Math 2.0

statement proposition

The complex exponential is entire

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.

Explanations

L1 human

The exponential function has a complex derivative at every point of the plane. So it is a holomorphic function without any exceptional points.

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To read this node, read these 14 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 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 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. 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

Sources

  1. Web https://en.wikipedia.org/wiki/Exponential_function, Complex exponential
  2. Web https://en.wikipedia.org/wiki/Entire_function, lead section

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