Struve H_ν (normalized)

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This function is certified starlike - the sharp, proven-exact coefficient bounds for the starlike class apply to it. See the certified proofs →

Canonical Key

struve_h_normalized

Parameter Space

nu ∈ R (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=-0.5

nu=-0.5 nu=4.0

Instance

Taylor series

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

Facts (34)

Property	Kind	Params	Holds	Value / r	Status	Confidence
K	membership	{"nu": 0.5}	✓	0.8287	🔬 screened	numerical_strong
K	membership	{"nu": 0.0}	✓	0.7625	🔬 screened	numerical_strong
K	membership	{"nu": 1.0}	✓	0.8657	🔬 screened	numerical_strong
K	membership	{"nu": 2.0}	✓	0.9060	🔬 screened	numerical_strong
K	membership	{"nu": 3.0}	✓	0.9275	🔬 screened	numerical_strong
K	membership	{"nu": -0.5}	✓	0.6071	🔬 screened	numerical_strong
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": 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": -0.5}	-	r ≤ 0.960	🔬 screened	numerical_strong
S*	membership	{"nu": 1.75}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": 4.0}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": -0.75}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": -0.25}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": 0.1}	✓	r ≤ 0.999	✓ proven	rigorous_verified
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": 1.25}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": 1.5}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": 2.5}	✓	r ≤ 0.999	✓ proven	rigorous_verified
S*	membership	{"nu": 0.5}	✓	0.9187	🔬 screened	numerical_weak
S*	membership	{"nu": 0.0}	✓	0.8900	🔬 screened	numerical_weak
S*	membership	{"nu": 1.0}	✓	0.9354	🔬 screened	numerical_weak
S*	membership	{"nu": 2.0}	✓	0.9541	🔬 screened	numerical_weak
S*	membership	{"nu": 3.0}	✓	0.9644	🔬 screened	numerical_weak
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": 1.0}	✓	-	🔬 screened	numerical_strong
S	membership	{"nu": 2.0}	✓	-	🔬 screened	numerical_strong
S	membership	{"nu": 3.0}	✓	-	🔬 screened	numerical_strong
S	membership	{"nu": -0.5}	✓	-	🔬 screened	numerical_strong

Transitively Implied Properties

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

Verification Runs (27)

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_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_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
boundary_scan_none	starlike	proves	not_applicable	-	-	boundary_v3.0
boundary_scan_none	starlike	proves	not_applicable	-	-	boundary_v3.0
boundary_scan_polynomial	starlike	proves	pass	0.999	-	boundary_v3.0