The elliptic integral of second kind

The variables q and a are called the amplitude and the modular angle, respectively the complete elliptic integral of second kind is E a). 

The reduction of the smn (17) to the complete elliptic integral of the second kind is performed as follows. By using equation (10) and = cos we can rearrange equation (12) as 

This implies Af3 IjSl this condition is certainly fulfilled, otherwise either /3+ or /3 would vanish or become positive. Consider a chain of 2N = 4r + 2 

The Hiickel matrix of order 2N is symmetrical and tridiagonal, and therefore has exactly 2N non-degenerate real eigenvalues. Since equation (25) is, a part from a constant factor, the secular determinant of the Hiickel matrix [6], it has exactly N non-zero distinct roots. 

First of all we observe that cos Z must be real since it is connected to the orbital energy by 

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Appendix A. The elliptic integral of second kind Appendix B. The roots of equation (25)... 

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