Conformal Mapping of the Domain Exterior to a Thin Region
Dorel Homentcovschi · SIAM Journal on Mathematical Analysis · 1979
This paper gives the asymptotic expansion of the function $Z = F(z,\varepsilon )$ which maps conformally the domain exterior to a thin region into the plane Z with a cut on the real axis. The function $F(z,\varepsilon )$ is represented as a superposition of singularities on the segment $[\alpha (\varepsilon ),\beta (\varepsilon )]$ inside the thin region. The singularity intensities $f(x,\varepsilon )$ and $f(x,\varepsilon )$ satisfy a system of linear integral equations. This system is asymptotically integrated by using the method of Handelsman and Kelley (Axially symmetric potential flow around a slender body, J. Fluid Mech., 28, (1967) pp.131–142). The explicit solution is constructed for some classes of domains.