On bounded oscillation and asymptotic expansion of conformal strip mappings

Abstract:

Relations between the boundary parameters ${\phi _ - },{\phi _ + }$ of a strip $S = \{ {\phi _ - }(x) < y < {\phi _ + }(x)\}$ and the values $f(x)$ of its canonical conformal mapping onto a horizontal strip $H = \{ |\upsilon | < 1\}$ are studied. Bounded oscillation $({\max _y}\operatorname {Re} f(x + iy) - {\min _y}\operatorname {Re} f(x + iy) = \omega (x) = O(1))$ is characterized in terms of ${\phi _ - },{\phi _ + }$. A formal series expansion $\upsilon = \sum {y^m}{a_{m,n}}(x)$ is derived for the solution to the Dirichlet problem on $S$ and its partial sums are used to obtain formulas for the asymptotic expansion of $f$ in terms of ${\phi _ + },{\phi _ - }$.