Exterior points of Schwarz--Christoffel images

If all finite prevertices are integrable and the total turning exponent is less than 1, the closure of the Schwarz--Christoffel image is a proper subset of the plane. This provides an exterior point about which to invert an unbounded polygonal image, reducing boundary separation questions to bounded images. Boundary simplicity is not required.

For total exponent greater than -1, the leading power has opening strictly less than 2π. The uniform power asymptotic confines the image at infinity to a slightly wider cone, while continuity up to the real axis bounds the remaining compact part. At total exponent -1, the real part tends to positive infinity, so the image has a global lower bound on its real part. For total exponent less than -1, the image is bounded and its closure is compact.

At total exponent 1 the leading term z ^ 2 / 2 maps the upper half-plane onto a slit plane, whose closure contains a full neighbourhood of infinity, and the logarithmic coefficient C = ((∑ i, e i * a i) ^ 2 - ∑ i, e i * a i ^ 2) / 2 decides what is left out. When C < 0, for each δ > 0 the closure of the image misses the points of height strictly between im c + π * C + δ and im c - δ that lie sufficiently far to the right, where c is the quadratic constant at infinity and how far to the right depends on δ. The heights im c + π * C and im c carry the outer sides of an end of opening 2π, so each such open sub-band supplies exterior points. Without the sign condition there need not be an exterior point, as for the slit plane z ↦ z ^ 2.

References

• L. Ahlfors, Complex Analysis, Chapter 6, Section 2.
• T. Driscoll and L. Trefethen, Schwarz--Christoffel Mapping, Chapter 2.

theorem TauCeti.exists_notMem_closure_image_schwarzChristoffelPrimitive_of_sum_lt_one {ι : Type u_1} [Fintype ι] (a e : ι → ℝ) (z₀ : UpperHalfPlane) (hfinite : ∀ (j : ι), -1 < ∑ i : ι with a i = a j, e i) (hhigh : ∑ i : ι, e i < 1) : ∃ (q : ℂ), q ∉ closure (schwarzChristoffelPrimitive a e z₀ '' UpperHalfPlane.upperHalfPlaneSet)

A Schwarz--Christoffel image has an exterior point if the finite prevertices are integrable and the total exponent is less than 1. The point lies outside the closure of the image, not merely outside the image. No simplicity or sign assumptions are imposed on the finite turning data.

theorem TauCeti.exists_forall_notMem_closure_image_schwarzChristoffelPrimitive_of_sum_eq_one {ι : Type u_1} [Fintype ι] (a e : ι → ℝ) (z₀ : UpperHalfPlane) (hfinite : ∀ (j : ι), -1 < ∑ i : ι with a i = a j, e i) (hsum : ∑ i : ι, e i = 1) (hC : (∑ i : ι, e i * a i) ^ 2 < ∑ i : ι, e i * a i ^ 2) {δ : ℝ} (hδ : 0 < δ) : ∃ (R : ℝ), ∀ (w : ℂ), R < w.re → (schwarzChristoffelQuadraticConstantAtInfinity a e z₀).im + Real.pi * (((∑ i : ι, e i * a i) ^ 2 - ∑ i : ι, e i * a i ^ 2) / 2) + δ < w.im → w.im < (schwarzChristoffelQuadraticConstantAtInfinity a e z₀).im - δ → w ∉ closure (schwarzChristoffelPrimitive a e z₀ '' UpperHalfPlane.upperHalfPlaneSet)

The far-right band missed by an image with an end of opening 2π. Suppose the finite prevertices are integrable, the total exponent is 1, and the logarithmic coefficient C = ((∑ i, e i * a i) ^ 2 - ∑ i, e i * a i ^ 2) / 2 is negative. Then for each δ > 0, sufficiently far to the right, no point whose height lies strictly between im c + π * C + δ and im c - δ is in the closure of the image, where c is the quadratic constant at infinity. No simplicity or ordering assumption on the finite data is imposed.

theorem TauCeti.exists_notMem_closure_image_schwarzChristoffelPrimitive_of_sum_eq_one {ι : Type u_1} [Fintype ι] (a e : ι → ℝ) (z₀ : UpperHalfPlane) (hfinite : ∀ (j : ι), -1 < ∑ i : ι with a i = a j, e i) (hsum : ∑ i : ι, e i = 1) (hC : (∑ i : ι, e i * a i) ^ 2 < ∑ i : ι, e i * a i ^ 2) : ∃ (q : ℂ), q ∉ closure (schwarzChristoffelPrimitive a e z₀ '' UpperHalfPlane.upperHalfPlaneSet)

A Schwarz--Christoffel image with an end of opening 2π has an exterior point if the finite prevertices are integrable, the total exponent is 1, and the logarithmic coefficient ((∑ i, e i * a i) ^ 2 - ∑ i, e i * a i ^ 2) / 2 is negative. The point lies outside the closure of the image. No simplicity or ordering assumption on the finite data is imposed.