scholarly journals IV. Tables of log K 0 (x) over the range x = 2 to x = 12 at intervals of 0·001

1927 ◽  
Vol 226 (636-646) ◽  
pp. 135-155
Keyword(s):  
Differential Equation ◽  
Potential Energy ◽  
Electrostatic Potential ◽  
Bessel Functions ◽  
Zero Order ◽  
The Third ◽  
Periodic Linear ◽  
Physical Importance ◽  
Significant Figures ◽  
Electrostatic Potential Energy

The differential equation d 2 y / dx 2 + 1 dy / x dx — y = 0, which differs from Bessel’s Equation of zero order only in the sign of the third term ( — y ), has two solutions denoted by I 0 ( x ) and K 0 ( x ): these solutions tend exponentially to infinity and zero respectively as x → ∞ by positive values. The function K 0 ( x ) is of physical importance, particularly in connection with the electrostatic potential of a periodic linear series of charges. It has been much used recently in calculations of the electrostatic potential energy of certain crystals, for which it was found necessary to construct the following tables. It appears that the earliest tables of K 0 ( x ) are due to Aldis (‘Roy. Soc. Proc.,’ vol. 64, pp. 219-221 (1899)), who gave the values of K 0 ( x ) to 21 decimal places for values of x from x = 0 to x = 6·0 at intervals of 0·1, and also to between 7 and 13 significant figures from x = 5·0 to x = 12·0 at intervals of 0·1. These tables are reprinted in the ‘Treatise on Bessel Functions’ by Gray, Mathews and MacRobert (Macmillan, 1922). Jahnke and Emde, ‘Functionentafeln,’ pp. 135-6 (Leipzig, 1923), also tabulate the function K 0 ( x )—there denoted by ½ i πH 0 ( 1 ) ( ix )—over the same range of x , but only to four significant figures.

A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order

1951 ◽  
Vol 47 (4) ◽  
pp. 699-712 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equation ◽  
Bessel Functions ◽  
Numerical Technique ◽  
Third Order ◽  
New Approach ◽  
The Third ◽  
Zeros Of Bessel Functions ◽  
Zeros Of Functions ◽  
Linear Differential ◽  
Image Position

In a recent paper (1) I described a method for the numerical evaluation of zeros of the Bessel functions Jn(z) and Yn(z), which was independent of computed values of these functions. The essence of the method was to regard the zeros ρ of the cylinder functionas a function of t and to solve numerically the third-order non-linear differential equation satisfied by ρ(t). It has since been successfully used to compute ten-decimal values of jn, s, yn, s, the sth positive zeros* of Jn(z), Yn(z) respectively, in the ranges n = 10 (1) 20, s = 1(1) 20. During the course of this work it was realized that the least satisfactory feature of the new method was the time taken for the evaluation of the first three or four zeros in comparison with that required for the higher zeros; the direct numerical technique for integrating the differential equation satisfied by ρ(t) becomes unwieldy for the small zeros and a different technique (described in the same paper) must be employed. It was also apparent that no mere refinement of the existing methods would remove this defect and that a new approach was required if it was to be eliminated. The outcome has been the development of the method to which the first part (§§ 2–6) of this paper is devoted.

scholarly journals Mach principle based explanation for the ‘cosmological red-shift’ and it’s evidence

2016 ◽  
Vol 4 (1) ◽  
pp. 27
Author(s):  
Hasmukh K. Tank
Keyword(s):  
Potential Energy ◽  
Gravitational Potential ◽  
Electrostatic Potential ◽  
Gravitational Effect ◽  
Red Shift ◽  
Gravitational Potential Energy ◽  
Supportive Evidence ◽  
Mach Principle ◽  
Electrostatic Potential Energy ◽  
Pioneer 10

<p>We first find here that the ratio of: (loss in energy of cosmologically red-shifting photon) and (loss in electrostatic potential-energy of an electron at the same distance <em>D</em>) remains equal to the famous ratio (G m<sub>e </sub>m<sub>p</sub>) / e<sup>2</sup> leading us towards a possibility that ‘cosmological red-shift’ may be due to gravitational effect. Also the ratio <em>h H<sub>0</sub> / m<sub>e</sub> c<sup>2 </sup>= (G m<sub>e </sub>m<sub>p</sub>) / e<sup>2</sup></em>. Starting with Mach’s principle, that ‘mass’ of an object is because of its ‘cosmic gravitational potential energy’, we arrive at a possibility that every moving chunk of matter and energy should experience a fixed value of acceleration <em>H<sub>0 </sub>c</em>. For the purpose of comparison, we express the ‘cosmological red shift’ as deceleration of the photon, and find that the deceleration experienced by the photon matches perfectly with the expected value. Then it is argued that if such a deceleration is true for a chunk of energy called photon, then it must be true for every particle of matter too. Strikingly, the decelerations experienced by the space-probes Pioneer-10, Pioneer-11, Galileo and Ulysses, as carefully measured by Anderson J.D. ET. Al. Match perfectly with the deceleration of the ‘cosmologically red-shifting photons’; thus providing supportive evidence for the new explanation proposed here.</p>

Structure and Properties of Hydrotalcite Using Electrostatic Potential Energy Model

2006 ◽  
Vol 19 (3) ◽  
pp. 277-280 ◽  
Author(s):  
Zhe-ming Ni ◽  
Guo-xiang Pan ◽  
Li-geng Wang ◽  
Wei-hua Yu ◽  
Cai-ping Fang ◽  
...  
Keyword(s):  
Potential Energy ◽  
Electrostatic Potential ◽  
Energy Model ◽  
Structure And Properties ◽  
Electrostatic Potential Energy

scholarly journals On the form and potential energy of the isomorphous crystals, ruby (Al 2 O 3 ) and hæmatite (Fe 2 O 3 )

1929 ◽  
Vol 122 (789) ◽  
pp. 251-273 ◽  
Keyword(s):  
Potential Energy ◽  
Electrostatic Potential ◽  
Equilibrium Configuration ◽  
Ionic Charge ◽  
X Ray ◽  
Inverse Power Law ◽  
Exact Position ◽  
Infinite Array ◽  
Repulsive Forces ◽  
Electrostatic Potential Energy

1. 1. Recently, much work has been done in attempting to explain the observed size and shape of crystals in terms of their interionic forces. To do this, a highly idealised form of the crystal has been postulated in which the ions of the crystal (in accordance with Kossel’s theory) are regarded as pointcharges, of magnitude equal to the resultant ionic charge and situate at the points of the crystal lattice determined by X-ray analysis. For ions which are surrounded symmetrically by other ions, these points doubtless correspond to the nuclei of the respective atoms, but for unsymmetrically situated ions there is some uncertainty as to the exact position to which to assign the pointcharge. Moreover, in order that the crystal may be in equilibrium, other repulsive forces, in addition to the electrostatic forces of attraction and repulsion between the ionic point-charges, are assumed to be operative between the various ions. These are termed “intrinsic repulsive” forces, and have been represented by an inverse power law varying with the distance according to the n th power, and expressible in the form μ n r - n . Consequently, in order to determine the equilibrium configuration of the crystal, as given by the minimum value of the potential energy of the crystal, it is necessary to determine (i) the electrostatic potential energy of an infinite array of point-charges, arranged according to the crystal pattern, and (ii) the potential energy due to the intrinsic repulsive forces between the ions.

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