CALCULATION OF DOMAINS OF DEPENDENCE FOR POLE FIGURES WITH AN ULTRAHYPERBOLIC DIFFERENTIAL EQUATION

X-ray and neutron methods of texture investigation are used to get experimental pole figures (PF) of polcrystalline samples and geological materials (Bunge, 1982). Usually several PF are used, 1--4 for polycrystalline samples with cubic, hexagonal lattice symmetry (copper, iron, beryllium...), 6-19 for low-symmetry materials of geological samples (quartz, biotite...). Pole figures are the sum of solutions of two ultrahyperbolic equations (Savyolova, 1982). If the solutions of ultrahyperbolic equations are known in some domains we can determine them uniquely in other domains using the Asgeirsson’s theorem (Courant, 1962) and their generalizations by ultra-Lorentz transformations. We get the domains of dependence of pole figures and the methods of continuations of solutions of ultrahyperbolic equations.


INTRODUCTION
The orientation distribution function (ODF) in a polycrystalline sample f(g), where g |a, fl, } e SO(3) is the rotation group, 0 < a, ?' < 27r, 0 < fl < r, are the three Eulerian angles, is determined from experimental pole figures Pi () where h-' describes the crystal direction in space R and describes the sample direction in space R3. The where h O, }, {X, r/}, 0 < O, Z < r, 0 < , r/< 2r are the spherical coordinates of unit vectors h, y.
From equation (7) (Vilenkin, 1965). r pe (p ) Thus if the function F(r, tp, p, O) is known in domain 0 < r < a, 0 < p < e, then we can calculate the series (10) with the coefficients (1 1). But the right-hand part of expression (10) is known in the domain 0 < r + p < a because in formula (12) the variables r and /9 are equivalent (interchangeable).
Consequently we have Theorem 1. If the function F(x , -) is known in the domain (8) then it is determined in the cone (9) by formulae (10), (11). There are some difficulties with the choice of boundary condition for the function F(r, tp, p, ) when r a. Apparently the boundary condition -g-rl 0 is less restrictive.
In that case An in the formula (10) 2 The application of the results for continuation of solutions of ultrahyperbolic equations for pole figures is in . (Ivanova, Savyolova, 1993). We cannot distinquish every component F and Ffor all pole figures except one PF with h 0 {00, D0}, 00 . The equality F Fholds for this PF.
In Figures 2, 2a-2d the domain G with 2(2 + -) < a < +o,, and the domain with {r -+ + p-->_ a} are marked off like in the first case. By contrast to the Figures 1, la-ld for every PF with 0 < 0 < the domain G is not equal to zero (see figure lb), because in the second case the parameter a is greater than in the first case. The domain G has the form of two rings in Figure 2b. For every a > 2 for PF with h Oi, tYPi} 0 O, we have G 0. For example, for approximation of PF{ 10]0 for a simple axial texture of beryllium in Figure 3a  with r-+, p-+ (18).
We get for PF{ 0001 (see Figure 3b

CONCLUSION
In the present paper we get ultrahyperbolic differential equations that allow us to calculate the domains of dependence for pole figures with different lattice symmetries. We get the methods of continuation of solutions of ultrahyperbolic equations and domains of dependence of pole figures. Using these results we can calculate the domains of PF without measuring.