Evaluation of integrals and the mellin transform

Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I.

doi:10.1007/bf01373648

Cited by 16 publications

(15 citation statements)

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“…On MB-integrals, see [6], [15], [18], [48], and the comprehensive book [49]. [8] and [11]- [13] contain simple, informative discussions relevant to the MT-method, not too different from the general material in Sections II-A to II-G; more detailed expositions can be found in the pioneering (but readable) works [6], [2]. The origins of what we call the "MT-method" go far back: The idea of the Mellin inversion formula appeared in an 1876 memoir by Riemman, and the first accurate discussion was given by Mellin in 1896 and 1902.…”

Section: An Integral Arising In the Theory Of Biaxially Anisotropimentioning

confidence: 99%

“…Let us call this a "standard product." 2 Matlab is a registered trademark of The MathWorks, Natick, MA.…”

Section: E Table Lookup Of Mellin Transforms; Mellin-barnes Integralsmentioning

confidence: 99%

“…In particular, it forms an important part of Mathematica's 1 routine which, according to S. Wolfram [1], "can evaluate most definite integrals listed in standard books of tables." Furthermore, the technique "has been used in an essential manner" [2] for the creation of the integral tables in the monumental, three-volume reference work [3]- [5]. We call the technique the "Mellin-transform method" (MT-method), because taking a Mellin transform (MT) is the method's initial and key step.…”

Section: Introductionmentioning

confidence: 99%

“…We call the technique the "Mellin-transform method" (MT-method), because taking a Mellin transform (MT) is the method's initial and key step. But it is known in the literature [1] with other names, such as the Marichev-Adamchik method [2], [6]- [8].…”

Section: Introductionmentioning

confidence: 99%

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Integral Evaluation Using the Mellin Transform and Generalized Hypergeometric Functions: Tutorial and Applications to Antenna Problems

Fikioris

2006

IEEE Trans. Antennas Propagat.

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This is a tutorial presentation of the Mellin-transform (MT) method for the exact calculation of one-dimensional definite integrals, and an illustration of the application of this method to antenna/electromagnetics problems. Once the basics have been mastered, one quickly realizes that the MT-method is extremely powerful, often yielding closed-form expressions very difficult to come up with other methods or to deduce from the usual tables of integrals. Yet, as opposed to other methods, the MT-method is very straightforward to apply; it usually requires laborious calculations, but little ingenuity. In fact, the MT-method is used by Mathematica to symbolically calculate definite integrals. The first part of this paper is a step-by-step tutorial, proceeding from first principles. It includes basic information on Mellin-Barnes integrals and generalized hypergeometric functions, and summarizes the key ideas of the MT-method. In the remaining parts, the MT-method is applied to three examples from the antenna area. The results here are believed to be new, at least in the antenna/electromagnetics literature. In our first example, we obtain a closed-form expression, as a generalized hypergeometric function, for the power radiated by a constant-current circular-loop antenna; this quantity has been extensively discussed recently. Our second example concerns the admittance of a 2-D slot antenna. In both these examples, the exact closed-form expressions are applied to improve upon existing formulas in standard antenna textbooks. In our third example, finally, we obtain a very simple expression for an integral arising in recent, unpublished studies of unbounded, biaxially anisotropic media.

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Section: An Integral Arising In the Theory Of Biaxially Anisotropimentioning

confidence: 99%

“…Let us call this a "standard product." 2 Matlab is a registered trademark of The MathWorks, Natick, MA.…”

Section: E Table Lookup Of Mellin Transforms; Mellin-barnes Integralsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Integral Evaluation Using the Mellin Transform and Generalized Hypergeometric Functions: Tutorial and Applications to Antenna Problems

Fikioris

2006

IEEE Trans. Antennas Propagat.

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“…The following constraints can be taken as the latter [31]. Let f (x) be absolutely integrable in any finite interval (ε 1 , ε 2 ), 0 < ε 1 < ε 2 < ∞, and satisfy the constraints |f (x)| < Ax −σ 1 for 0 < x ε 1 and |f (x)| < Ax −σ 2 for x ε 2 , σ 1 < σ 2 , A = const > 0.…”

Section: The Mellin Transformmentioning

confidence: 99%

Absorption of gamma-ray photons in a vacuum neutron star magnetosphere: I. Electron-positron pair production

Istomin

Sob’yanin²

2011

J. Exp. Theor. Phys.

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The production of electron-positron pairs in a vacuum neutron star magnetosphere is investigated for both low (compared to the Schwinger one) and high magnetic fields. The case of a strong longitudinal electric field where the produced electrons and positrons acquire a stationary Lorentz factor in a short time is considered. The source of electron-positron pairs has been calculated with allowance made for the pair production by curvature and synchrotron photons. Synchrotron photons are shown to make a major contribution to the total pair production rate in a weak magnetic field. At the same time, the contribution from bremsstrahlung photons may be neglected.The existence of a time delay due to the finiteness of the electron and positron acceleration time leads to a great reduction in the electron-positron plasma generation rate compared to the case of a zero time delay. The effective local source of electron-positron pairs has been constructed. It can be used in the hydrodynamic equations that describe the development of a cascade after the absorption of a photon from the cosmic gamma-ray background in a neutron star magnetosphere.

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Rapidly convergent series and closed-form expressions for a class of integrals involving products of spherical Bessel functions

Lovat,

Celozzi

2024

Journal of Computational and Applied Mathematics

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Evaluation of integrals and the mellin transform

Cited by 16 publications

References 482 publications

Integral Evaluation Using the Mellin Transform and Generalized Hypergeometric Functions: Tutorial and Applications to Antenna Problems

Integral Evaluation Using the Mellin Transform and Generalized Hypergeometric Functions: Tutorial and Applications to Antenna Problems

Absorption of gamma-ray photons in a vacuum neutron star magnetosphere: I. Electron-positron pair production

Rapidly convergent series and closed-form expressions for a class of integrals involving products of spherical Bessel functions

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