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Mittag-Leffler E_α

Mittag-Leffler · 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
mittag_leffler_normalized
Generating Definition
f(z) = Σ_{n>=1} z^n / Γ(α(n-1)+1)
Parameter Space
alpha ∈ (0, 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.

alpha=1.0
alpha=1.0 alpha=4.0
Taylor series
r* ≈ 0.990
Radius of starlikeness r* ≈ 0.9900
disproven f is starlike on |z| < r*; the colored disk shows the certified region.

Unit disk explorer - hover to evaluate

Hover over the disk

Related

→
33
Counterexamples
 →
78
Facts
 →
50
Verification runs
 →
38
Instances
 →
3
Cited in papers

Facts (78)

Property Kind Params Holds Value / r Status Confidence
K membership {"alpha": 0.25} ✗ -0.4990 unscreened -
K membership {"alpha": 0.75} ✗ -1.9710 unscreened -
K membership {"alpha": 1.5} ✗ -607.8305 unscreened -
K membership {"alpha": 3.0} ✓ 0.5432 🔬 screened numerical_strong
K membership {"alpha": 5.0} ✓ 0.9844 unscreened -
K membership {"alpha": 1.0} ✗ -23.9600 unscreened -
K membership {"alpha": 2.0} ✗ -3.9467 unscreened -
K membership {"alpha": 0.1} ✗ -30.1357 unscreened -
K membership {"alpha": 0.3} ✗ -0.3165 unscreened -
K membership {"alpha": 0.6} ✗ -0.7724 unscreened -
K membership {"alpha": 0.9} ✗ -4.6377 unscreened -
K membership {"alpha": 1.25} ✗ -348.4066 unscreened -
K membership {"alpha": 2.5} ✗ -0.0957 unscreened -
r* radius {"alpha": 0.25} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"alpha": 0.75} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"alpha": 1.5} - r ≤ 0.898 🔬 screened numerical_strong
r* radius {"alpha": 3.0} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"alpha": 5.0} - r ≤ 0.920 unscreened -
r* radius {"alpha": 1.0} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"alpha": 2.0} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"alpha": 0.1} - r ≤ 0.865 unscreened -
r* radius {"alpha": 0.3} - r ≤ 0.920 unscreened -
r* radius {"alpha": 0.6} - r ≤ 0.920 unscreened -
r* radius {"alpha": 0.9} - r ≤ 0.920 unscreened -
r* radius {"alpha": 1.25} - r ≤ 0.883 unscreened -
r* radius {"alpha": 2.5} - r ≤ 0.920 unscreened -
S* membership {"alpha": 1.75} ✗ r ≤ 0.990 ✗ disproven proven
S* membership {"alpha": 1.9} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 2.1} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 2.25} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 2.75} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 4.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.15} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"alpha": 1.2} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"alpha": 1.3} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"alpha": 1.4} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"alpha": 1.6} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"alpha": 0.95} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.1} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"alpha": 0.85} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.8} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.7} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.65} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.55} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.45} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.4} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.35} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.2} ✗ r ≤ 0.950 ✗ disproven proven
S* membership {"alpha": 0.15} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"alpha": 0.05} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.25} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.75} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.5} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"alpha": 3.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 5.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.1} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.3} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.6} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.9} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.25} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"alpha": 2.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 5.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S membership {"alpha": 0.25} ✗ - unscreened -
S membership {"alpha": 0.75} ✗ - unscreened -
S membership {"alpha": 1.5} ✗ - unscreened -
S membership {"alpha": 3.0} ✓ - 🔬 screened numerical_strong
S membership {"alpha": 5.0} ✗ - unscreened -
S membership {"alpha": 1.0} ✗ - unscreened -
S membership {"alpha": 2.0} ✗ - unscreened -
S membership {"alpha": 0.1} ✗ - unscreened -
S membership {"alpha": 0.3} ✗ - unscreened -
S membership {"alpha": 0.6} ✗ - unscreened -
S membership {"alpha": 0.9} ✗ - unscreened -
S membership {"alpha": 1.25} ✗ - unscreened -
S membership {"alpha": 2.5} ✗ - 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 disproves pass 0.800 0.9*exp(i*3.153864) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*3.129321) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.990 0.999*exp(i*3.147729) 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*3.178408) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*3.117049) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.800 0.9*exp(i*3.132389) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.800 0.9*exp(i*3.153864) 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*5.075942) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*2.988195) 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 disproves pass 0.800 0.9*exp(i*3.153864) 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 0.800 0.9*exp(i*3.141976) 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
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 0.990 [-0.49500000000000044, -0.857365149746594] gft-0.1.0-reaudit-20260610
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 0.990 [-0.49500000000000044, -0.857365149746594] 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 [0.33859994189241216, 0.9302956945780492] gft-0.1.0-reaudit-20260610
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 0.990 [-0.99, 1.2124003311558797e-16] gft-0.1.0-reaudit-20260610
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 0.990 [-0.99, 1.2124003311558797e-16] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.99, 1.2124003311558797e-16] gft-0.1.0-reaudit-20260610

Parameter Sweep (90 verified instances, starlikeness × disk radius vs parameter)

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

Instances (38)

{"alpha": 1.0} {"alpha": 5.0} {"alpha": 2.0} {"alpha": 0.25} {"alpha": 0.75} {"alpha": 1.5} {"alpha": 3.0} {"alpha": 0.1} {"alpha": 0.3} {"alpha": 0.6} {"alpha": 0.9} {"alpha": 1.25} {"alpha": 2.5} {"alpha": 0.05} {"alpha": 0.15} {"alpha": 0.2} {"alpha": 0.35} {"alpha": 0.4} {"alpha": 0.45} {"alpha": 0.5} {"alpha": 0.55} {"alpha": 0.65} {"alpha": 0.7} {"alpha": 0.8} {"alpha": 0.85} {"alpha": 0.95} {"alpha": 1.1} {"alpha": 1.15} {"alpha": 1.2} {"alpha": 1.3} {"alpha": 1.4} {"alpha": 1.6} {"alpha": 1.75} {"alpha": 1.9} {"alpha": 2.1} {"alpha": 2.25} {"alpha": 2.75} {"alpha": 4.0}

Cited in Papers (3)

Exponential radii of starlikeness and convexity of some special functions
Unknown authors · 2024
Radii of starlikeness and convexity of generalized Mittag-Leffler functions
Árpád Baricz, Anuja Prajapati · 2019
Starlikeness and convexity of integral operators involving Mittag-Leffler functions
B. A. Frasin · 2018
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