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Registry› Rational› Koebe Kα: z/(1-z)^(2α)
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Koebe Kα: z/(1-z)^(2α)

Rational · parametric, arity 1

Canonical Key
z/(1-z)**(2*alpha)
Closed Form
f(z) = z/(1-z)^(2*alpha)
Parameter Space
alpha ∈ (0, inf) (real)

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.

alpha=0.5
alpha=0.5 alpha=3.0
Taylor series
r* ≈ 0.000
Radius of starlikeness r* ≈ 0.0000
disproven f is starlike on |z| < r*; the colored disk shows the certified region.

Unit disk explorer - hover to evaluate

Hover over the disk

Related

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25
Counterexamples
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25
Facts
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48
Verification runs
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16
Instances
0
Cited in papers

Facts (25)

Property Kind Params Holds Value / r Status Confidence
K membership {"alpha": 0.5} ✗ -702.9535 unscreened -
K membership {"alpha": 1.5} ✗ -224.8189 unscreened -
K membership {"alpha": 2.5} ✗ -167.5027 unscreened -
r* radius {"alpha": 0.5} - r ≤ 0.955 🔬 screened numerical_strong
r* radius {"alpha": 1.5} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"alpha": 2.5} - r ≤ 0.960 🔬 screened numerical_strong
S* membership {"alpha": 3.0} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 2.25} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 2.0} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 1.75} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 1.25} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 1.1} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 1.0} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.9} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.75} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.6} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.4} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.25} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"alpha": 0.1} ✗ r ≤ 0.950 ✗ disproven proven
S* membership {"alpha": 0.5} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"alpha": 1.5} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 2.5} ✗ r ≤ 0.000 ✗ disproven proven
S membership {"alpha": 0.5} ✗ - unscreened -
S membership {"alpha": 1.5} ✗ - unscreened -
S membership {"alpha": 2.5} ✗ - unscreened -

Verification Runs (48)

Verifier Property Direction Outcome Domain r Witness Engine
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*0.266913) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.950 0.99*exp(i*6.04235) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*0.266913) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
pointwise_starlike_iv starlike disproves pass 0.990 [0.9605927690132365, -0.2395026766436712] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9516490789789357, -0.27288098225882895] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9516490789789357, 0.2728809822588292] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.966 [0.5949152356949368, -0.761456777791907] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.919 [0.31428489640892354, -0.8634906561591519] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.872 [-0.2108379814507693, 0.8456249560679293] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.840 [-0.624179204256013, -0.5620134798787216] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.816 [-0.6065685453665486, 0.5461567713918984] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.840 [0.6618628136385869, 0.5171039030151222] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.769 [0.5889529924450025, 0.4941902387413569] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.690 [0.19014344990770973, 0.6631090136098873] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.761 [0.6951390015927522, -0.3094958237858294] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.532 [-0.5266728241231426, -0.07401903831733232] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9516490789789357, -0.27288098225882895] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.919 [0.31428489640892354, -0.8634906561591519] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.840 [0.6618628136385869, 0.5171039030151222] gft-0.1.0-reaudit-20260610
boundary_scan_polynomial starlike disproves pass 0.000 0.3*exp(i*3.005835) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.000 0.3*exp(i*3.203719) boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.300 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.300 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.300 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.300 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.600 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.600 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.600 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.600 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.800 - boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.900 0.95*exp(i*6.00937) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.950 0.99*exp(i*0.241218) boundary_v3.0
boundary_scan_closed_form starlike proves pass 0.999 - boundary_v3.0
boundary_scan_closed_form starlike disproves pass 0.300 0.6*exp(i*2.847068) boundary_v3.0
boundary_scan_closed_form starlike disproves pass 0.000 0.3*exp(i*2.945243) boundary_v3.0

Parameter Sweep (48 verified instances, starlikeness × disk radius vs parameter)

Proven on full disk (r ≥ 0.999)   ·   Proven on partial disk   ·   Disproven (witness)

Instances (16)

{"alpha": 0.5} {"alpha": 1.5} {"alpha": 2.5} {"alpha": 0.1} {"alpha": 0.25} {"alpha": 0.4} {"alpha": 0.6} {"alpha": 0.75} {"alpha": 0.9} {"alpha": 1.0} {"alpha": 1.1} {"alpha": 1.25} {"alpha": 1.75} {"alpha": 2.0} {"alpha": 2.25} {"alpha": 3.0}
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