Elliptical integrals

This equation can be solved exactly in terms of elliptic integrals, but this solution is somewhat complex. However, by a slight modification of the boundary condition at the end of the pore, it is possible to obtain a good engineering approximation that is useful for fast reactions. In this approximation, we replace the boundary condition at the end of the pore by... 

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... 

These two equations show that the pad deformation is proportional to the applied load and inversely proportional to the pad material stiffness (represented by the Young s modulus). The equations involve complete elliptic integrals that can be readily evaluated numerically. The relative deformation represented by Eqs. (24) and (25) is plotted in Fig. 13, where the inner region corresponds to Eq. (24) and the outer corresponds to Eq. (25). 

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). 

Unfortunately, integration of this equation is rather difficult and leads to elliptic integrals which only have numerical solutions. (A relatively simple numerical solution to (A.3), without the use of elliptic integrals, was developed by Chan et al., 1980.)... 

This integral was analytically simplified to a one-dimensional integral of a complete elliptic integral, which admits numerical evaluation with an arbitrary precision [20]... 

K and E are the full elliptic integrals, respectively, of the first and second kind with the modulus 1 /s/2 K was defined above, 2E K = 0.847. 

E Complete elliptic integral of the te Contact time duration... 

K Complete elliptic integral of the first kind, defined by Eq. (2.116) Center approaching distance under maximum deformation... 

Procedure for Transformation of the Phase-Integral Formulas into Formulas Involving Complete Elliptic Integrals... 

The phase-integral quantities in the formulas obtained in Chapter 5 can be expressed in terms of complete elliptic integrals. One thereby achieves the result that well-known properties of complete elliptic integrals, such as for instance series expansions, can be exploited for analytic studies. Furthermore, complete elliptic integrals can be evaluated very rapidly by means of standard computer programs. 

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... 

K, E and II being complete elliptic integrals of the first, second and third kind, respectively. According to Section 117.03 on Page 14 in Byrd and Friedman (1971) we have the formula... 

In this chapter we collect and present, without derivation, in explicit, Anal form the relevant phase-integral quantities and their partial derivatives with respect to E and Z expressed in terms of complete elliptic integrals for the first, third and fifth order of the phase-integral approximation. For the first- and third-order approximations some of the formulas were first derived by means of analytical calculations, and then all formulas were obtained by means of a computer program. In practical calculations it is most convenient to work with real quantities. For the phase-integral quantities associated with the r -equation we therefore give different formulas for the sub-barrier and the super-barrier cases. As in Chapter 6 we use instead of L2 , L2n, K2n the notations LAn+1 >, L( 2n+1 KAn+l). 

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