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q-exponential e_q(z)

q-Series · parametric, arity 1

This function is certified starlike - the sharp, proven-exact coefficient bounds for the starlike class apply to it. See the certified proofs →
Canonical Key
q_exp
Parameter Space
q ∈ (0,1) (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.

q=0.1
q=0.1 q=0.99
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

→
12
Counterexamples
 →
71
Facts
 →
50
Verification runs
 →
38
Instances
0
Cited in papers

Facts (71)

Property Kind Params Holds Value / r Status Confidence
K membership {"q": 0.1} ✗ -548.6284 unscreened -
K membership {"q": 0.3} ✗ -1.1192 unscreened -
K membership {"q": 0.5} ✗ -1.9828 unscreened -
K membership {"q": 0.7} ✗ -3.9577 unscreened -
K membership {"q": 0.9} ✗ -10.1157 unscreened -
K membership {"q": 0.05} ✗ -21.8681 unscreened -
K membership {"q": 0.2} ✗ -3.5040 unscreened -
K membership {"q": 0.4} ✗ -1.2205 unscreened -
K membership {"q": 0.6} ✗ -2.3366 unscreened -
K membership {"q": 0.8} ✗ -4.5240 unscreened -
K membership {"q": 0.95} ✗ -8.5937 unscreened -
r* radius {"q": 0.1} - r ≤ 0.941 🔬 screened numerical_strong
r* radius {"q": 0.3} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"q": 0.5} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"q": 0.7} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"q": 0.9} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"q": 0.05} - r ≤ 0.874 unscreened -
r* radius {"q": 0.2} - r ≤ 0.920 unscreened -
r* radius {"q": 0.4} - r ≤ 0.920 unscreened -
r* radius {"q": 0.6} - r ≤ 0.920 unscreened -
r* radius {"q": 0.8} - r ≤ 0.920 unscreened -
r* radius {"q": 0.95} - r ≤ 0.920 unscreened -
S* membership {"q": 0.99} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.925} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.875} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.85} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.825} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.775} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.75} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.725} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.675} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.65} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.625} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.575} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.55} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.525} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.475} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.45} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.425} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.375} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.35} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.325} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.275} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.225} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.175} ✗ r ≤ 0.950 ✗ disproven proven
S* membership {"q": 0.15} ✗ r ≤ 0.950 ✗ disproven proven
S* membership {"q": 0.125} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"q": 0.075} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"q": 0.025} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"q": 0.1} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"q": 0.3} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.7} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.9} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.05} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"q": 0.2} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.4} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.6} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.8} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"q": 0.95} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S membership {"q": 0.1} ✗ - unscreened -
S membership {"q": 0.3} ✗ - unscreened -
S membership {"q": 0.5} ✗ - unscreened -
S membership {"q": 0.7} ✗ - unscreened -
S membership {"q": 0.9} ✗ - unscreened -
S membership {"q": 0.05} ✗ - unscreened -
S membership {"q": 0.2} ✗ - unscreened -
S membership {"q": 0.4} ✗ - unscreened -
S membership {"q": 0.6} ✗ - unscreened -
S membership {"q": 0.8} ✗ - unscreened -
S membership {"q": 0.95} ✗ - 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 disproves pass 0.950 0.99*exp(i*2.992797) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.950 0.99*exp(i*2.988195) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*3.291923) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.800 0.9*exp(i*2.992797) 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 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 disproves pass 0.900 0.95*exp(i*2.988195) boundary_v3.1
pointwise_starlike_iv starlike disproves pass 0.990 [-0.9803653880541546, 0.13778136995046467] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.8898061058361754, 0.4339874353211865] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.49500000000000044, -0.857365149746594] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.8898061058361754, 0.4339874353211865] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.9803653880541546, 0.13778136995046467] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.9803653880541546, 0.13778136995046467] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.9803653880541546, 0.13778136995046467] 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

Parameter Sweep (83 verified instances, starlikeness × disk radius vs parameter)

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

Instances (38)

{"q": 0.1} {"q": 0.3} {"q": 0.5} {"q": 0.7} {"q": 0.9} {"q": 0.05} {"q": 0.2} {"q": 0.4} {"q": 0.6} {"q": 0.8} {"q": 0.95} {"q": 0.025} {"q": 0.075} {"q": 0.125} {"q": 0.15} {"q": 0.175} {"q": 0.225} {"q": 0.275} {"q": 0.325} {"q": 0.35} {"q": 0.375} {"q": 0.425} {"q": 0.45} {"q": 0.475} {"q": 0.525} {"q": 0.55} {"q": 0.575} {"q": 0.625} {"q": 0.65} {"q": 0.675} {"q": 0.725} {"q": 0.75} {"q": 0.775} {"q": 0.825} {"q": 0.85} {"q": 0.875} {"q": 0.925} {"q": 0.99}
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