Math 2.0

Functions of a complex variable Definition

Holomorphic function

Builds on Complex differentiability

Definition

Let U⊆ℂU \subseteq \mathbb{C} be an open set and let f:U→ℂf : U \to \mathbb{C} be a function. The function ff is holomorphic on UU if ff is complex differentiable at each point of UU. Let z0∈ℂz_0 \in \mathbb{C}. A function ff is holomorphic at z0z_0 if ff is holomorphic on some open disc with center z0z_0.

Remarks

Polynomials in zz are holomorphic on ℂ\mathbb{C}, and z↦1/zz \mapsto 1/z is holomorphic on ℂ∖{0}\mathbb{C} \setminus \{0\}. Sums, products, quotients with a nonzero denominator and compositions of holomorphic functions are holomorphic. Each holomorphic function is analytic, see holomorphic functions are analytic. For this reason, many authors use the words holomorphic and analytic for the same functions. A function that is holomorphic on all of ℂ\mathbb{C} is an entire function.

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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 functiondefinition
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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

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205 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Holomorphic_function, Definition
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DefinesHolomorphic function
UsesComplex differentiability

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