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

1998 ◽  
Vol 41 (5) ◽  
pp. 489-491
Author(s):  
D. K. Ershov
Keyword(s):  
Fine Structure ◽  
Beta Function ◽  
Regularity Properties ◽  
Euler Beta Function ◽  
Structure Formula

COMPARISON BETWEEN POLYNOMIAL, EULER BETA-FUNCTION AND EXPO-RATIONAL B-SPLINE BASES

2011 ◽  
Author(s):  
Arnt R. Kristoffersen ◽  
Lubomir T. Dechevsky ◽  
Arne Lakså ◽  
Bo̸rre Bang ◽  
George Venkov ◽  
...  
Keyword(s):  
Beta Function ◽  
B Spline ◽  
Euler Beta Function

scholarly journals Deviation from the Coulomb law for the proton

1939 ◽  
Vol 171 (945) ◽  
pp. 269-280 ◽  
Keyword(s):  
Wave Equation ◽  
Fine Structure ◽  
Hydrogen Atom ◽  
Anomalous Scattering ◽  
Relativistic Wave Equation ◽  
Relativistic Wave ◽  
Heavy Particles ◽  
Theoretical Predictions ◽  
Structure Formula ◽  
Coulomb Law

In a recent paper R. C. Williams (1938) has found that the fine structure of the H α line in the spectrum of the hydrogen atom is not quite in agreement with the theoretical predictions (Sommerfeld’s fine structure formula). In discussing these experiments, Pasternack (1938) has pointed out that these deviations can be described by a simple shift of the 2 2 S level of hydrogen by an amount of 0.03 cm. -1 in the direction of higher energies. At the present state of our knowledge it seems conceivable that such a departure from the theory may be ascribed to a deviation from the Coulomb law of force at small distances rather than to a breakdown of the relativistic wave equation for the electron. A departure from the Coulomb law of force has often been discussed in connexion with the anomalous scattering of heavy particles. We know now, however, that this anomalous scattering is due to the internuclear forces and has no direct connexion with a possible departure from the Coulomb law. In view of the above experiments a new examination of the validity of the Coulomb law seems to be desirable.

scholarly journals An Investigation on the Beta Function III: Explicit Evaluation of the Harmonic Number H3/2

2014 ◽  
Vol 9 ◽  
pp. 1-2
Author(s):  
Edigles Guedes ◽  
K. Raja Rama Gandhi
Keyword(s):  
Closed Form ◽  
Beta Function ◽  
Harmonic Number ◽  
Explicit Evaluation ◽  
Euler Beta Function

In previous paper, we developed new versions of the Euler beta function, which given a closed form for the harmonic number H3/2.

Successes

2019 ◽  
pp. 259-299
Author(s):  
Anthony Duncan ◽  
Michel Janssen
Keyword(s):  
Fine Structure ◽  
Quantum Theory ◽  
Stark Effect ◽  
Principal Quantum Number ◽  
Classical Mechanics ◽  
Spectral Lines ◽  
Arnold Sommerfeld ◽  
X Ray ◽  
Old Quantum Theory ◽  
Structure Formula

The set of principles formulated in 1915-1918, and now collectively called the old quantum theory, were successfully applied to a number of problems in atomic and X-ray spectroscopy. The three most notable successes are all associated with the Munich school headed by Arnold Sommerfeld. First, there was the derivation of a relativistic fine-structure formula which predicted splittings of stationary state energies for orbits of varying eccentricity at a given principal quantum number. These splittings were empirically verified by Paschen for ionized helium, and constituted the first quantitative confirmation of the special relativistic mechanics introduced by Einstein a decade earlier. The relativistic fine-structure formula was also applied successfully to the splitting of lines in the X-ray spectra of atoms of widely varying atomic number. Finally, the principles of the old quantum theory (in particular, the use of Schwarzschild quantization in combination with Hamilton-Jacobi methods of classical mechanics) were successfully applied to explain the first order splitting spectral lines in the presence of an external electric field (Stark effect).

scholarly journals ENCODING THE SCALING OF THE COSMOLOGICAL VARIABLES WITH THE EULER BETA FUNCTION

2005 ◽  
Vol 20 (20n21) ◽  
pp. 4917-4924 ◽  
Author(s):  
M. A. PER ◽  
A. J. SEGUÍ
Keyword(s):  
Equation Of State ◽  
Beta Function ◽  
Cosmological Models ◽  
Physical Variable ◽  
Scaling Exponent ◽  
Singular Behavior ◽  
Scaling Exponents ◽  
Euler Beta Function

We study the scaling exponents for the expanding isotropic flat cosmological models. The dimension of space, the equation of state of the cosmic fluid and the scaling exponent for a physical variable are related by the Euler Beta function that controls the singular behavior of the global integrals. We encounter dual cosmological scenarios using the properties of the Beta function. When we study the integral of the density of entropy we reproduce the Fischler–Susskind holographic bound.

A generalization of the Euler beta function and applications

2011 ◽  
Vol 18 (2) ◽  
pp. 271-298
Author(s):  
Ilia Lomidze
Keyword(s):  
Special Functions ◽  
Beta Function ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Euler Formula ◽  
Euler Beta Function ◽  
Dimensional Variable

Abstract A generalization of the Euler beta function to the case of a multi-dimensional variable is defined. In this context, the original beta function is a function of a two-dimensional variable. An analogue of the Euler formula for this new function is derived for the case of a three-dimensional variable. Based on the derived formula, a number of relations for the Gauss hypergeometrical function are obtained. Moreover, the analytic formulae for some new integrals of special functions are obtained.

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