The driving function of the Loewner equation generating a tangential slit emanating from the corner of a digon

The driving function X of the Loewner equation that generates a non-tangential slit is Holder continuous with exponent 1/2. For a tangential slit emanating from a corner, the behavior of the driving function X(t) in a neighborhood of т = 0 depends on the tan-gency order of the slit and on the angle of the corner. In this paper we investigate a family of mappings f = f (z, r) , re [0,Г ]. For a fixed т, the mapping f takes the half-plane onto a digon with a slit (the length of the slit depends on t) along a circular arc emanating tangentially from a vertex of the digon with an angle an. We obtain the form of the expansion of the driving function X at the point т = 0, which generates the slit in the digon. We construct a mapping of the half-plane onto a triangle with tangential slit emanating from a corner of the triangle, assuming that the driving function has the same form as for the digon. We propose the following conjecture: if X generates in a simply connected domain D a slit along a circular arc emanating tangentially from a corner with an interior 1+ka 2+a angle ап, then the function X expands into the series X(r) = к =0