Math 2.0

Functions of a complex variable Definition

Entire function

Builds on Holomorphic function

Definition

An entire function is a function f:ℂ→ℂf : \mathbb{C} \to \mathbb{C} that is holomorphic on all of ℂ\mathbb{C}.

Remarks

Polynomials, the complex exponential and the complex sine and cosine are entire. Sums, products and compositions of entire functions are entire. An entire function is equal to one power series ∑n≥0anzn\sum_{n \ge 0} a_n z^n that converges for every z∈ℂz \in \mathbb{C}. By Liouville's theorem, each bounded entire function is constant.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

Show the path as a graph
Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionEntire function, definition, no evidence levelEntire functiondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Real and imaginary parts definition
  5. Supremum definition
  6. Completeness axiom statement
  7. Existence of square roots of nonnegative reals statement
  8. Square root definition
  9. Modulus of a complex number definition
  10. Complex differentiability definition
  11. Holomorphic function definition
  12. Entire function definition

Where this leads

28 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Entire_function, lead section
About this page
Tag0419794R0T
URIm2://0419794R0T
Kinddefinition
WritingW1 stated
MSC30D20
DefinesEntire function
UsesHolomorphic function

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by

Results about this