Potential at infinity of a polynomial image of the disk
Let \(q(t)= a_0t+\cdots+ a_nt^{n+1}\), \(a_0\neq 0\) real, be a polynomial univalent in the unit disc \(\Delta\). Let \(D= q(\Delta)\) and \(p(z)= \frac{1}{\pi} \int_D \frac{d\mu(\xi)}{z-\xi}\) be the potential of the domain \(D\). Then for large \(z\), \(p(z)= \frac{c_0}{z}+ \frac{c_1}{z^2} +\cdots+ \frac{c_n}{z^{n+1}}\) with \(c_0\neq 0\) real and \(c_k\) polynomials in \(a_0, a_1,\dots, a_n, \overline{a}_1,\dots, \overline{a}_n\). Thus, one sets up a map \(\eta\) by defining \(\eta(q)= p\). In this note, by calculating the Jacobian of \(\eta\), the author makes some interesting remarks about the nature of \(\eta\), useful in the inverse problem of recovering \(D\) from \(p(z)\).
scientific article; zbMATH DE number 1452682