Airy functions

The Airy equation is very often encountered in the resolution of electrochemical mass transport problems in dimensionless variables, and has the form 

The Airy functions, Ai(z) and Bi(z), are independent solutions of this equation, where 

From tables listing values of T between 0 and 1 it is possible to calculate T(z) for any positive value. Some values for fractional z are given in Table A1.2, and the variation of the gamma function with z is shown in Fig. A1.2. 

As is easily seen by inspection, (A1.25) represents the integral of a curve of the same type as the Gaussian distribution—hence its name. It should be noted that erf (0) = 0 and that erf (oo) = 1, as shown in Fig. A1.3. This function is frequently encountered in diffusion problems. 

Values of the error function and its first derivative (the Gauss distribution) are tabulated1. 

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The next source we investigate is the surface of an extended, limb-darkened star whose apparent diameter increases from 1 to 25 milli-arcseconds. The visibility function of such a source resembles the Airy function, varying periodically between zero and a maximum value which decreases with increasing frequency. Note how the fringe contrast vanishes repeatedly to rise again without reaching its previous maximum value as the source s apparent diameter... 

When an external electric field is applied along the periodicity axis of the polymer, the potential becomes non periodic (Fig. 2), Bloch s theorem is no longer applicable and the monoelectronic wavefunctions can not be represented under the form of crystalline orbitals. In the simple case of the free electron in a one-dimensional box with an external electric field, the solutions of the Schrddinger equation are given as combinations of the first- and second-species Airy functions and do not show any periodicity [12-16],... 

Indefinite integrals involving products of Airy functions and/or their derivatives can be evaluated without many difficulties [34]. Thus,... 

By a simple variable change the integration in pz is expressible in terms of the Airy function of proper argument [see eq. (B.3)j, so that... 

We first follow the flow chart for the simple case of elastic scattering of structureless atoms. The number of internal states, Nc, is one, quantum scattering calculations are feasible and recommended, for even the smallest modem computer. The Numerov method has often been used for such calculations (41), but the recent method based on analytic approximations by Airy functions (2) obtains the same results with many fewer evaluations of the potential function. The WKB approximation also requires a relatively small number of function evaluations, but its accuracy is limited, whereas the piecewise analytic method (2) can obtain results to any preset, desired accuracy. 

The argument z, of the Airy function depends on / only through the velocity v,(R ") at the minimum of the difference potential. If the well depth of V+(R) is large compared to the collision energy, /) 2> )(oo), this velocity is nearly independent on / [see relation (II.9)], and we can write... 

Instead of FC-region values. As mentioned above, this approximation enables one to obtain a qualitative description of the distribution. The good quantitative agreement with experiment obtained in the paper shows that asymptotic and FC region vibrational frequencies and bond lengths do not differ significantly. In this calculation the Airy function was used to describe the translational dependence. As noted by Freed and Band (2), the use of the uniform semiclassical approximation... 

The Airy function was used to describe translational motion. The calculations are in a good agreement with experimental data (51). 

Deslouis et al. [10, 38], at the beginning of the eighties gave an analytical treatment for both potentiostatic and galvanostatic regulations by using Airy functions. 

In the vicinity of x1 a shift in energy means only a redefinition of x, so one may suppose without the loss of generality that E = 0. Then the solution of the Schrodinger equation, near the right turning point is expressed via the Airy functions ... 

In the time-independent formulation, the absorption cross section is proportional to (4>/(.R .E) i(R] E )) 2. Approximate expressions may be derived in several ways. One possibility is to employ the semiclas-sical WKB approximation of the continuum wavefunction (Child 1980 Tellinghuisen 1985 Child 1991 ch.5). Alternatively, one may linearly approximate the excited-state potential around the turning point and solve the Schrodinger equation for the continuum wavefunction in terms of Airy functions (Freed and Band 1977). Both approaches yield rather accurate but quite involved expressions for bound-free transition matrix elements. Therefore, we confine the subsequent discussion to a merely qualitative illustration as depicted in Figure 6.2. 

Let us imagine that p(x) is real, saving suitable modifications for later. We know that we will have to distinguish between the non-classical region p.(x ) < 0, in which n(x) typically decays monotonically, and the classical region p(x) > in which n(x) oscillates. We also know from standard WKB or Airy function analysis that n(x) will have an amplitude p, x) 1 2 with a phase argument going as f1 /i(a ) 1, 2dx. 

If there is only one turning point, the solutions of Eq. (1-0.32) can be well approximated in terms of the uniform regular and irregular Airy functions A, and B, [330, 331],... 

Here S E) is the classical action at the energy E, x = (q,p), X= (Q,P) represent the coordinates transverse to the trajectory on a PSS, M is the mapping matrix on the same PSS, I is a unit matrix, n is the dimensionality of the system, J is the 2(m — 1) by 2 n—l) unit matrix, s is a spectral parameter that will eventually be set to zero, T is the period of the periodic trajectory, Ai(x) is the Airy function, and the phase factor jj is a multiple of ti/4 determined by the focusing behavior of classical trajectories close to the jth periodic orbit. Note that on the closed periodic trajectory we have X = 0. In the semiclassical limit i.e., (H —> 0), these scar terms reduce to... 

When applied to isotope effects, the main weakness of the reflection method is the assumption that the transition dipole moment is constant for all isotopologues. This weakness remains in the improved model presented below. Only ab initio calculations are able to go beyond this approximation. However, the dependence of the transition dipole moment along the nuclear coordinates can be introduced (numerically or analytically) in the model below, even if a less compact analytic form is expected. This paper is organized as follows in Section 2 the "standard" reflection model is improved by taking into account the curvature of the upper state potential (in addition to its slope). In Section 3, the quantum character of the final state is taken into account by replacing the Dirac function by an Airy function. In Section 4 the model is applied to the CI2 molecule. In Section 5 the model is adapted and applied to the O3, SO2 and CO2 triatomic molecules. Conclusions and perspectives are presented in Section 6. 

Note that the upper limit a oo) of the condition (20), corresponds to the Condon approximation for which the Airy function can be approximated by a Dirac function. 

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 ... 

Many authors have given analytic solutions with differing degrees of accuracy. Deslouis et al. developed a method that, after an approximation, reduces the problem to the canonical equation for Airy functions. Tribollet and Newman gave a solution under the form of two series one for K < 10 and one for K > 10. The two series overlapped well. 

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