Decay of a reflected pre-Schwarzian at infinity #
Suppose that the inverse coordinate g(w) = f(-1 / w) of a conformal map extends continuously
and injectively to a straight boundary edge through w = 0. Normalize the target edge to the
real axis, with the interior on its upper side. Schwarz reflection extends g holomorphically
across zero with nonzero derivative. The pre-Schwarzian chain rule then gives
z * f''(z) / f'(z) → -2 along the upper half-plane, and in particular f'' / f' → 0.
TauCeti.tendsto_zero_cobounded_of_eqOn_logDeriv_deriv transfers this decay to any
conjugation-symmetric continuation of the pre-Schwarzian which is continuous near infinity. It
supplies the full-plane limit needed in the partial-fraction characterization of the
Schwarz--Christoffel differential equation.
The straight-edge hypotheses concern the map in the inverse coordinate, rather than assuming
any differentiability of that map on the boundary. The asymptotic and the decay theorem are also
stated for the original map, whose normalized inverse coordinate is the one assumed to extend
across zero; in that form the asymptotic is what forces the exponents of a Schwarz--Christoffel
map to sum to -2.
The point at infinity may instead be sent to a vertex at infinity of the target: the map tends to
infinity there, and far out its image is a sector of opening β * π, 0 < β < 2, with vertex
some point c. Inverting the target about c turns this into a corner at 0 in the inverse
coordinate, and the power coordinate at that corner gives z * f''(z) / f'(z) → β - 1 instead
(TauCeti.tendsto_mul_logDeriv_deriv_upperHalfPlaneSet_of_eqOn_div_neg_inv). If instead far out
its image is a half-strip between two parallel rays, the exponential of the normalized map has a
straight edge through 0 in the inverse coordinate, and the map is a logarithm of its Schwarz
reflection, which gives z * f''(z) / f'(z) → -1
(TauCeti.tendsto_mul_logDeriv_deriv_upperHalfPlaneSet_of_eqOn_exp_neg_inv).
More generally, the limit of z * f''(z) / f'(z) at infinity does not change when f is
replaced by f ∘ φ for a map φ that fixes infinity and is conformal across it
(TauCeti.tendsto_mul_logDeriv_deriv_upperHalfPlaneSet_of_eqOn_comp_neg_inv). This compares an
end of a domain that has no elementary straightening coordinate with an explicit model map.
References #
- L. Ahlfors, Complex Analysis, Ch. 6, Section 2.
- T. Driscoll and L. Trefethen, Schwarz--Christoffel Mapping, Ch. 2.
At a straight boundary edge in the inverse coordinate, the pre-Schwarzian has the
asymptotic z * f'' / f' → -2. The edge is normalized to the real axis.
A continuation of a polygon map's pre-Schwarzian that is conjugation-symmetric near infinity tends to zero there when the inverse coordinate maps a neighborhood of zero to a straight edge. Continuity near infinity holds, in particular, for a continuation holomorphic off finitely many prevertices.
Holomorphy is preserved when a half-plane map is read in the coordinate w ↦ -1 / w
and affinely normalized.
The pre-Schwarzian of a map with a straight side at infinity. Read the map f of the
upper half-plane in the coordinate w ↦ -1 / w at infinity and normalize the target by
w ↦ (w - q) / b. If the result extends to a function g which is continuous and injective up to
a real segment through 0, holomorphic and upper half-plane valued above it, and real on it, then
z * f''(z) / f'(z) → -2 as z tends to infinity in the upper half-plane.
Decay of a continued pre-Schwarzian derivative at infinity. Read the map f of the upper
half-plane in the coordinate w ↦ -1 / w at infinity and normalize the target by w ↦ (w - q) / b.
If the result extends to a function g which is continuous and injective up to a real segment
through 0, holomorphic and upper half-plane valued above it, and real on it, then every
conjugation-symmetric continuation φ of the pre-Schwarzian derivative of f which is continuous
near infinity on the real axis tends to 0 at infinity.
This is the form the Schwarz--Christoffel converse uses: φ is holomorphic off the finitely many
prevertices, so it is automatically continuous near infinity, and the hypotheses on g say that
the point at infinity is an interior point of a side of the polygon.
The pre-Schwarzian of a map with a vertex at infinity. Read the map f of the upper
half-plane in the coordinate w ↦ -1 / w at infinity and invert the target about c by
w ↦ b / (w - c). Suppose the result extends to a function g with g 0 = 0 which is continuous
and injective up to a real segment through 0, takes the upper part of that neighbourhood into the
sector |arg w| < β * π / 2 of opening β * π, where 0 < β < 2, and takes the other real points
to the two bounding rays of that sector. Then z * f''(z) / f'(z) → β - 1 as z tends to
infinity in the upper half-plane.
In terms of f, the hypotheses say that f tends to infinity at infinity and that far out it
fills the sector of opening β * π with vertex c, with the far parts of the real axis carried to
the two bounding rays. For a Schwarz--Christoffel map the limit is the sum of the turning
exponents, so the finite vertices then
turn through (β - 1) * π in total.
The pre-Schwarzian of a map with a parallel-sided end at infinity. Read the map f of
the upper half-plane in the coordinate w ↦ -1 / w at infinity, normalize the target by
w ↦ (w - c) / b, and exponentiate. If the result extends to a function g with g 0 = 0
which is continuous and injective up to a real segment through 0, upper half-plane valued
above it, and real on it, then z * f''(z) / f'(z) → -1 as z tends to infinity in the upper
half-plane.
The hypotheses are local at w = 0, that is near infinity in the source: they constrain f only
through g on the upper half of the ball of radius r, and say nothing about the rest of the
image of f. The typical source of such a g is a map f which far out fills a half-strip
between two parallel rays, with the far parts of the real axis carried to those rays, where c
and b are chosen so that w ↦ (w - c) / b carries that half-strip to
{w | w.re < 0 ∧ 0 < w.im ∧ w.im < π}, which the exponential maps onto the upper half of the unit
disc; the polygonal-domain theorems derive the hypotheses on g from such geometry. This is
the opening β = 0 counterpart of
TauCeti.tendsto_mul_logDeriv_deriv_upperHalfPlaneSet_of_eqOn_div_neg_inv.
The pre-Schwarzian asymptotic at infinity is invariant under maps conformal across
infinity. Let q be holomorphic with nonvanishing derivative on the upper half-plane, with
ζ * q''(ζ) / q'(ζ) → L as ζ tends to infinity there. Suppose that f ∘ φ = q near infinity,
where φ fixes infinity and is conformal across it: in the coordinate w ↦ -1 / w, the map
φ reads as a function g with g 0 = 0 which is continuous and injective up to a real
segment through 0, holomorphic and upper half-plane valued above it, and real on it. Then
also z * f''(z) / f'(z) → L as z tends to infinity in the upper half-plane.
This compares a far end of a domain with an explicit model map q whose asymptotic is computed
directly.