Elliptic singular point

One of the two integrals (4.67) may be evaluated analytically the second one (elliptic) may be calculated numerically by the trapezoidal method with the introduction of a small imaginary part in (4.67) to avoid spurious oscillations due to the finite number of integration points. The resulting density-of-states function is shown in Fig. 4.9. The band is asymmetric because of the equivalent term Vx j it exhibits three van Hove singularity points two discontinuities at the boundaries, and one logarithmic divergence corres-... 

The ZA(f) are chosen so that they tend to 1 in the vicinity of atom A but drop to zero in the direction of all other nuclei. Thus, even for integrals involving atomic basis functions of two different atoms the integrand of each contribution la has no more than one singular point. The integration can be further simplified by suitable transformations to intrinsic coordinates, e.g. elliptic-hyperbolic coordinates for diatomic molecules or spherical coordinates for polyatomic systems. 

As pointed out in Section II-3, numerical modelling of viscoelastic flows leads to numerical difficulties related to the mixed character (elliptic - hypeiholic) of the constitutive equation, to the propagation of "stress singularities" and to the so-called "High Weissenberg Number" problem. 

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