Exact expansions of Hankel transforms and related integrals

Kisselev, A. V.

doi:10.1007/s11139-020-00274-x

Cited by 4 publications

(4 citation statements)

References 16 publications

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“…where k = k/ and Γ (z) denotes the Gamma function. In passing from (D1a) to (D1b), we used the Taylor series expansion of the function 1 Kisselev (2021). Using the Taylor series expansion given in (D1b), we obtain…”

Section: Author Orcidsmentioning

confidence: 99%

“…We begin by expanding the function about : where and denotes the Gamma function. In passing from (D1 a ) to (D1 b ), we used the Taylor series expansion of the function about based on the expression given in (70) in Kisselev (2021). Using the Taylor series expansion given in (D1 b ), we obtain From standard tables, we have the Taylor series expansion of as Using the series expansions in (D2) and (D3) in the expression for in (B1), we obtain the series expansion of as where .…”

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confidence: 99%

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Self-induced flow over a cylinder in a stratified fluid

Thomas

Camassa

2023

J. Fluid Mech.

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In this paper we study the self-induced low-Reynolds-number flow generated by a cylinder immersed in a stratified fluid. In the low Péclet limit, where the Péclet number is the ratio of the radius of the cylinder and the Phillips length scale, the flow is captured by a set of linear equations obtained by linearising the governing equations with respect to the prescribed far field conditions. We specifically focus on the low Péclet regime and develop a Green's function approach to solve the linearised equations governing the flow over the cylinder. We cross check our analytical solution against numerical solution of the nonlinear equations to obtain the range of the Péclet numbers for which the linear solution is valid. We then take advantage of the analytical solution to find explicit far-field decay rates of the flow. Our detailed analysis points out that the streamfunction and the velocity field decays algebraically in the far field. Intriguingly, this algebraic decay of the flow is much slower when compared with the exponential decay of the flow generated by a slow moving cylinder in the homogeneous Stokes regime, in the absence of stratification. Consequently, the flow generated by a cylinder in the stratified Stokes regime will have a larger domain of influence when compared with the flow generated by a cylinder in the homogeneous Stokes regime.

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Section: Author Orcidsmentioning

confidence: 99%

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confidence: 99%

Self-induced flow over a cylinder in a stratified fluid

Thomas

Camassa

2023

J. Fluid Mech.

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“…Some of these integrals cannot be solved manually and need computer software such as Mathematica and Maple to be solved. In addition, sometimes numerical methods can be used to solve some improper integrals that cannot be solved using previous methods [12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

General Master Theorems of Integrals with Applications

2022

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Many formulas of improper integrals are shown every day and need to be solved in different areas of science and engineering. Some of them can be solved, and others require approximate solutions or computer software. The main purpose of this research is to present new fundamental theorems of improper integrals that generate new formulas and tables of integrals. We present six main theorems with associated remarks that can be viewed as generalizations of Cauchy’s results and I.S. Gradshteyn integral tables. Applications to difficult problems are presented that cannot be solved with the usual techniques of residue or contour theorems. The solutions of these applications can be obtained directly, depending on the proposed theorems with an appropriate choice of functions and parameters.

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“…However, the order of Bessel function is required to be non-negative, and the accuracy is low. Kisselev [27] proved the Hankel transform can be expressed by the absolutely and uniformly convergent series in reciprocal powers of parameter r, and can be used to obtain analytic expressions for the Hankel transform of an arbitrary integer order with the positive parameter r. But the conditions for the non-oscillating item are very stringent. The G-S inverse Laplace transform method (G-SILTM) uses a pure real number operation, and only a small number of Laplace transform variable values need to be calculated, so the efficiency of the algorithm is relatively high.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Algorithm for Arbitrary Real-Order Hankel Transform

Yang

Ding

et al. 2022

Wuhan Univ. J. Nat. Sci.

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The Hankel transform is widely used to solve various engineering and physics problems, such as the representation of electromagnetic field components in the medium, the representation of dynamic stress intensity factors, vibration of axisymmetric infinite membrane and displacement intensity factors which all involve this type of integration. However, traditional numerical integration algorithms cannot be used due to the high oscillation characteristics of the Bessel function, so it is particularly important to propose a high precision and efficient numerical algorithm for calculating the integral of high oscillation. In this paper, the improved Gaver-Stehfest (G-S) inverse Laplace transform method for arbitrary real-order Bessel function integration is presented by using the asymptotic characteristics of the Bessel function and the accumulation of integration, and the optimized G-S coefficients are given. The effectiveness of the algorithm is verified by numerical examples. Compared with the linear transformation accelerated convergence algorithm, it shows that the G-S inverse Laplace transform method is suitable for arbitrary real order Hankel transform, and the time consumption is relatively stable and short, which provides a reliable calculation method for the study of electromagnetic mechanics, wave propagation, and fracture dynamics.

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Exact expansions of Hankel transforms and related integrals

Cited by 4 publications

References 16 publications

Self-induced flow over a cylinder in a stratified fluid

Self-induced flow over a cylinder in a stratified fluid

General Master Theorems of Integrals with Applications

A Numerical Algorithm for Arbitrary Real-Order Hankel Transform

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