Math 2.0

Functions of a complex variable Proposition

Power series of the complex logarithm

Builds on Principal branchPower series in the complex planeDisc of convergence

Statement

Let Log⁡\operatorname{Log} be the principal branch of the logarithm. For each z∈ℂz \in \mathbb{C} with ∣z∣<1|z| < 1, the series ∑n≥1zn/n\sum_{n \ge 1} z^n / n converges absolutely, and −Log⁡(1−z)=∑n=1∞znn=z+z22+z33+⋯ .-\operatorname{Log}(1 - z) = \sum_{n=1}^{\infty} \frac{z^n}{n} = z + \frac{z^2}{2} + \frac{z^3}{3} + \cdots.

Remarks

Replacing zz by −z-z gives Log⁡(1+z)=∑n≥1(−1)n+1zn/n\operatorname{Log}(1 + z) = \sum_{n \ge 1} (-1)^{n+1} z^n / n for ∣z∣<1|z| < 1. The radius of convergence is 11. For ∣z∣<1|z| < 1, the number 1−z1 - z has positive real part, so it is not on the branch cut. Proof idea: both sides vanish at 00, and both have the derivative 1/(1−z)=∑n≥0zn1/(1 - z) = \sum_{n \ge 0} z^n on the unit disc. At z=−1z = -1 the series still converges, and gives ln⁡2=1−12+13−⋯\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \cdots.

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To understand this page from the ground up, read these 18 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 exponentialdefinitionPower series in the complex plane, definition, no evidence levelPower series in the c…definitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionDisc of convergence, definition, no evidence levelDisc of convergencedefinitionCompleteness 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 logarithmdefinitionPrincipal value of the argument, statement, no evidence levelPrincipal value of th…statementPrincipal branch, definition, no evidence levelPrincipal branchdefinitionPower series of the complex logarithm, statement, no evidence levelPower series of the c…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.
  • no evidence level
  1. Complex numbers object
  2. Multivalued function concept
  3. Real numbers object
  4. Set bounded above definition
  5. Complex exponential definition
  6. Power series in the complex plane definition
  7. Real and imaginary parts definition
  8. Supremum definition
  9. Disc of convergence definition
  10. Completeness axiom statement
  11. Existence of square roots of nonnegative reals statement
  12. Square root definition
  13. Modulus of a complex number definition
  14. Argument of a complex number definition
  15. Complex logarithm definition
  16. Principal value of the argument statement
  17. Principal branch definition
  18. Power series of the complex logarithm statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Mercator_series, Complex series
About this page
Tag09BG0Y7SZD
URIm2://09BG0Y7SZD
Aliasesmercator-series, newton-mercator-series
Kindstatement, proposition
WritingW0 stub
MSC30B10
StatusThis copy of the site has no build data. Run m2 status 09BG0Y7SZD.
UsesPrincipal branch, Power series in the complex plane, Disc of convergence

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