Function Elliptic

For a second order reaction in a slab or a sphere, analytical solutions proceed in terms of elliptic functions beyond the solution of P7.03.ll, although a numerical solution throughout may be preferable. Such a numerical procedure is adopted in P7.03.19 for a second order reversible reaction. 

Properties of the elliptic function of the first kind, F( ct), are given by Abramowitz Stegun (1964). Equations (10) and (11) corresponding values of ys and 0L. 

Theta functions are special functions related to Jacobian elliptic functions (Morse and Feshbach, 1953 Widder, 1975) with special properties that make then extremely useful to calculate solutions to diffusion problems for small values of time. Three of the four theta functions will be used in the present context... 

Elliptic Functions of the Worst Kind Non-linear Quantisation of the Classical Spherical Pendulum... 

These two systems are examples from non-linear physics, where the equations can be solved in terms of elliptic functions and elliptic integrals. The reader who is not familiar with these functions, which do not arise in the same way as the previously mentioned special functions, is referred to the excellent book by Whittaker and Watson [6]. In that book, the reader will see that there are two flavours of elliptic functions, the Weierstrass and Jacobi representations, three kinds of elliptic integrals, and six kinds of pseudo-periodic functions, the Weierstrass zeta and sigma functions and the four kinds of Jacobi theta functions. Of historical interest for theoretical chemists is the fact that Jacobi s imaginary transformation of the theta functions is the same as the Ewald transformation of crystal physics [7]. 

So how is the spherical pendulum quantised The answer is that its motion is generally chaotic, except for discrete values of the initial projection speed, for which it is periodic. The precise details of this phenomenon are difficult to get a handle on, because, although the vertical motion is always described by periodic elliptic functions, the horizontal motion is described in terms of Lame functions, which are very difficult to study and for which periodicity is difficult to diagnose. 

There is a considerable literature [10-13] devoted to finding approximate formulas for the frequency of the simple pendulum for non-zero amplitudes, usually based on mathematical arguments designed to approximate elliptic functions. 

Whittaker [3] has given an expression for the azimuthal angle of the spherical pendulum in terms of elliptic functions of the third kind, so that, not surprisingly, there has been very little numerical discussion of its motion. Instead, we see if our approximate theory can be used to obtain a simpler picture of the motion. 

Stationary (i.e. for dA/ dz = 0) localized solutions to Eq.(3.2) represent nonlinear modes in the planar waveguide and may be found in an analytical form via matching the partial solutions of Eq.(3.2) at the core/cladding boundary. The partial solutions are Jacobi elliptical function in the core and 2l rccosh — )E]/E in the cladding (the functional dependence similar to a fundamental soliton in a uniform nonlinear medium). Here is a parameter which depends on the boundary conditions. Contrary to the modes of a linear waveguide, the transverse profile of a nonlinear mode depends on the power in the mode. 

This ambiguity may be removed by introducing the Weierstrassian elliptic function (Jones, 1964, p. 32). Fortunately, such a drastic step is not necessary in the problem at hand, a homogeneous ellipsoid in a uniform electrostatic field aligned along the z axis. In this instance the potential has the symmetry properties... 

In these formulas the symbol Za(co) stands for the Bessel function, sn (to), dn ( ), cn ( ) are the Jacobi elliptic functions having the module /(xvxv) is the general solution of the ordinary differential equation... 

The solution was not entirely original, as the paper made clear, but the application and asymptotics were. Moreover, it was great fun to work with elliptic functions again. I had not encountered them since first studying them in Edinburgh. 

The solution of this differential equation could be expressed in terms of elliptic functions. The transverse-vibration frequency v is determined by the formula analogous to Eqs. (431)-(432b) ... 

Appearing in the above are the elliptic functions, cr(z), o (z) and (z). The reader is referred to the original paper [81] or the book by Whittaker and Watson [14] for a review of these functions. Despite its complexity Eq. (83) bears some resemblance to the form taken by the free energy in the linear model. However, little more can be said of the above in its present form. Greater insight is obtained through the asymptotic expression below. 

Some of the functions used in the expressions for the uniform and nonuniform potentials are defined in this Appendix. The uniform solution, for example, relies on the elliptic function, sn, defined by... 

K is a complete elliptic function of first kind [96]. The following symbols are also represented in (78) and subsequent equations,... 

In this chapter we shall describe the procedure for expressing the phase-integral formulas derived in Chapter 5 in terms of complete elliptic integrals. The integral in question is first expressed in terms of a Jacobian elliptic function and then in terms of complete elliptic integrals. Different elliptic functions are appropriate for different phase-integrals. For practical calculations it is most convenient to work with real quantities. For the phase-integrals associated with the r -equation it is therefore appropriate to use different formulas for the sub-barrier case and for the super-barrier case. We indicate in this chapter, where we use the notations L 2n+l L 2n+1 K< 2n+1 ) instead of the notations 2 , 2n used previously, the main... 

Considering first L in the sub-barrier case in Fig. 5.1b, we introduce the Jacobian elliptic function snu by the transformation... 

Recalling (5.15a,b) and (5.7a), and using (6.1), (6.3) and general properties of Jacobian elliptic functions, we obtain after some calculations... 

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