共查询到20条相似文献，搜索用时 15 毫秒
1.
《Integral Transforms and Special Functions》2012,23(12):927-941
ABSTRACTAn earlier method for obtaining indefinite integrals of special function from the second-order linear equations which define them has been reformulated in terms of Riccati equations, which are nonlinear and first-order. For this application the nonlinearity is not a problem and the first-order property is a great advantage. Integrals can be derived using fragments of these Riccati equations and here only two specific fragment types are examined in detail. These fragments allow general integration formulas to be derived using quadrature. Other results will be presented separately. Results are presented here for Airy functions, Bessel functions, complete elliptic integrals, associated Legendre functions and Gauss hypergeometric functions. All results have been checked by differentiation using Mathematica. 相似文献
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John T. Conway 《Integral Transforms and Special Functions》2018,29(10):805-819
Sequences of new recurrence relations are presented for Bessel functions, parabolic cylinder functions and associated Legendre functions. The sequences correspond to values of an integer variable r and are generalizations of each conventional recurrence relation, which correspond to r=1. The sequences can be extended indefinitely, though the relations become progressively more intricate as r increases. These relations all have the form of a first-order linear inhomogeneous differential equation, which can be solved by an integrating factor. This gives a very general indefinite integral for each recurrence. The method can be applied to other special functions which have conventional recurrence relations. All results have been checked numerically using Mathematica. 相似文献
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John T. Conway 《Integral Transforms and Special Functions》2020,31(4):253-267
ABSTRACTElementary linear first and second order differential equations can always be constructed for twice differentiable functions by explicitly including the function's derivatives in the definition of these equations. If the function also obeys a conventional differential equation, information from this equation can be introduced into the elementary equations to give blended linear equations which are here called hybrid equations. Integration theorems are derived for these hybrid equations and several universal integrals are also derived. The paper presents integrals derived with these methods for cylinder functions, associated Legendre functions, and the Gegenbauer, Chebyshev, Hermite, Jacobi and Laguerre orthogonal polynomials. All the results presented have been checked using Mathematica. 相似文献
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《Integral Transforms and Special Functions》2012,23(3-4):261-274
Fourier transforms on finite intervals (finite Fourier transforms) are considered here, not as Fourier coefficients, but as functions of a continuous variables. The tables of properties of finite Fourier exponential, Fourier sine, and Fourier consine transformations are composed. The use of these properties is illustrated by means of an extension of Graf's theorem for Bessel functions, a series of the parabolic cylinder functions, and spectral relationships for Chebyshev and Legendre polynomials. 相似文献
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Vector displacements expressed in spherical coordinates are proposed. They correspond to electromagnetic fields in vacuum that globally rotate about an axis and display many circular patterns on the surface of a ball. The fields satisfy the set of Maxwell's equations, and some connections with magnetohydrodynamics can also be established. The solutions are extended with continuity outside the ball. In order to avoid peripheral velocities of arbitrary magnitude, as it may happen for a rigid rotating body, they are organized to form successive encapsulated shells, with substructures recalling ball-bearing assemblies. A recipe for the construction of these solutions is provided by playing with the eigenfunctions of the vector Laplace operator. Some applications relative to astronomy are finally discussed. 相似文献
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John T. Conway 《Integral Transforms and Special Functions》2017,28(3):166-180
A method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals of products of Gauss hypergeometric functions are also presented, which seem to be the first integrals of this type. All results presented have been numerically checked with Mathematica. 相似文献
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《Integral Transforms and Special Functions》2012,23(11):845-858
A substantial number of indefinite integrals of special functions are presented, which have been obtained using a new method presented in a companion paper [Conway JT. A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015; submitted to]. The method was originally derived from the Euler–Lagrange equations but an elementary proof is also presented in [Conway JT. A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015; submitted to]. Sample results are presented here for Bessel functions, Airy functions and hypergeometric functions. More extensive results are given for the complete elliptic integrals of the first and second kinds. Results are presented which link integrals of some products of associated Legendre functions and the complete elliptic integral of the second kind with the Golden Ratio. The method is applicable to any elementary or special function which satisfies a linear ordinary differential equation of the second order. 相似文献
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《Integral Transforms and Special Functions》2012,23(3):229-244
We consider multiple orthogonal polynomials corresponding to two Macdonald functions (modified Bassel functions of the second kind), with emphasis on the polynomials on the diagonal of the Hermite-Padé table. We give some properties of these polynomials: differential properties, a Rodrigues type formula and explicit formulas for the third order linear recurrence relation. 相似文献
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《Integral Transforms and Special Functions》2012,23(4):243-262
We consider a modification of moment functionals for the Hahn classical polynomials of a discrete variable by adding two mass points at the ends of the interval, i.e., in x = 0 and x = N-1. We obtain the resulting orthogonal polynomials and identify them as hypergeometric functions. The corresponding three term recurrence relation and tridiagonal matrices are also studied. 相似文献
13.
