Math 2.0

Functions of a complex variable Definition

Branch cut

Builds on Multivalued functionBranch of the logarithm

Definition

Let U⊆ℂU \subseteq \mathbb{C} be an open set and let FF be a multivalued function on UU. A branch cut for FF is a closed subset KK of UU such that FF has a continuous single-valued branch on the open set U∖KU \setminus K. Usually a branch cut is a curve or a union of curves. A branch on U∖KU \setminus K is a continuous function ff on U∖KU \setminus K with f(z)∈F(z)f(z) \in F(z) for each z∈U∖Kz \in U \setminus K. The standard branch cut for the logarithm on ℂ∖{0}\mathbb{C} \setminus \{0\} is the ray (−∞,0)(-\infty, 0). The open set ℂ∖(−∞,0]\mathbb{C} \setminus (-\infty, 0] is the slit plane.

Remarks

A branch cut usually joins branch points. A branch point of FF is a point z0z_0 with the following property. When zz goes once around a small circle with center z0z_0, the values of FF do not come back to their start. For log⁡z\log z and z\sqrt z, the branch points are 00 and ∞\infty, so a cut is a curve from 00 to ∞\infty. The principal branches of the logarithm and of the square root use the slit plane. Each other ray from 00 is also a valid branch cut and gives other branches. The slit plane is {z∈ℂ:Re⁡z>0 or Im⁡z≠0}\{z \in \mathbb{C} : \operatorname{Re} z > 0 \text{ or } \operatorname{Im} z \neq 0\}.

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Complex numbers, object, no evidence levelComplex numbersobjectMultivalued function, concept, no evidence levelMultivalued functionconceptReal 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 …definitionArgument of a complex number, definition, no evidence levelArgument of a complex…definitionComplex logarithm, definition, no evidence levelComplex logarithmdefinitionBranch of the logarithm, definition, no evidence levelBranch of the logarit…definitionBranch cut, definition, no evidence levelBranch cutdefinition
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  1. Complex numbers object
  2. Multivalued function concept
  3. Real numbers object
  4. Set bounded above definition
  5. Complex exponential definition
  6. Real and imaginary parts 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. Argument of a complex number definition
  13. Complex logarithm definition
  14. Branch of the logarithm definition
  15. Branch cut definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Branch_point, Branch cuts
  2. Web https://en.wikipedia.org/wiki/Complex_logarithm, Branch cuts
About this page
Tag06JYC06JHY
URIm2://06JYC06JHY
Aliasesslit-plane
Kinddefinition
WritingW1 stated
MSC30A99
DefinesBranch cut
UsesMultivalued function, Branch of the logarithm

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