If we use the same type of considerations as in the 3D case, we receive
where is a smooth function,
Let us expand the class of discontinuities
with arbitrary (noninteger) q, and
R(-)q(t) = R(+)q(-t).We must also expand the notion of q-equivalence (which was introduced in the Chapter 1) to a noninteger q: if .The operator of noninteger differentiation was considered above for q=1/2 (see Chapter 8). For arbitrary q it can be defined by spectra response .
It can be shown that
It is easy to see that
(it is proposed that d(-)>0).
If d(-)<0, then
and general formula
Let d(-)=0. We introduce the order of touching of curves and :the order = p if
If the order p is even, then
If the order is uneven
The point is a special one if, for given and ,.Each special point of the order p=2 is the point on caustics.