Math 2.0

Special functions Proposition

The principal logarithm is holomorphic off the branch cut

Builds on Principal branchBranch cutHolomorphic functionComplex exponential

Statement

Let S=ℂ∖(−∞,0]S = \mathbb{C} \setminus (-\infty, 0] be the slit plane, the complement of the standard branch cut. Let Log⁡\operatorname{Log} be the principal branch of the logarithm. Then Log⁡\operatorname{Log} is holomorphic on SS, and Log⁡′(z)=1z\operatorname{Log}'(z) = \frac{1}{z} for each z∈Sz \in S.

Remarks

Proof idea: on SS, Log⁡\operatorname{Log} is continuous and it is the inverse of the complex exponential on the strip {w:−π<Im⁡w<π}\{w : -\pi < \operatorname{Im} w < \pi\}. The inverse function rule gives Log⁡′(z)=1/eLog⁡z=1/z\operatorname{Log}'(z) = 1 / e^{\operatorname{Log} z} = 1/z. Log⁡\operatorname{Log} is not continuous at the points of (−∞,0)(-\infty, 0). For x<0x < 0, its values tend to ln⁡∣x∣+iπ\ln|x| + i\pi from above the axis and to ln⁡∣x∣−iπ\ln|x| - i\pi from below.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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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 differentiability, definition, no evidence levelComplex differentiabi…definitionComplex logarithm, definition, no evidence levelComplex logarithmdefinitionPrincipal value of the argument, statement, no evidence levelPrincipal value of th…statementHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionBranch of the logarithm, definition, no evidence levelBranch of the logarit…definitionPrincipal branch, definition, no evidence levelPrincipal branchdefinitionBranch cut, definition, no evidence levelBranch cutdefinitionThe principal logarithm is holomorphic off the branch cut, statement, no evidence levelThe principal logarit…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. 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 differentiability definition
  14. Complex logarithm definition
  15. Principal value of the argument statement
  16. Holomorphic function definition
  17. Branch of the logarithm definition
  18. Principal branch definition
  19. Branch cut definition
  20. The principal logarithm is holomorphic off the branch cut statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Complex_logarithm, The derivative of the complex logarithm
  2. Web https://en.wikipedia.org/wiki/Complex_logarithm, The principal value as an inverse function
About this page
Tag0XVGJE7Q04
URIm2://0XVGJE7Q04
Aliasesderivative-of-the-logarithm
Kindstatement, proposition
WritingW0 stub
MSC33B10
StatusThis copy of the site has no build data. Run m2 status 0XVGJE7Q04.
UsesPrincipal branch, Branch cut, Holomorphic function, Complex exponential

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