Laplace method

It is easy to verify that the first above polynomial, equal to 1 (it corresponds to the residue at infinity in the integration by the Laplace method), gives, in combination with the second polynomial, and with the help of the following... 

Abstract. This chapter gives exact calculation of the matrix elements of the transition by the use of hypergeometric series and its verification by using the Laplace method. The incidence of the diverse approximations with regard to the exact solution are drawn out numerically. A conclusion is the necessity of the use of retardation and considerable divergence between relativistic and nonrelativistic approachs for the high values of energy in the continuum. 

For a long time it has been the only way, by the use of the Laplace method. Such a method is based on the representation in the complex plane of the confluent hypergeometric functions and the use the residues theorem. But it needs the approximation Z2a2 C ft2, which allows one to replace in (12.16)... 

The calculation gives results obtained by the Laplace method and their comparison with the ones obtained by the use of the hypergeometric series. 

For small values of Z the line (b) must be close to 1 as that can be seen on table I (Z=l) and II (Z = 11). So, for these values, the formulas (12.27)-(12.34) whose running time on a computer is shorter than for the ones of the hypergeometric series, can be used. But for large values of Z, the table III shows that the Laplace method gives wrong results. 

We have a numerical confirmation of the validity of the formulas giving the exact relativistic matrix elements with retardation of the photoeffect of hy-drogenic atoms, by the good concordance, for small walues of Z, of the results in the two ways of calculation which have been employed, the exact one, based on the use of the hypergeomtric series, and the Laplace method which implies the approximation c Z2 [Pg.84]

The values of a hr (expressed in barn) in function of T (expressed in eV) are rounded off at three numerals in [I/E]. We give some values, rounded off at four numerals, obtained by the way of calculation based on the use of hypergeometric series (Sect. 12.2) and verified (for Z = 1,2 only, for larger values of Z the verification is less legitimated) by the Laplace method ... 

We evaluate the integral G(0) for large Z using the Laplace method to obtain... 

Boudhabhay presented the applieation of the Pade-Laplace method (PLM) to N-NMR while Tellier applied it to situations of signal overlap to... 

The resolution of the differential equations in order to obtain the time dependent photocurrent has been performed via the Laplace method. In this approach, the concentration of the intermediates and products can be expressed as ... 

With respect to the fact that f(R) has maximum at 7 = Ro, it is easy to see, that for narrow size distribution function the only one line remains in the spectrum. In this case, the conditions AR/Ro < C 1 and AR/Ro 1 correspond to bulk ferroic (core part) or to the surfaee layer (shell part) respectively. In the intermediate case the spectrum consists of two lines if cOc — oo (/ ) > (A + Aj)/2. The above criteria have been confirmed by the detailed numerical calculations of integral (3.98) in Ref. [100]. It has also been shown that the linewidth increases with the size decrease that makes it difficult to meet the criteria of core and shell lines simultaneous observation. The approximate calculations of I((o) by Laplace method [100] for Rq/o > k/(RoAs) lead to the following analytical expression ... 

For calculations one can use the quantitative values < (A)=/o presented in Table 8.1. However, in the case of d < 4 space, an approximated analytical expression /o can be formulated, using the Laplace method [11]. With this aim we have rewritten (8.32) as a d-multipled Laplace integral... 

Scattering kernel with the dynamic structiural factor (3.1.6) is properly normalized (if the integral over P( is evaluated by the Laplace method) ... 

As mentioned briefly in the introduction, two methods can be used to predict the shape and size of a bubble in contact with a horizontal flat plate of poor wettability. One is the Laplace method which focuses on local pressure balance around a gas-liquid interface [10-13], The other, referred to as the potential method, calculates the shape of the bubble through energy minimization. Details of the Laplace... 

Figures 4.6. 9 show the relationship between the bubble volume, Fb, and the aspect ratio,, for = 96,104,116, and 131 ° in the water-air system, respectively. A similar relationship is shown in Fig. 4.10 for 0a = 130° in the mercury-air system. In each figure, thin lines denote predicted values of the aspect ratio based on the potential method, a thick line denotes predicted results based on the Laplace method, and open and solid circles denote the measured values [8]. The open circles represent bubbles long before detachment and the solid circles denote bubbles just before detachment. The measured data for 0a = 96° were obtained by observing bubbles on the upper surface of a rectangular cylinder placed horizontally in a water-air bubbling jet [15], The remaining data were obtained by using the experimental apparatus shown in Figs. 4.1 and 4.2.

Models of Types 4 and 5 proposed in the preceding section could give the minimum energy for every bubble volume. The difference between Model 4 and the exact value calculated from the Laplace method ranged from —1 to - -35%, while... 

Figures 4.6-4.10 show that the measured aspect ratio, Ab, deviates appreciably (in the range —40 to 100%) from that predicted with the Laplace method. The deviation was —40 to +100%. A slightly smaller deviation of —37 to +5% is observed for the critical bubble volume, Vb, just before detachment. The deviation becomes large with an increase in 9. Such a trend can be attributed to the following ...

The error inherent with this method is 21% for 0, = 96° [15], The error inevitably increases with an increase in 9 because a small measurement error in H(r) near the edge of the bubble on the plate amplifies the error in Fb- This is primarily responsible for the observed deviation of the measured Fb from the value calculated with the Laplace method for an increase in... 

The shape and size of a bubble in contact with the upper surface of a flat plate of poor wettability placed horizontally in a water bath or a mercury bath were predicted. The error inherent in the potential method employed was evaluated by comparing the results with those calculated from the Laplace method. The results indicate the need for a more sophisticated model for predicting the bubble shape than could be... 

---