Functions, elliptic integral

FIGURE 15.C.1 Plot of the incomplete elliptic integral function of the second type, E[k, —0.6] vs. k (elliptic module), from a series approximation with cp =0.6. 

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]. 

Related to the elliptic integral of the third kind are the Lame functions, which arise in the generalisation of spherical harmonics to confocal ellipsoidal coordinates. Applications of these in molecular electrostatics can be found... 

The function on the right hand side of Eq. (34) consists of a series of elliptic integrals, which depend not only on the unknown electrostatic force but also on the surface charge densities, q and on the interface and protein surface, respectively, and on the inverse Debye screening length (1/K). 

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... 

In Equation (3) G12 and Ge are complicated functions of Sj involving elliptic integrals. Unlike previous equations, [22, 32, 33], this equation can be applied both to endohedral and exohedral adsorption. 

It is known [275] that the hypergeometric function F(a,b c z) with half-integral parameters a, b and an integral parameter c can be expressed in terms of the complete elliptic integrals... 

When tp = tt/2, the integrals of the first and second types are considered as complete elliptic integrals. The complete elliptic integral of the second type, illustrated above as a function of k, is defined by... 

It leads to an elliptic integral, containing the energy W as parameter. The calculation of W as a function of the quantum numbers m and n cannot be carried out explicitly, except in the case of rotational symmetry (A —A ) which we have already dealt with ( 6). 

The total surface area related to the semimajor axis is a function of the two aspect ratios (3 and y. The special functions F(ty, k) and E(ty, k) are incomplete elliptical integrals of the first and second kind, respectively. They depend on the amplitude angle <[> and the modulus k. These special functions can be computed quickly and accurately by means of computer algebra systems such as Mathematica [153]. Their properties are given in Abramowitz and Stegun [1], The relationship between the square root of the total surface area and the semimajor axis is [150] ... 

The elliptic functions are thus related to the elliptic integrals the same as the trigonometrical functions are related to the inverse trigonometrical functions, for, as we have seen, if... 

Consider a disk source parallel to a detector with a circular aperture (Fig. 8.9). Starting with Eq. 8.4, one may obtain an expression involving elliptic integrals or the following equation in terms of Bessel functions ... 

This integral leads to an elliptic integral of the first kind, for which tables are available it can therefore be solved numerically. The results are given in Table IX. Here, for a number of values of z, from z = 0.5 up to z = 10 (i.e., for monovalent ions, up to = 256 millivolts), we give the values of Jcd for a series of values of u. From these data a graphic representation of m as a function of the plate distance, for a given value of z, may easily be obtained. 

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