Abstract
For conformal maps defined in the unit disk one can define a certain Poisson bracket that involves the harmonic moments of the image domain. When this bracket is applied to the conformal map itself together with its conformally reflected map the result is identically one. This is called the string equation, and it is closely connected to the governing equation, the Polubarinova-Galin equation, for the evolution of a Hele-Shaw blob of a viscous fluid (or, by another name, Laplacian growth). In th
Results & Lemmas (4)
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Lemma 3.1.
Lemma 3.1. Assume that g(ζ, t) is analytic in ζ in a neighborhood of the closed unit disk and depends smoothly on t in such a way that…
Lemma 3.1. Assume that g(ζ, t) is analytic in ζ in a neighborhood of the closed unit disk and depends smoothly on t in such a way that (3.4) holds. Then 1 2πi d dt Z D g(ζ, t)|f(ζ, t)|2d¯ζdζ = 1 2π Z 2π 0 g(ζ, t){f, f ∗}t dθ, (3.5)
Corollary 3.2.
Corollary 3.2. If h(z) is analytic in a fixed domain containing the closure of f(D, t) then 1 2πi d dt Z D h(f(ζ, t))|f(ζ, t)|2d¯ζdζ = 1 2π…
Corollary 3.2. If h(z) is analytic in a fixed domain containing the closure of f(D, t) then 1 2πi d dt Z D h(f(ζ, t))|f(ζ, t)|2d¯ζdζ = 1 2π Z 2π 0 h(f(ζ, t)){f, f ∗}t dθ.
Theorem 5.1.
Theorem 5.1. With f a polynomial as in (5.1), the identity ∂( ¯ Mn,... ¯ M1, M0, M1,..., Mn) ∂(¯an,..., ¯a1, a0, a1,..., an) = 2an2+3n+1 0…
Theorem 5.1. With f a polynomial as in (5.1), the identity ∂( ¯ Mn, . . . ¯ M1, M0, M1, . . . , Mn) ∂(¯an, . . . , ¯a1, a0, a1, . . . , an) = 2an2+3n+1 0 R(f ′, f ′∗) (5.4) holds generally. It follows that the derivative ∂f/∂M0 makes sense whenever R(f ′, f ′∗) ̸= 0, and then also the string equation {f, f ∗} = 1 (5.5) holds. 16
Theorem 6.1.
Theorem 6.1. Consider functions f which are analytic in a neighborhood of the closed unit disk, are normalized by f(0) = 0, f ′(0) > 0 and…
Theorem 6.1. Consider functions f which are analytic in a neighborhood of the closed unit disk, are normalized by f(0) = 0, f ′(0) > 0 and satisfy f ′ ̸= 0 on ∂D. Let ω1, . . . , ωm denote the zeros of f ′ in D, these zeros assumed to be simple. Under these assumptions a quadrature identity of the kind (6.1) holds, for some choice of coefficients c0, c1, . . . , cn with cn ̸= 0, if and only if f is a rational function such that f has a pole of order n+1 at infinity and possibly finite poles at the r
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