arXiv subjects

Radoslaw Szmytkowski

Publications and source records attributed to Radoslaw Szmytkowski.

10 recordsLinked to original sources

arXiv2011

Electric and magnetic dipole shielding constants for the ground state of the relativistic hydrogen-like atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

The Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825; erratum 30 (1997) 2747] is exploited to derive closed-form expressions for electric ($σ_{\mathrm{E}}$) and magnetic ($σ_{\mathrm{M}}$) dipole shielding constants for the ground state of the relativistic hydrogen-like atom with a point-like and spinless nucleus of charge $Ze$. It is found that $σ_{\mathrm{E}}=Z^{-1}$ (as it should be) and $$σ_{\mathrm{M}}=-(2Zα^{2}/27)(4γ_{1}^{3}+6γ_{1}^{2}-7γ_{1}-12) /[γ_{1}(γ_{1}+1)(2γ_{1}-1)],$$ where $γ_{1}=\sqrt{1-(Zα)^{2}}$ ($α$ is the fine-structure constant). This expression for $σ_{\mathrm{M}}$ agrees with earlier findings of several other authors, obtained with the use of other analytical techniques, and is elementary compared to an alternative one presented recently by Cheng \emph{et al.} [J. Chem. Phys. 130 (2009) 144102], which involves an infinite series of ratios of the Euler's gamma functions.

physics.atom-ph↗

arXiv2011

Comment on "Four-component relativistic theory for NMR parameters: Unified formulation and numerical assessment of different approaches" [J. Chem. Phys. 130, 144102 (2009)]

In the paper commented on [J. Chem. Phys. 130 (2009) 144102], Cheng et al. derived a formula for the magnetic dipole shielding constant $σ$ for the Dirac one-electron atom in its ground state. That formula involves an infinite series of ratios of the Euler's gamma functions. We show that with some algebra the series may be expressed in terms of elementary functions. This leads to a simple closed-form expression for the shielding constant.

physics.atom-ph↗

arXiv2010

A note on parameter derivatives of classical orthogonal polynomials

Coefficients in the expansions of the form $\partial P_{n}(λ;z)}/\partialλ=\sum_{k=0}^{n}a_{nk}(λ)P_{k}(λ;z)$, where $P_{n}(λ;z)$ is the $n$th classical (the generalized Laguerre, Gegenbauer or Jacobi) orthogonal polynomial of variable $z$ and $λ$ is a parameter, are evaluated. A method we adopt in the present paper differs from that used by Fröhlich [Integral Transforms Spec. Funct. 2 (1994) 253] for the Jacobi polynomials and by Koepf [Integral Transforms Spec. Funct. 5 (1997) 69] for the generalized Laguerre and the Gegenbauer polynomials.

math.CA↗

arXiv2009

Some differentiation formulas for Legendre polynomials

In a series of recent works, we have provided a number of explicit expressions for the derivative of the associated Legendre function of the first kind with respect to its degree, $[\partial P_ν^{m}(z)/\partialν]_{ν=n}$, with $m,n\in\mathbb{N}$. In this communication, we use some of those expressions to obtain several, we believe new, explicit formulas for the derivatives $\mathrm{d}^{m}[P_{n}(z)\ln(z\pm1)]/\mathrm{d}z^{m}$, where $P_{n}(z)$ is the Legendre polynomial.

math.CA↗

arXiv2009

On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)

A relationship between partial derivatives of the associated Legendre function of the first kind with respect to its degree, $[\partial P_ν^{m}(z)/\partialν]_{ν=n}$, and to its order, $[\partial P_{n}^μ(z)/\partialμ]_{μ=m}$, is established for $m,n\in\mathbb{N}$. This relationship is used to deduce four new closed-form representations of $[\partial P_ν^{m}(z)/\partialν]_{ν=n}$ from those found recently for $[\partial P_{n}^μ(z)/\partialμ]_{μ=m}$ by the present author [R. Szmytkowski, J. Math. Chem. 46 (2009) 231]. Several new expressions for the associated Legendre function of the second kind of integer degree and order, $Q_{n}^{m}(z)$, suitable for numerical purposes, are also derived.

math.CA↗

arXiv2009

An orthogonality relation for the Whittaker functions of the second kind of imaginary order

