Modified Bessel I_ν

← Back to Families

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_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.

Parameter nu=0.0

nu=0.0 nu=8.0

Instance

Taylor series

Radius of starlikeness r* ≈ 0.3000
disproven f is starlike on |z| < r*; the colored disk shows the certified region.

Unit disk explorer - hover to evaluate

Hover over the disk

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}	✗	r ≤ 0.300	✗ disproven	proven
S*	membership	{"nu": 0.25}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": 0.75}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": 3.5}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": 6.0}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": 8.0}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": -0.9}	✗	r ≤ 0.000	✗ disproven	proven
S*	membership	{"nu": -0.5}	✓	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": 1.5}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": 2.0}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": 2.5}	✓	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": 10.0}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": 0.0}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": 1.0}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S	membership	{"nu": 0.0}	✓	-	🔬 screened	numerical_strong
S	membership	{"nu": 1.0}	✓	-	🔬 screened	numerical_strong

Transitively Implied Properties

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

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	0.000	-	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	0.6exp(i3.19068)	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	0.6exp(i3.043418)	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	0.000	0.3exp(i3.043418)	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