Math 2.0

definition

Entire 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.

Explanations

L1 human

An entire function is a function that is holomorphic at every complex number, with no exceptions. For example, z3−2z+5z^3 - 2z + 5 and eze^z are entire, but 1/z1/z is not, because it is not defined at 00.

What this node 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 node is. They do not measure its truth.

What this needs

To read this node, read these 12 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

What depends on this

3 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Entire_function, lead section

What links here

Used by