Math 2.0

Functions of a complex variable Proposition

Holomorphic functions with nonzero derivative are conformal

Builds on Holomorphic functionComplex differentiabilityConformal map

Statement

Let U⊆ℂU \subseteq \mathbb{C} be an open set, let f:U→ℂf : U \to \mathbb{C} be holomorphic, and let z0∈Uz_0 \in U. If f′(z0)≠0f'(z_0) \neq 0, then ff is conformal at z0z_0. In particular, if f′(z)≠0f'(z) \neq 0 for each z∈Uz \in U, then ff is conformal on UU.

Remarks

Write f′(z0)=reiφf'(z_0) = r e^{i\varphi} with r>0r > 0; then the real derivative of ff at z0z_0 is h↦reiφhh \mapsto r e^{i\varphi} h, a rotation by φ\varphi followed by a scaling by rr. The converse also holds: a map that is conformal at each point of an open set UU is holomorphic on UU with a nonzero derivative. The condition f′(z0)≠0f'(z_0) \neq 0 is necessary: z↦z2z \mapsto z^2 doubles the angles between curves through 00. The exponential function is conformal at each point but not injective, so a conformal map need not be injective.

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Complex numbers, object, no evidence levelComplex numbersobjectComplex plane, concept, no evidence levelComplex planeconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionConformal map, definition, no evidence levelConformal mapdefinitionSupremum, 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 functiondefinitionHolomorphic functions with nonzero derivative are conformal, statement, no evidence levelHolomorphic functions…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. Complex plane concept
  3. Real numbers object
  4. Set bounded above definition
  5. Real and imaginary parts definition
  6. Conformal map definition
  7. Supremum definition
  8. Completeness axiom statement
  9. Existence of square roots of nonnegative reals statement
  10. Square root definition
  11. Modulus of a complex number definition
  12. Complex differentiability definition
  13. Holomorphic function definition
  14. Holomorphic functions with nonzero derivative are conformal statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Conformal_map, In two dimensions
About this page
Tag0Q7M6K6VD7
URIm2://0Q7M6K6VD7
Aliasesholomorphic-maps-are-conformal
Kindstatement, proposition
WritingW0 stub
MSC30C35
StatusThis copy of the site has no build data. Run m2 status 0Q7M6K6VD7.
UsesHolomorphic function, Complex differentiability, Conformal map

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