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Normalized Bessel J_ν

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

This function is certified starlike - the sharp, proven-exact coefficient bounds for the starlike class apply to it. See the certified proofs →
Canonical Key
bessel_j_normalized
Generating Definition
f(z) = z + Σ_{n>=2} (-1)^(n-1) / (4^(n-1) (n-1)! Π_{k=1}^{n-1}(ν+k)) z^n
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=9.0
Taylor series
r* ≈ 0.999
Radius of starlikeness r* ≈ 0.9990
proven f is starlike on |z| < r*; the colored disk shows the certified region.

Unit disk explorer - hover to evaluate

Hover over the disk

Related

→
10
Counterexamples
 →
66
Facts
 →
50
Verification runs
 →
33
Instances
 →
7
Cited in papers

Facts (66)

Property Kind Params Holds Value / r Status Confidence
K membership {"nu": 0.5} ✓ 0.6071 🔬 screened numerical_strong
K membership {"nu": 0.0} ✓ 0.2913 🔬 screened numerical_strong
K membership {"nu": 10.0} ✓ 0.9576 unscreened -
K membership {"nu": 1.5} ✓ 0.7885 🔬 screened numerical_strong
K membership {"nu": 1.0} ✓ 0.7257 🔬 screened numerical_strong
K membership {"nu": 2.0} ✓ 0.8276 🔬 screened numerical_strong
K membership {"nu": 3.0} ✓ 0.8738 🔬 screened numerical_strong
K membership {"nu": 4.0} ✓ 0.9003 🔬 screened numerical_strong
K membership {"nu": 5.0} ✓ 0.9211 unscreened -
K membership {"nu": -0.25} ✗ -0.1215 unscreened -
K membership {"nu": -0.5} ✗ -3.9467 unscreened -
r* radius {"nu": 0.5} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"nu": 0.0} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"nu": 10.0} - r ≤ 0.920 unscreened -
r* radius {"nu": 1.5} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"nu": 1.0} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"nu": 2.0} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"nu": 3.0} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"nu": 4.0} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"nu": 5.0} - r ≤ 0.920 unscreened -
r* radius {"nu": -0.25} - r ≤ 0.920 unscreened -
r* radius {"nu": -0.5} - r ≤ 0.960 🔬 screened numerical_strong
S* membership {"nu": 0.4} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 0.6} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 0.8} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 1.2} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 1.4} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 1.6} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 1.8} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 2.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 6.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 7.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 8.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 9.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": -0.9} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"nu": -0.8} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"nu": -0.7} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"nu": -0.6} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"nu": -0.4} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": -0.3} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": -0.2} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": -0.1} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 0.1} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 0.2} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 0.5} ✓ 0.8287 🔬 screened numerical_weak
S* membership {"nu": 0.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 10.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 1.5} ✓ 0.9012 🔬 screened numerical_weak
S* membership {"nu": 1.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 3.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 4.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": 5.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": -0.25} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"nu": -0.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S membership {"nu": 0.5} ✓ - 🔬 screened numerical_strong
S membership {"nu": 0.0} ✓ - 🔬 screened numerical_strong
S membership {"nu": 10.0} ✗ - unscreened -
S membership {"nu": 1.5} ✓ - 🔬 screened numerical_strong
S membership {"nu": 1.0} ✓ - 🔬 screened numerical_strong
S membership {"nu": 2.0} ✓ - 🔬 screened numerical_strong
S membership {"nu": 3.0} ✓ - 🔬 screened numerical_strong
S membership {"nu": 4.0} ✓ - 🔬 screened numerical_strong
S membership {"nu": 5.0} ✗ - unscreened -
S membership {"nu": -0.25} ✗ - unscreened -
S membership {"nu": -0.5} ✗ - unscreened -

Transitively Implied Properties

Because starlikeness is proven, these properties follow by classical implications (unconditional edges in the lattice):

S C

Verification Runs (50)

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 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 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 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*0.049087) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*0.024544) 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 0.413 [0.4133613445378151, 0.0] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.872 [0.8715126050420168, 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
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
boundary_scan_polynomial starlike disproves pass 0.000 0.3*exp(i*6.185011) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.300 0.6*exp(i*6.185011) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.600 0.8*exp(i*6.185011) boundary_v3.0

Parameter Sweep (68 verified instances, starlikeness × disk radius vs parameter)

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

Instances (33)

{"nu": 0.0} {"nu": 1.0} {"nu": 2.0} {"nu": -0.5} {"nu": 0.5} {"nu": 1.5} {"nu": 3.0} {"nu": 4.0} {"nu": 5.0} {"nu": 10.0} {"nu": -0.25} {"nu": -0.9} {"nu": -0.8} {"nu": -0.7} {"nu": -0.6} {"nu": -0.4} {"nu": -0.3} {"nu": -0.2} {"nu": -0.1} {"nu": 0.1} {"nu": 0.2} {"nu": 0.4} {"nu": 0.6} {"nu": 0.8} {"nu": 1.2} {"nu": 1.4} {"nu": 1.6} {"nu": 1.8} {"nu": 2.5} {"nu": 6.0} {"nu": 7.0} {"nu": 8.0} {"nu": 9.0}

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
Unknown authors · 2024
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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