Integral, Elliptic, Complete

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). 

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

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). 

E and fl are complete elliptic integrals of the second and third kind in the standard notation of Byrd and Friedman [101]. This completes the explicit form of the Gross-Kohn approximation (191). 

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... 

The value of T(0) is of course 1/2, and the value of A(0) is also known in terms of complete elliptic integrals of the first kind ... 

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... 

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