Parametric Characteristics of High-Order Tangential Loewner’s Slits

Abstract

We describe the chordal Loewner evolution with a driving function generating the slit in the upper half-plane which is high-order tangential to the real axis at 0. In this case, the slit is a joint boundary for the analytic corner and the analytic cusp. We give an asymptotic expansion of the slit and of the conformal mapping from the upper half-plane onto the upper half-plane with the slit both in the corner and the cusp. We find also a relation between harmonic measures of the two sides of the slit.