Math 2.0

Special functions Proposition

The complex exponential is entire

Builds on Complex exponentialEntire function

Statement

The complex exponential exp⁡:ℂ→ℂ\exp : \mathbb{C} \to \mathbb{C} is an entire function: it is complex differentiable at each z∈ℂz \in \mathbb{C}.

Remarks

Proof: exp⁡\exp is given by a power series with infinite radius of convergence, and power series are holomorphic in their disc of convergence. Its derivative is exp⁡\exp itself.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionComplex exponential, definition, no evidence levelComplex exponentialdefinitionReal 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 functiondefinitionThe complex exponential is entire, statement, no evidence levelThe complex exponenti…statement
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. Complex exponential definition
  5. Real and imaginary parts definition
  6. Supremum definition
  7. Completeness axiom statement
  8. Existence of square roots of nonnegative reals statement
  9. Square root definition
  10. Modulus of a complex number definition
  11. Complex differentiability definition
  12. Holomorphic function definition
  13. Entire function definition
  14. The complex exponential is entire statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Exponential_function, Complex exponential
  2. Web https://en.wikipedia.org/wiki/Entire_function, lead section
About this page
Tag0G2YJN0H5S
URIm2://0G2YJN0H5S
Kindstatement, proposition
WritingW0 stub
MSC33B10
StatusThis copy of the site has no build data. Run m2 status 0G2YJN0H5S.
UsesComplex exponential, Entire function

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