Math 2.0

Special functions Proposition

Derivative of the complex exponential

Builds on Complex exponentialComplex differentiability

Statement

For each z∈ℂz \in \mathbb{C}, the complex exponential is complex differentiable at zz, and exp⁡′(z)=exp⁡z.\exp'(z) = \exp z.

Remarks

Proof: differentiate the series ∑n≥0zn/n!\sum_{n \ge 0} z^n / n! term by term. The derivative of zn/n!z^n / n! is zn−1/(n−1)!z^{n-1} / (n-1)! for n≥1n \ge 1, so the series of derivatives is the same series. With the chain rule, the derivative of eaze^{a z} is aeaza e^{a z} for each constant a∈ℂa \in \mathbb{C}. It follows that exp⁡\exp is entire.

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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…definitionDerivative of the complex exponential, statement, no evidence levelDerivative of the com…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.
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  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. Derivative of the complex exponential statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Exponential_function, Complex exponential
  2. Web https://en.wikipedia.org/wiki/Exponential_function, Differential equation
About this page
Tag019H1Z6419
URIm2://019H1Z6419
Kindstatement, proposition
WritingW0 stub
MSC33B10
StatusThis copy of the site has no build data. Run m2 status 019H1Z6419.
UsesComplex exponential, Complex differentiability

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