Koebe function
E898494
The Koebe function is a specific univalent holomorphic function on the unit disk that extremizes several classical bounds in geometric function theory, notably serving as the extremal example in the Koebe quarter theorem.
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
extremal function
ⓘ
holomorphic function ⓘ object of geometric function theory ⓘ univalent function ⓘ |
| alternativeFormula | k(z) = z + 2z^2 + 3z^3 + 4z^4 + \ ⓘ |
| attainsEqualityIn |
Koebe quarter theorem lower bound 1/4
ⓘ
distortion theorem for univalent functions ⓘ growth theorem for univalent functions ⓘ |
| belongsToClass | S (class of normalized univalent functions on unit disk) ⓘ |
| definedOn | open unit disk ⓘ |
| extremalFamily | {e^{-i\theta} k(e^{i\theta} z) : \theta \in \mathbb{R}} ⓘ |
| field |
complex analysis
ⓘ
geometric function theory ⓘ |
| hasAsymptoticBehavior |
k(-r) \to -1/4 as r \to 1^- along negative real axis
ⓘ
k(r) \to +\infty as r \to 1^- along positive real axis ⓘ |
| hasCoveringProperty | k(\mathbb{D}) \supset \{w : |w| < 1/4\} GENERATED ⓘ |
| hasDerivative | k'(z) = (1 + z) / (1 - z)^3 ⓘ |
| hasFormula | k(z) = z / (1 - z)^2 ⓘ |
| hasGrowthProperty | |k(z)| \ge |z| / (1 + |z|)^2 for |z| < 1 ⓘ |
| hasMaclaurinSeries | z + 2z^2 + 3z^3 + 4z^4 + 5z^5 + \cdots GENERATED ⓘ |
| hasRealBoundaryValuesOn |
(-1,0)
GENERATED
ⓘ
(0,1) GENERATED ⓘ |
| hasRotationFamily | e^{-i\theta} k(e^{i\theta} z) ⓘ |
| hasSecondDerivative | k''(z) = 4 / (1 - z)^4 ⓘ |
| hasSingularity | pole of order 2 at z = 1 ⓘ |
| hasSlitType | slit along the negative real axis ⓘ |
| hasTaylorCoefficient | a_n = n for n \ge 1 ⓘ |
| image | \mathbb{C} \setminus (-\infty,-1/4] ⓘ |
| isExampleOf | normalized schlicht function ⓘ |
| isExtremalFor |
Bieberbach conjecture for n = 2
ⓘ
Koebe quarter theorem NERFINISHED ⓘ covering theorems in geometric function theory ⓘ distortion estimates for univalent functions ⓘ growth estimates for univalent functions ⓘ |
| isPrototypeFor | sharp coefficient bounds in univalent function theory ⓘ |
| isUnivalentOn | {z \in \mathbb{C} : |z| < 1} ⓘ |
| maps | unit disk onto complex plane minus a slit ⓘ |
| namedAfter | Paul Koebe NERFINISHED ⓘ |
| normalizedBy |
k'(0) = 1
ⓘ
k(0) = 0 ⓘ |
| powerSeriesExpansion | k(z) = \sum_{n=1}^{\infty} n z^n ⓘ |
| symbol | k(z) ⓘ |
| usedAs | extremal example in many univalence criteria ⓘ |
| usedToShow | sharpness of coefficient estimates for univalent functions ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.