An orthogonality relation for the Whittaker functions of the second kind of imaginary order, $W_{κ,\mathrm{i}μ}(x)$, with $μ\in\mathbb{R}$, is investigated. The integral $\int_{0}^{\infty}\mathrm{d}x\: x^{-2}W_{κ,\mathrm{i}μ}(x)W_{κ,\mathrm{i}μ'}(x)$ is shown to be proportional to the sum $δ(μ-μ')+δ(μ+μ')$, where $δ(μ\pmμ')$ is the Dirac delta distribution. The proportionality factor is found to be $π^{2}/[μ\sinh(2πμ)Γ({1/2}-κ+\mathrm{i}μ) Γ({1/2}-κ-\mathrm{i}μ)]$. For $κ=0$ the derived formula reduces to the orthogonality relation for the Macdonald functions of imaginary order, discussed recently in the literature.

math.CA↗

arXiv2009

Comment on the orthogonality of the Macdonald functions of imaginary order

Recently, Yakubovich [Opuscula Math. 26 (2006) 161--172] and Passian et al. [J. Math. Anal. Appl. doi:10.1016/j.jmaa.2009.06.067] have presented alternative proofs of an orthogonality relation obeyed by the Macdonald functions of imaginary order. In this note, we show that the validity of that relation may be also proved in a simpler way by applying a technique occasionally used in mathematical physics to normalize scattering wave functions to the Dirac delta distribution.

math.CA↗

arXiv2009

On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree

In our recent works [R. Szmytkowski, J. Phys. A 39 (2006) 15147; corrigendum: 40 (2007) 7819; addendum: 40 (2007) 14887], we have investigated the derivative of the Legendre function of the first kind, $P_ν(z)$, with respect to its degree $ν$. In the present work, we extend these studies and construct several representations of the derivative of the associated Legendre function of the first kind, $P_ν^{\pm m}(z)$, with respect to the degree $ν$, for $m\in\mathbb{N}$. At first, we establish several contour-integral representations of $\partial P_ν^{\pm m}(z)/\partialν$. They are then used to derive Rodrigues-type formulas for $[\partial P_ν^{\pm m}(z)/\partialν]_{ν=n}$ with $n\in\mathbb{N}$. Next, some closed-form expressions for $[\partial P_ν^{\pm m}(z)/\partialν]_{ν=n}$ are obtained. These results are applied to find several representations, both explicit and of the Rodrigues type, for the associated Legendre function of the second kind of integer degree and order, $Q_{n}^{\pm m}(z)$; the explicit representations are suitable for use for numerical purposes in various regions of the complex $z$-plane. Finally, the derivatives $[\partial^{2}P_ν^{m}(z)/\partialν^{2}]_{ν=n}$, $[\partial Q_ν^{m}(z)/\partialν]_{ν=n}$ and $[\partial Q_ν^{m}(z)/\partialν]_{ν=-n-1}$, all with $m>n$, are evaluated in terms of $[\partial P_ν^{-m}(\pm z)/\partialν]_{ν=n}$.

math.CA↗

arXiv2008

On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order

The derivative of the associated Legendre function of the first kind of integer degree with respect to its order, $\partial P_{n}^μ(z)/\partialμ$, is studied. After deriving and investigating general formulas for $μ$ arbitrary complex, a detailed discussion of $[\partial P_{n}^μ(z)/\partialμ]_{μ=\pm m}$, where $m$ is a non-negative integer, is carried out. The results are applied to obtain several explicit expressions for the associated Legendre function of the second kind of integer degree and order, $Q_{n}^{\pm m}(z)$. In particular, we arrive at formulas which generalize to the case of $Q_{n}^{\pm m}(z)$ ($0\leqslant m\leqslant n$) the well-known Christoffel's representation of the Legendre function of the second kind, $Q_{n}(z)$. The derivatives $[\partial^{2} P_{n}^μ(z)/\partialμ^{2}]_{μ=m}$, $[\partial Q_{n}^μ(z)/\partialμ]_{μ=m}$ and $[\partial Q_{-n-1}^μ(z)/\partialμ]_{μ=m}$, all with $m>n$, are also evaluated.

math.CA↗

arXiv1999

Addendum to "The Dirac-Coulomb Sturmians and the series expansion of the Dirac-Coulomb Green function: application to the relativistic polarizability of the hydrogen-like atom" [J. Phys. B: At. Mol. Opt. Phys. 30 (1997) 825-61, (E) 30 (1997) 2747]

Closure relations satisfied by the radial Dirac-Coulomb Sturmians are proved analytically. The Sturmian expansion of the Dirac-Coulomb Green function is transformed to the form containing only series with summations running over non-negative indices. The main paper was published in J. Phys. B: At. Mol. Opt. Phys. 30 (1997) 825-61 [Erratum: 30 (1997) 2747], see also J. Phys. A: Math. Gen. 31 (1998) 4963--90 [Erratum: 31 (1998) 7415-6].

physics.atom-ph↗