Logarithmic Expansions Research Articles

Logarithmic Perturbation Expansions for a Central Potential

In this paper, the logarithmic perturbation expansions for the central potential are studied in details. Perturbation expansions of various orders of the perturbing potential of any order are treated. The unified recurrent formulas of corrections of various orders for the energies and wavefunctions of the ground state and low-lying excited states are given. The formulas for nodal positions of the wavefunctions of the excited states are also given.

The flow at a rear stagnation point is eventually determined by exponentially small values of the velocity

It is suggested that current conceptions about unsteady rear-stagnation-point flow do not fully describe the physics, since they show discrepancies from recent numerical results. The previously neglected exponentially small rotational perturbation velocity above the boundary-layer proves to have a dominating influence on the final boundary-layer development. An asymptotic analysis reveals possible difficulties for common computational schemes for viscous flows. Failure of the usual asymptotic matching rule in the analysis is in accordance with Fraenkel's warning on logarithmic expansions.

Logarithmic perturbation expansions in nonrelativistic quantum mechanics

We discuss an alternative method of performing coupling constant perturbation expansions in nonrelativistic quantum mechanics. This method, called logarithmic perturbation theory, yields new expressions for any-order corrections En to an unperturbed bound-state energy which do not involve cumbersome sums over intermediate unperturbed states. In one dimension, these corrections En can be evaluated using a simple explicit form containing a small number of integrals. In more than one dimension the approach is systematic but computations require the solution of well-defined partial differential equations. For order n=2 this equation is identical to that appearing in the method of Dalgarno and Lewis. Numerous illustrative examples are presented.

Bound state perturbation theory for the one-space and one-time dimension Klein–Gordon equation

We present a perturbation theory for an arbitrary bound state in the one-space and one-time dimension Klein–Gordon equation in the presence of a scalar potential and a vector (fourth component only) potential by reducing it to a Ricatti equation with the method of logarithmic perturbation expansions. All corrections to the energies and wavefunctions, including corrections to the positions of the nodes in excited states, are expressed in quadratures in a hierarchical scheme, without the use of either the Green’s function or the sum over intermediate states.

The asymptotic expansion of solutions of the differential equation u iv +λ 2 [(z 2 + c ) u" + azu '+ bu ]=0 for large IzI

The asymptotic expansion of solutions of the fourth order differential equation u iv + λ 2 [( z 2 + c ) u " + azu ' + bu ] = 0 are investigated for | z | -> oo where the parameters a, b, c and λ are supposed to be arbitrary complex constants with λ, b ^ 0. Exact solutions in the form of Laplace and Mellin-Barnes integrals, involving a Whittaker function and a Gauss hypergeometric function respectively, are used to define a fundamental system of solutions. The asymptotic expansion of these solutions is obtained in a full neighbourhood of the point at infinity and their asymptotic character is found to be either exponentially large or algebraic in certain sectors of the z -plane. The expansions corresponding to certain special values of the parameters a and b which yield logarithmic expansions are also treated. Linear combinations of these fundamental solutions which possess an exponentially small expansion for | z |->oo in a certain sector are discussed.

Logarithmic perturbation expansions

A method previously developed for one-dimensional nonrelativistic perturbation theory is extended to three-dimensional problems. This method essentially consists of performing the perturbation expansion on the logarithm of the wave function instead of on the wave function itself. It is shown that, for the first-order corrections in problems that are not reducible to one dimension, this method is equivalent to that of Sternheimer and to that of Dalgarno and Lewis. In the present approach, the higher-order corrections can be obtained in a hierarchical scheme and there exists an isomorphisim between the equation for the first-order correction and the equation for the $i\mathrm{th}$-order correction. As an illustration of the technique developed, the authors consider the hydrogen atom in an external multipole field and in two different spherically symmetric perturbation potentials, $\ensuremath{\delta}(r\ensuremath{-}a)$ and ${e}^{\ensuremath{-}\ensuremath{\alpha}r}$. The last potential is related to the problem of the screened Coulomb potential. By considering the $\ensuremath{\delta}(r\ensuremath{-}a)$-type potential, two interesting sum rules are obtained.

A Note on the Logarithmic and Binomial Expansions

(1971). A Note on the Logarithmic and Binomial Expansions. Mathematics Magazine: Vol. 44, No. 1, pp. 23-24.

A Note on the Logarithmic and Binomial Expansions

A Note on the Logarithmic and Binomial Expansions

Explaining the Construction of Logarithm Tables to 3rd Formers and a Note on e

Whether it is because of the utility aspect or whether it is because of some other reason, boys are invariably interested in logarithms. Whenever I take up the subject with a form, at least one boy will ask: “Please, Sir, how were logarithm tables constructed?” When a master is confronted with this question, he usually thinks of logarithmic expansions, and he is obliged to tell the boy that the explanation is beyond his understanding.

Explaining the Construction of Logarithm Tables to 3rd Formers and a Note on e

Whether it is because of the utility aspect or whether it is because of some other reason, boys are invariably interested in logarithms. Whenever I take up the subject with a form, at least one boy will ask: “Please, Sir, how were logarithm tables constructed?” When a master is confronted with this question, he usually thinks of logarithmic expansions, and he is obliged to tell the boy that the explanation is beyond his understanding.

Approximating tau-functions by theta-functions

We prove that the logarithm of an arbitrary tau-function of the KdV hierarchy can be approximated, in the topology of graded formal series by the logarithmic expansions of hyperelliptic theta-functions of finite genus, up to at most quadratic terms. As an example we consider theta-functional approximations of the Witten--Kontsevich tau-function.