Math 2.0

Functions of a complex variable Definition

Zero of a holomorphic function

Builds on Holomorphic function

Definition

Let U⊆ℂU \subseteq \mathbb{C} be an open set and let f:U→ℂf : U \to \mathbb{C} be holomorphic on UU. A zero of ff is a point z0∈Uz_0 \in U with f(z0)=0f(z_0) = 0.

Remarks

The zeros of sin⁡\sin are the numbers kπk\pi with k∈ℤk \in \mathbb{Z}. The complex exponential has no zeros. Each zero has an order, a positive integer or ∞\infty. If ff is not the zero function on a domain, its zeros are isolated.

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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 functiondefinitionZero of a holomorphic function, definition, no evidence levelZero of a holomorphic…definition
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  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. Zero of a holomorphic function definition

Where this leads

43 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Zeros_and_poles, Definitions
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Tag0CEGEASY49
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WritingW0 stub
MSC30C15
DefinesZero of a holomorphic function
UsesHolomorphic function

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