Some Representations of Unified Voigt Functions 总被引：1，自引：0，他引：1
M. KAMARUJJAMA Dinesh SINGH 《数学学报(英文版)》2005,21(4):865-868
The authors derive a set of unified representations of the Voigt functions in terms of familiar special functions of Mathematical Physics. Some deductions from these representations are also considered. 相似文献
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John T. Conway 《Integral Transforms and Special Functions》2018,29(4):269-283
A new method is presented for deriving indefinite integrals involving quotients of special functions. The method combines an integration formula given previously with the recursion relations obeyed by the function. Some additional results are presented using an elementary method, here called reciprocation, which can also be used in combination with the new method to obtain additional quotient integrals. Sample results are given here for Bessel functions, Airy functions, associated Legendre functions and the three complete elliptic integrals. All results given have been numerically checked with Mathematica. 相似文献
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Howard S. Cohl Jessica E. Hirtenstein Hans Volkmer 《Integral Transforms and Special Functions》2016,27(10):767-774
In 1946, Magnus presented an addition theorem for the confluent hypergeometric function of the second kind U with argument x+y expressed as an integral of a product of two U's, one with argument x and another with argument y. We take advantage of recently obtained asymptotics for U with large complex first parameter to determine a domain of convergence for Magnus' result. Using well-known specializations of U, we obtain corresponding integral addition theorems with precise domains of convergence for modified parabolic cylinder functions, and Hankel, Macdonald, and Bessel functions of the first and second kind with order zero and one. 相似文献
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《Integral Transforms and Special Functions》2012,23(4):331-337
Orthogonality relations for the associated Legendre functions of imaginary order are derived. They are expressed in terms of the Dirac delta function. The method is based on some known properties of the associated Legendre functions and the Dirac delta distribution. A special case of one of the relations has appeared in some recent applications. 相似文献
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John T. Conway 《Integral Transforms and Special Functions》2017,28(6):488-503
A method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete elliptic integrals. This gives a total of 27 integrals for the complete integral of the third kind, including the single integral given in the literature. Some typographical errors in a previous related paper are corrected. 相似文献
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Arcadii Z. Grinshpan 《Journal of Mathematical Analysis and Applications》2006,314(2):724-735
Several integral inequalities for the classical hypergeometric, confluent hypergeometric, and confluent hypergeometric limit functions are given. The related results for Bessel and Whittaker functions as well as for Laguerre, Hermite, and Jacobi polynomials are discussed. 相似文献
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《Integral Transforms and Special Functions》2012,23(4):277-287
In a recent paper we introduced a multivariable generalization of the Bessel polynomials, depending on one extra parameter, and related to the so-called hyperbolic Calogero–Moser–Sutherland model with external Morse potential. In this paper, we obtain a corresponding multivariable generalization of a well-known orthogonality relation for the (one-variable) Bessel polynomials due to Krall and Frink [H.L. Krall and O. Frink, A new class of orthogonal polynomials: the Bessel polynomials, Trans. Amer. Math. Soc. 65 (1949), pp. 100–115]. 相似文献