🧭 New here?
Take a guided tour of the site.
Registry› Laguerre› Laguerre L_n
← Back to Families

Laguerre L_n

Laguerre · parametric, arity 1 · polynomial

Canonical Key
laguerre_polynomial
Parameter Space
n ∈ Z+ (integer)

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.

n=1
n=1 n=10
Taylor series

Unit disk explorer - hover to evaluate

Hover over the disk

Related

0
Counterexamples
 →
19
Facts
 →
23
Verification runs
 →
10
Instances
0
Cited in papers

Facts (19)

Property Kind Params Holds Value / r Status Confidence
K membership {"n": 2} ✗ -144.7841 unscreened -
K membership {"n": 3} ✗ -69.8850 unscreened -
K membership {"n": 4} ✗ -377.6525 unscreened -
r* radius {"n": 2} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"n": 3} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"n": 4} - r ≤ 0.960 🔬 screened numerical_strong
S* membership {"n": 10} - - retracted -
S* membership {"n": 1} - - retracted -
S* membership {"n": 5} - - retracted -
S* membership {"n": 6} - - retracted -
S* membership {"n": 7} - - retracted -
S* membership {"n": 8} - - retracted -
S* membership {"n": 9} - - retracted -
S* membership {"n": 2} ✗ -198.6724 🔬 screened numerical_weak
S* membership {"n": 3} ✗ -85.4612 🔬 screened numerical_weak
S* membership {"n": 4} ✗ -141.9342 🔬 screened numerical_weak
S membership {"n": 2} ✗ - unscreened -
S membership {"n": 3} ✗ - unscreened -
S membership {"n": 4} ✗ - unscreened -

Verification Runs (23)

Verifier Property Direction Outcome Domain r Witness Engine
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*0.073631) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*6.135923) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.300 0.6*exp(i*0.049087) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.300 0.6*exp(i*6.037748) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.300 0.6*exp(i*6.037748) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.300 0.6*exp(i*0.343612) boundary_v3.1
pointwise_starlike_iv starlike disproves pass 0.990 [0.99, 0.0] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.99, 0.0] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.99, 0.0] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.99, 0.0] gft-0.1.0-reaudit-20260610
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 [0.941545951132202, 0.3059268244311979] gft-0.1.0-reaudit-20260610
boundary_scan_polynomial starlike disproves pass 0.300 0.6*exp(i*6.025477) boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.600 0.8*exp(i*0.073631) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.600 0.8*exp(i*0.049087) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.300 0.6*exp(i*0.122718) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.300 0.6*exp(i*0.122718) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.300 0.6*exp(i*0.122718) boundary_v3.0
boundary_scan_none starlike proves not_applicable - - boundary_v3.0
boundary_scan_none starlike proves not_applicable - - boundary_v3.0
boundary_scan_none starlike proves not_applicable - - boundary_v3.0

Parameter Sweep (23 recorded instances, starlikeness evidence × disk radius vs parameter)

Legacy “proven” label (scope not recorded)   ·   Recorded finite-radius evidence   ·   Disproven (witness)

Instances (10)

{"n": 2} {"n": 3} {"n": 4} {"n": 1} {"n": 5} {"n": 6} {"n": 7} {"n": 8} {"n": 9} {"n": 10}
↑↓ navigate↵ openesc closeall results
✦ 279 explorers have wandered the registry - thanks for stopping by. Tell us what you'd like to see →
Usage & privacy

We use first-party site metrics to improve navigation: page visits, internal clicks, search result counts (not search terms), and approximate active time. A random tab ID groups events for seven days; GPC/DNT turns this off. The separate visitor count uses a self-hosted pseudonymous fingerprint and also respects GPC/DNT.

💬 Feedback