function; definition of operational orthogonality by analogy with the Hilbert space and identification of the nth derivative at a point based on the improper integral on the positive semiaxis ; deduction of inversion integrals for Laplace transforms for analytical functions

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Rocking Response of a Surface-Supported Strip Foundation under a Harmonic Swaying Force

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.226-228.1453 ◽

2012 ◽

Vol 226-228 ◽

pp. 1453-1457

Author(s):

Jue Wang ◽

Ding Zhou ◽

Wei Qing Liu ◽

Shu Guang Wang

Keyword(s):

Integration Method ◽

Vertical Displacement ◽

Elastic Half Space ◽

Infinite Length ◽

Rigid Foundation ◽

Strip Foundation ◽

Impedance Function ◽

Improper Integral ◽

Layer Method ◽

Thin Layer Method

This paper presents an accurate analytical method to obtain the rocking impedance function of a surface-supported strip foundation. The Green’s functions of the elastic half-space under concentrated or uniform loads with infinite length are derived and an elaborate integration method is used to calculate the multi-value improper integral. The interface between the foundation and the supporting medium is divided into a number of strip units. The rocking impedance function is solved by adding the moments in every strip, based on the fact that the vertical displacement of each unit can be uniquely determined by the rotation amplitude of the rigid foundation. Excellent convergence has been observed. Comparing the numerical results to those obtained by the thin layer method, good agreements are achieved. Finally, the effect of the Poisson’s ratio on the rocking impedance function of the strip foundation is discussed in detail.

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An Improper Integral with a Square Root

10.20944/preprints201610.0089.v1 ◽

2016 ◽

Cited By ~ 2

Author(s):

Feng Qi

Keyword(s):

Catalan Numbers ◽

Integral Representations ◽

Square Root ◽

Improper Integral ◽

Improper Integrals

In the paper, the author presents explicit and unified expressions for a sequence of improper integrals in terms of the beta functions and the Wallis ratios. Hereafter, the author derives integral representations for the Catalan numbers originating from combinatorics.

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Analytical Solution to Improper Integral of Divergent Power Functions Using The Riemann Zeta Function

10.31219/osf.io/9h6ab ◽

2019 ◽

Author(s):

Siamak Tafazoli ◽

Farhad Aghili

Keyword(s):

Closed Form Solution ◽

Zeta Functions ◽

Bernoulli Numbers ◽

Form Solution ◽

Binomial Coefficients ◽

Solution Technique ◽

Power Functions ◽

Definite Integrals ◽

Improper Integral ◽

Riemann Zeta

This paper presents an analytical closed-form solution to improper integral $\mu(r)=\int_0^{\infty} x^r dx$, where $r \geq 0$. The solution technique is based on splitting the improper integral into an infinite sum of definite integrals with successive integer limits. The exact solution of every definite integral is obtained by making use of the binomial polynomial expansion, which then allows expression of the entire summation equivalently in terms of a weighted sum of Riemann zeta functions. It turns out that the solution fundamentally depends on whether or not $r$ is an integer. If $r$ is a non-negative integer, then the solution is manifested in a finite series of weighted Bernoulli numbers, which is then drastically simplified to a second order rational function $\mu(r)=(-1)^{r+1}/(r+1)(r+2)$. This is achieved by taking advantage of the relationships between Bernoulli numbers and binomial coefficients. On the other hand, if $r$ is a non-integer real-valued number, then we prove $\mu(r)=0$ by the virtue of the elegant relationships between zeta and gamma functions and their properties.

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On a Well-Known Improper Integral

American Mathematical Monthly ◽

10.2307/2315818 ◽

1967 ◽

Vol 74 (7) ◽

pp. 846 ◽

Cited By ~ 1

Author(s):

Hiroshi Haruki

Keyword(s):

Improper Integral

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On Fuzzy Improper Integral and Its Application for Fuzzy Partial Differential Equations

International Journal of Differential Equations ◽

10.1155/2016/7246027 ◽

2016 ◽

Vol 2016 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

ElHassan ElJaoui ◽

Said Melliani

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Laplace Transforms ◽

Partial Differential ◽

First Order ◽

Improper Integral ◽

Linear Partial Differential Equations ◽

Riemann Integrals ◽

Generalized Hukuhara Differentiability

We establish some important results about improper fuzzy Riemann integrals; we prove some properties of fuzzy Laplace transforms, which we apply for solving some fuzzy linear partial differential equations of first order, under generalized Hukuhara differentiability.

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