Related papers: Integral and Series Representations of the Dirac D…
We calculate some infinite sums containing the digamma function in closed-form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as…
We formulate the Lorentz-Dirac equation in the plane wave and in the Dirac delta-function pulse. The discussion on the relation of the Dirac delta-function to the ultrashort laser pulse is involved.
Multiple-integral representations of the (skew-)Macdonald symmetric functions are obtained. Some bosonization schemes for the integral representations are also constructed.
This paper uses the convolution theorem of the Laplace transform to derive new inverse Laplace transforms for the product of two parabolic cylinder functions in which the arguments may have opposite sign. These transforms are subsequently…
We present an integral representation formula for a Dirichlet series whose coefficients are the values of the Liouville's arithmetic function.
We present a generalization of the representation in plane waves of Dirac delta, $\delta(x)=(1/2\pi)\int_{-\infty}^\infty e^{-ikx}\,dk$, namely $\delta(x)=(2-q)/(2\pi)\int_{-\infty}^\infty e_q^{-ikx}\,dk$, using the…
The Digamma and Polygamma functions are important tools in mathematical physics, not only for its many properties but also for the applications in statistical mechanics and stellar evolution. In many textbooks is found its develop almost by…
Exact analytic solutions are found to the Dirac equation for a combination of Lorentz scalar and vector Coulombic potentials with additional non-Coulombic parts. An appropriate linear combination of Lorentz scalar and vector non-Coulombic…
Using the Dirichlet integrals, which are employed in the theory of Fourier series, this paper develops a useful method for the summation of series and the evaluation of integrals.
On the one hand the Fermi-Dirac and Bose-Einstein functions have been extended in such a way that they are closely related to the Riemann and other zeta functions. On the other hand the Fourier transform representation of the gamma and…
We give a half-page proof of the Lagrange-Good formula, using the Fourier representation of Dirac delta function.
Motivated by the substantial development of the special functions, we contribute to establish some rigorous results on the general series identities with bounded sequences and hypergeometric functions with different arguments, which are…
A representation of solutions of the one-dimensional Dirac equation is obtained. The solutions are represented as Neumann series of Bessel functions. The representations are shown to be uniformly convergent with respect to the spectral…
Using the language of the Geometric Algebra, we recast the massive Dirac bispinor as a set of Lorentz scalar, vector, bivector, pseudovector, and pseudoscalar fields that obey a generalized form of Maxwell's equations of electromagnetism.…
Smooth, highly accurate analytical representations of Fermi-Dirac (FD) integral combinations important in free-energy density functional calculations are presented. Specific forms include those that occur in the local density approximation…
In the paper, the authors establish integral representations of some functions related to the remainder of Burnside's formula for the gamma function and find the (logarithmically) complete monotonicity of these and related functions. These…
Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $\eta(s)$, and hence Riemann's function $\zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(i\kappa)$ of…
Generating functions and functional equations of Dickson polynomials of the first and second kind are derived and continued analytically. These formulae are expressed in terms of the incomplete gamma function over complex variables of the…
By using the theory of the elliptic integrals a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect…
Matrix elements of spinor and principal series representations of the Lorentz group are studied in the basis of complex angular momentum (helicity basis). It is shown that matrix elements are expressed via hyperspherical functions…