Airy formulas

Airy formula, intensity distribution of the Fraunhofer diffraction pattern, 148... 

Eliminating the constants A, C, and D yields the Airy formulae for the reflectance and transmission amplitudes ... 

Equations (4.49,4.50) are called the Airy formulas. Since we have neglected absorption, we should have Ir + Ij — Iq, as can easily be verified from (4.49,4.50). 

Since the laser resonator is a Fabry-Perot interferometer, the spectral distribution of the transmitted intensity follows the Airy formula (4.57). According to (4.53b), the halfwidth Avr of the resonances, expressed in terms of the free spectral range 5y, is Avr = Sy/F. If diffraction losses can be neglected, the finesse F is mainly determined by the reflectivity R of the mirrors, therefore the halfwidth of the resonance becomes... 

If the sample near-surface region consists of a single film (see Figure 3), the composite reflection coefficients can be calculated from the Airy formula ... 

METHYL ALCOHOL. ICAS 67-56-11. CH,OH. formula weight 32.04. colorless, mobile liquid with mild characteristic odor, mp —97 (VC. bp 64.6 C. sp gr 0.792. Also known as mi iluiitnl. the compound is miscible in all proportions with H l). ethyl alcohol, orerher. When ignited, methyl alcohol bums in air with a pale blue, transparent II a I lie. producing fUO and CO . The vapor forms an explosive mixture with air. The upper explosive limit (% by volume in airi is 36.5 and the lower limit is 6.0. 

The Formula (27 ) can be approximated, a product of a Gaussian, correction function, yq E) where the subscript "q" stands for "quantum" because Formula (27 ) and then yq E) describes the quantum effect linked to the Airy function of the final state of the photon absorption ... 

The formula (6) can be refined. With much more work, one can show that T /z[3-2In2] + 20 +. .., where a = 2.338 is the smallest root of Ai(-a) = 0. Here Ai(x) is a special f unction called the Airy function. This correction term comes from an estimate of the time required to turn the comer between... 

Analytic formulas can be very useful if linewidths have been measured for many vibrational levels (for example, Child, 1974). It is convenient to represent Xv,J and xe,j in a uniform semiclassical approximation (Section 5.1.1). As for the bound bound case, the overlap integral between bound and continuum wavefunctions can be expressed as an Airy function (taking into account the proper normalization factor). The linewidth is then... 

The numerical problems arise because, individually, either Ai, Ai, Bi, or Bi may become very large in the upper complex plane. It therefore becomes essential to work with F + iG if a well-behaved function is to be evaluated. The asymptotic forms of the Airy functions permit us to evaluate F + iG for various ranges of the argument x, though the complexity of the resultant formulae inevitably impels us to use computational techniques. 

The ordinary Airy function A,(z) corresponds to this solution with A = 0. Equation (85) represents the famous connection formula for the WKB solutions crossing the turning point. As can now be easily understood, once we know all the Stokes constants the connections among asymptotic solutions are known and the physical quantities, such as the scattering matrix, can be derived. However, the Airy function is exceptionally simple and the Stokes constants are generally not known except for some special cases (40). 

From the approximation that the static polarizability is given by the variational formula a= (4/9ao)2(Airi ) Nis the number of electrons, is the electron mass a crude approximation is Xra=(Mi/4m c )a, where Ejis the ionization energy... 

The expression (7) for the moving tsunami wave height is consistent with the well-known Airy-Green formula the wave amplitude increases as a depth of the bottom H decreases. 

In ref. 151 the author studies the piecewise perturbation methods to solve the Schrodinger equation and the two form of this approach, i.e. the LP and CP methods. On each stepsize the potential is numerically approximated by a constant (in the case of CP) or by a linear function (in the case of LP). After that the deviation of the true potential from this numerical approximation is obtained by the perturbation theory. The main idea of the author is that an LP algorithm can be made computationally more efficient if expressed in a CP-like form. The author produces a version of order 12 whose the two main parts are a new set of formulae for the computation of the zeroth-order solution which replaces the use of the Airy functions, and an efficient way of obtained the formulae for the perturbation corrections. The main remark for this paper is that from our experience for these methods the computational cost is considerably higher than for the finite difference methods. 

This is the well-known Gamov formula of tunneling probability. It is now well understood that the connection formulas of Equations (2.49) and (2.50) are crucial. These formulas can be obtained from the Airy function, since the potential in the vicinity of the turning point can be approximated by a linear function of x for which the Airy function gives the exact analytical solution. 

The theoretical resolution, defined as half the width of the Airy disk, is determined by the wavelength of the light (2) and the numerical aperture (NA) of the system according to the formula [7] ... 

Fig. 2. Intensity of light scattered from a water drop as predicted by geometrical optics and Airy s formula for wave optics (redrawn from reference 29).

Ford and Wheelerin their semiclassical treatment of quantum mechanical rainbow scattering remark that the necessary mathematics is not essentially different from Airy s treatment of the reflection and refraction of light. Indeed, their semiclassical formula for the cross section contains the Airy function or rainbow integral. The semiclassical cross section is illustrated schematically in Fig. 3c. The supernumerary rainbows of the wave mechanical theory are clearly resolved in the Na-Hg cross section measurements of Buck and Pauly, reproduced here in Fig. 5. 

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