Euler Beta Research Articles

Inequalities for the beta function of n variables

AbstractWe present various inequalities for Euler's beta function of n variables. One of our theorems states that the inequalitieshold for all xi ≥ (i = 1,… n; n ≥ 3) with the best possible constants an = 0 and bn = 1 − 1/(n − 1)!. This extends a recently published result of Dragomir et al., who investigated (*) for the special case n = 2.

Relative orientation of quadrupole tensors from two-dimensional multiple-quantum MAS NMR.

The Multiple-Quantum Magic Angle Spinning (MQMAS) technique enables the resolution of crystallographically inequivalent nuclei and also yield the quadrupolar coupling constant C-Q, quadrupolar asymmetry eta, and isotropic chemical shift delta(CS) of each crystallographic site. However, no information is obtained that relates one quadrupole tensor to another, such as internuclear distances or relative orientations. Here, we discuss a recently developed modification of the MQMAS NMR experiment that utilizes two-dimensional correlation of second-order (rank I = 4) broadened MAS line shapes to obtain the relative orientation of quadrupole tensors. This new method involves the insertion of a mixing time, tau(m), into the MQMAS experiment such that magnetization transfer between two distinct nuclei within a crystallite during this period will result in a two-dimensional cross-peak correlating the two line shapes. The shapes of these cross-peaks are characteristic of the three Euler angles, alpha', beta', and gamma', that describe the relative orientation of the two quadrupole tensors.

Pion form factor in large Nc QCD

The electromagnetic form factor of the pion is obtained using a particular realization of QCD in the large $N_c$ limit, which sums up the infinite number of zero-width resonances to yield an Euler's Beta function of the Veneziano type. This form factor agrees with space-like data much better than single rho-meson dominance. A simple unitarization ansatz is illustrated, and the resulting vector spectral function in the time-like region is shown to be in reasonable agreement with the ALEPH data from threshold up to about 1.3 ${GeV}^2$.

On the Ostrowski's integral inequality for mappings with bounded variation and applications

A generalization of Ostrowski's inequality for mappings with bounded variation and applications in Numerical Analysis for Euler's Beta function is given.

Inequalities for Beta and Gamma functions via some classical and new integral inequalities

In this survey paper we present the natural application of certain integral inequalities such as, Chebychev's inequality for synchronous and asynchronous mappings, Holder's inequality and Gruss' and Ostrowski's inequalities for the celebrated Euler's Beta and Gamma functions. Natural applications dealing with some adaptive quadrature formulae which can be deduced from Ostrowski's inequality are also pointed out.

The Ostrowski's integral inequality for Lipschitzian mappings and applications

A generalization of Ostrowski's inequality for Lipschitzian mappings and applications in numerical analysis and for Euler's Beta function are given.

Remarks on a generalized beta function

We show that a certain generalized beta function B( x, y; b) which reduces to Euler's beta functions B( x, y) when its variable b vanishes and preserves symmetry in its parameters may be represented in terms of a finite number of well known higher transcendental functions except (possibly) in the case when one of its parameters is an integer and the other is not. In the latter case B( x, y; b) may be represented as an infinite series of either Wittaker functions or Laguerre polynomials. As a byproduct of this investigation we deduce representations for several infinite series containing Wittaker functions, Laguerre polynomials, and products of both.

Derivation of the fine-structure formula utilizing the regularity properties of the Euler beta function

The method utilizing the integral Fourier transformation of the wave function of the ground state of the hydrogen-like atom is used to find its momentum representation, which is then used to derive the formula for the electron energy levels. Useful regularity properties are found for the Euler beta function appearing in the calculation of some formally diverging integrals.

An Integral with Three Parameters

In this paper we give an exact expression for the integral \begin{eqnarray} I(a, b; r) = \int_{0}^{\infty} \left[ \frac{x^{2}}{x^{4} + 2a x^{2} + 1} \right]^{r} \cdot \frac{x^{2} + 1}{x^{b} + 1} \cdot \frac{dx}{x^{2}} \nonumber \end{eqnarray} in terms of Euler's beta function. Several classical integrals are deduced from it. The expression for $I(a,b;r)$ provides a method for unifying a large class of integrals.

Extension of Euler's beta function

An extension of Euler's beta function, analogous to the recent generalization of Euler's gamma function and Riemann's zeta function, for which the usual properties and representation are naturally and simply extended, is introduced. It is proved that the extension is connected to the Macdonald, error and Whittaker functions. In addition, the extended beta distribution is introduced.

Algebraic independence of the periods of abelian integrals

Abelian integrals of the first and the second kind are proved to have two algebraically independent periods. Some corollaries concerning the algebraic independence of the values of Euler's beta and gamma functions at rational points are derived.

NORMALIZATION OF THE WAVEFUNCTION FOR THE CALOGERO-SUTHERLAND MODEL WITH INTERNAL DEGREES OF FREEDOM

The exact normalization of a multicomponent generalization of the ground state wave-function of the Calogero-Sutherland model is conjectured. This result is obtained from a conjectured generalization of Selberg’s N-dimensional extension of the Euler beta integral, written as a trigonometric integral. A new proof of the Selberg integral is given, and the method is used to provide a proof of the multicomponent generalization in a special two-component case.

On some generalizations of the Vandermonde matrix and their relations with the Euler beta-function

A multiple Vandermonde matrix which, besides the po- wers of variables, also contains their derivatives is introduced and an explicit expression of its determinant is obtained. For the case of arbitrary real powers, when the variables are positive, it is proved that such generalized multiple Vandermonde matrix is positive def- inite for appropriate enumerations of rows and columns. As an ap- plication of these results, some relations are obtained which in the one-dimensional case give the well-known formula for the Euler beta- function.

Orthogonal polynomials and some q-beta integrals of Ramanujan

Two integrals of Ramanujan are used to define a q-analogue of the Euler beta integral on the real line and of the Cauchy beta-integral on the complex unit circle. Such integrals are connected to orthogonal, biorthogonal and Laurent polynomials. Explicit examples of Laurent orthogonal polynomials are given on the real line and on the circle.