Elliptic integral of the first kind

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

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 ... 

The ellipsoidal integral can be expressed in terms of the incomplete elliptical integral of the first kind F(k, < >) [10,60] ... 

The square/square problem has a complex analytical solution that requires the numerical computation of complete elliptical integrals of the first kind (see Ref. 8). For a large range of the parameter dID, the analytical solution provides an accurate asymptotic expression ... 

The complete elliptic integral of the first kind K(k) is recognized with the result... 

Hogg, Healy, and Fuerstenau [7] developed their HHF theory to describe the interactions of two particles of different size. In 1985, Matijevi and Barouch [8] evaluated the validity of the HHF theory for the electrostatic interaction between two surfaces of different sizes for both unlike particles with potentials opposite in sign, and for particles with same sign potentials. The computational calculations overcame the problem of the accuracy in the evaluation of incomplete elliptic integrals of the first kind, which is a direct consequence of a non-linearity of the Poisson-Boltzmann equation. They concluded that for systems with dissimilar particles with either opposite signs or the same sign, the approximation of the HHF theory achieved good results. However, when potential differences increased, marked deviations from the HHF theory were found. 

As we show in Chapter 4, the above result can be re-expressed as an elliptic integral of the first kind for certain realizable boundary conditions. 

The integral term is called the complete Elliptical integral of the first kind and is widely tabulated (Byrd and Friedman 1954). In general, the incomplete elliptic integrals are defined by... 

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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