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Li et al. z + ieiγ z − ieiγ sin γ sin γ log log + iγ 2(1 − sin 2γ ) ie 2(1 + sin 2γ ) −ieiγ u = Im − z − e−iγ sin γ log cos2 2γ −e−iγ z − e−iγ cos γ log 2 cos 2γ −e−iγ F = ±Re − if γ ∈ / cos γ cos γ e−iγ − Re , cos 2γ cos 2γ z − e−iγ cos γ cos γ z + ieiγ z − ieiγ + log log 2(1 − sin 2γ ) ieiγ 2(1 + sin 2γ ) −ieiγ v = Im − − − sin γ sin γ e−iγ − Re , cos 2γ cos 2γ z − e−iγ log (z + ieiγ ) log (z − ieiγ ) sin 2γ − − log (z − e−iγ ) 2(1 − sin 2γ ) 2(1 + sin 2γ ) cos2 2γ ie−iγ (z − e−iγ ) cos 2γ π 3π 5π 7π , , 4, 4 4 4 + c, .

Equivalently, we say that a function is sense-preserving if the left-hand side of a curve is mapped to the left-hand side of its image. The following theorem formalizes this intuition. 8 (Lewy [22]) f (z) = h(z) + g(z) is locally univalent and sensepreserving if and only if |ω(z)| = |g (z)/ h (z)| < 1, for all z ∈ D. The function ω = g / h is known as the dilatation of f = h + g. Observe that in the harmonic case, terms involving z are permissible, but terms involving zz are not. Also, the graphics highlight the fact that the images of radial and circular lines intersect at right angles in the conformal case, but not in the harmonic case.

Let’s define this idea of inner mapping radius precisely. 1 For f ∈ SHO , the inner mapping radius, ρO (f ), of the domain f (D) is the real number F (0), where • F is the analytic function that maps D onto f (D) • F (0) = 0 • F (0) > 0. Notice that the existence of such a function F is guaranteed by the Riemann Mapping Theorem. The functions in S are normalized by requiring that F (0) = 1. The Riemann Mapping Theorem does not guarantee that there is a schlicht mapping to any simply-connected domain but does guarantee that we can multiply a schlicht function by some positive real number in order to map onto that domain.