z/(1-z)

Canonical Key

z/(1-z)

Closed Form

f(z) = z/(1-z)

Coefficient Formula

a_n = 1

Image domain \(f(\mathbb{D})\)

The image of the unit disk under \(f\): concentric circles \(|z|=r\) and radial spokes mapped through \(f(z)=z+a_2z^2+\dots\); the bold curve is the boundary \(f(e^{i\theta})\). Rendered in your browser from the certified Taylor coefficients - nothing is computed server-side.

Taylor series

Unit disk explorer - hover to evaluate

Hover over the disk

Univalence Criteria

Schwarzian ladder estimate ||S(f)||

0.000

Screen only; univalence is inconclusive

Pre-Schwarzian ladder estimate ||T(f)||

3.980

Screen only; univalence is inconclusive

Facts (8)

Property	Kind	Params	Holds	Value / r	Status	Confidence
B	membership	{}	-	3.9800	retracted	-
K	membership	{}	-	-132.3225	retracted	-
N	membership	{}	-	0.0000	retracted	-
\|\|T(f)\|\|	bound	{}	-	3.9800	retracted	-
r*	radius	{}	-	r ≤ 0.874	🔬 screened	numerical_strong
\|\|S(f)\|\|	bound	{}	-	0.0000	retracted	-
S*	membership	{}	-	-11.4339	retracted	-
S	membership	{}	✗	-	unscreened	-

Verification Runs (5)

Verifier	Property	Direction	Outcome	Domain r	Witness	Engine
boundary_scan_closed_form_v31	starlike	proves	pass	0.999	-	boundary_v3.1
alexander_koebe_closed_form	convex	proves	pass	0.999	-	gft-0.1.0-reaudit-20260610
pointwise_becker_univalent_iv	becker_univalent	disproves	pass	0.872	[0.5122622564306923, 0.7050685082909731]	gft-0.1.0-reaudit-20260610
boundary_scan_centered_v31	starlike	disproves	pass	-	-	boundary_v3.1
boundary_scan_closed_form	starlike	proves	pass	0.999	-	boundary_v3.0

Instances (1)

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