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Modified Bessel I_ν

Bessel · parametric, arity 1 · entire · decay: factorial

Canonical Key
bessel_i_normalized
Parameter Space
nu ∈ (-1, 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.

nu=0.0
nu=0.0 nu=8.0
Taylor series

Unit disk explorer - hover to evaluate

Hover over the disk

Related

0
Counterexamples
 →
25
Facts
 →
40
Verification runs
 →
19
Instances
 →
7
Cited in papers

Facts (25)

Property Kind Params Holds Value / r Status Confidence
K membership {"nu": 0.0} ✓ 0.2913 🔬 screened numerical_strong
K membership {"nu": 1.0} ✓ 0.7257 🔬 screened numerical_strong
r* radius {"nu": 0.0} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"nu": 1.0} - r ≤ 0.960 🔬 screened numerical_strong
S* membership {"nu": -0.75} - - retracted -
S* membership {"nu": 0.25} - - retracted -
S* membership {"nu": 0.75} - - retracted -
S* membership {"nu": 3.5} - - retracted -
S* membership {"nu": 6.0} - - retracted -
S* membership {"nu": 8.0} - - retracted -
S* membership {"nu": -0.9} - - retracted -
S* membership {"nu": -0.5} - - retracted -
S* membership {"nu": -0.25} - - retracted -
S* membership {"nu": 0.5} - - retracted -
S* membership {"nu": 1.5} - - retracted -
S* membership {"nu": 2.0} - - retracted -
S* membership {"nu": 2.5} - - retracted -
S* membership {"nu": 3.0} - - retracted -
S* membership {"nu": 4.0} - - retracted -
S* membership {"nu": 5.0} - - retracted -
S* membership {"nu": 10.0} - - retracted -
S* membership {"nu": 0.0} - 0.7257 retracted -
S* membership {"nu": 1.0} - 0.8749 retracted -
S membership {"nu": 0.0} ✓ - 🔬 screened numerical_strong
S membership {"nu": 1.0} ✓ - 🔬 screened numerical_strong

Verification Runs (40)

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 proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.300 0.6*exp(i*3.19068) boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
pointwise_starlike_iv starlike disproves pass 0.990 [-0.99, 1.2124003311558797e-16] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.413 [-0.4133613445378151, 5.062216474794815e-17] gft-0.1.0-reaudit-20260610
boundary_scan_polynomial starlike disproves pass 0.300 0.6*exp(i*3.043418) boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.300 0.3*exp(i*3.043418) boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0

Parameter Sweep (40 recorded instances, starlikeness evidence × disk radius vs parameter)

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

Instances (19)

{"nu": 0.0} {"nu": 1.0} {"nu": -0.9} {"nu": -0.5} {"nu": -0.25} {"nu": 0.5} {"nu": 1.5} {"nu": 2.0} {"nu": 2.5} {"nu": 3.0} {"nu": 4.0} {"nu": 5.0} {"nu": 10.0} {"nu": -0.75} {"nu": 0.25} {"nu": 0.75} {"nu": 3.5} {"nu": 6.0} {"nu": 8.0}

Cited in Papers (7)

Bounds and asymptotic expansions for the radii of convexity and uniform convexity of normalized Bessel functions
Árpád Baricz, Pranav Kumar, Sanjeev Singh · 2025
Exponential radii of starlikeness and convexity of some special functions
Adiba Naz; Sumit Nagpal; V. Ravichandran · 2024 · The Ramanujan Journal
Sufficiency for Nephroid Starlikeness using Hypergeometric Functions
A. Swaminathan, Lateef Ahmad Wani · 2021
Radii of starlikeness and convexity of generalized $k-$Bessel functions
Evrim Toklu · 2019
Radii of starlikeness and convexity of generalized Mittag-Leffler functions
Árpád Baricz, Anuja Prajapati · 2019
Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity
Christophe Charlier, Alfredo Deaño · 2017
Certain results on $q$-starlike and $q$-convex error functions
S. Kanas, C. Ramachandran, L. Vanitha · 2016
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