*-Logarithm for Slice Regular Functions

In this paper, we study the (possible) solutions of the equation exp_∗(f)=g, where g is a slice regular never vanishing function on a circular domain of the quaternions H and exp_∗ is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function f which satisfies exp_∗(f)=g is called a ∗-logarithm of g. We provide necessary and sufficient conditions, expressed in terms of the zero set of the “vector” part g_v of g, for the existence of a ∗-logarithm of g, under a natural topological condition on the domain Ω. By this way, we prove an existence result if g_v has no non-real isolated zeroes; we are also able to give a comprehensive approach to deal with more general cases. We are thus able to obtain an existence result when the non-real isolated zeroes of g_v are finite, the domain is either the unit ball, or H, or D (the solid torus obtained by circularization in H of the disc contained in C and centered in 2i with radius 1), and a further condition on the “real part” g_0 of g is satisfied (see Theorem 6.19 for a precise statement). We also find some unexpected uniqueness results, again related to the zero set of g_v, in sharp contrast with the complex case. A number of examples are given throughout the paper in order to show the sharpness of the required conditions.