Statements (50)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
Complex function |
| gptkbp:compatibleWith |
Meromorphic function (unless constant)
|
| gptkbp:definedIn |
gptkb:Complex_plane
|
| gptkbp:example |
gptkb:Exponential_function
gptkb:Cosine_function gptkb:Sine_function Hyperbolic cosine Hyperbolic sine Polynomial function |
| gptkbp:growthClassifiedBy |
gptkb:state_order
Type |
| gptkbp:heldBy |
Holomorphic function
Analytic function Function holomorphic everywhere |
| https://www.w3.org/2000/01/rdf-schema#label |
Entire Function
|
| gptkbp:property |
If bounded, must be constant (Liouville's theorem)
Can be classified by growth rate at infinity If of finite order, growth is controlled If of order 0, is a polynomial If of order less than 1, has special properties If real on real axis, called real entire function May have essential singularity at infinity No singularities in finite complex plane Zeros are isolated Zeros can be counted with Jensen's formula Can be written as infinite product (Weierstrass factorization) If not a polynomial, has essential singularity at infinity Can be represented by power series convergent everywhere If of order 1, called entire function of exponential type Can be constructed from prescribed zeros (Weierstrass theorem) |
| gptkbp:relatedTo |
gptkb:Weierstrass_factorization_theorem
gptkb:Hadamard's_factorization_theorem gptkb:Liouville's_theorem gptkb:Picard's_theorem gptkb:Exponential_function gptkb:Cosine_function gptkb:Sine_function Meromorphic function Polynomial Holomorphic function Analytic function Entire transcendental function |
| gptkbp:studiedIn |
Complex analysis
|
| gptkbp:subjectOf |
gptkb:Weierstrass_factorization_theorem
gptkb:Hadamard's_factorization_theorem gptkb:Liouville's_theorem gptkb:Picard's_theorem |
| gptkbp:bfsParent |
gptkb:Meromorphic_Functions
|
| gptkbp:bfsLayer |
7
|