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The principal logarithm fails to turn multiplication into addition at $(- 1, - 1)$

Example

For the pointwise principal logarithm,

$Log ((- 1) (- 1)) \neq Log (- 1) + Log (- 1) .$

Indeed,

$Log (1) = 0, Log (- 1) = i π,$

$Log ((- 1) (- 1)) = 0 but Log (- 1) + Log (- 1) = 2 π i .$

This is exactly the branch-cut warning from Dictionary for holomorphic logarithm branches, the principal logarithm, and principal powers: principal logarithms do not satisfy a global product-to-sum law across the negative axis.

Facts & Assumptions

Given: The principal logarithm conventions of Dictionary for holomorphic logarithm branches, the principal logarithm, and principal powers.

[F1]

The pointwise principal logarithm is defined by $Log z = \log r + i θ, z = r (\cos θ + i \sin θ), - π < θ \leq π,$ and on the negative real axis one has $Log (- 1) = i π$ (Dictionary for holomorphic logarithm branches, the principal logarithm, and principal powers).

Verification

1.1F1given

Applying [F1] to $1 = 1 (\cos 0 + i \sin 0)$ gives $Log (1) = 0$ .

2.1F1step 1.1algebra∎

Applying [F1] to $- 1 = 1 (\cos π + i \sin π)$ gives $Log (- 1) = i π$ , hence $Log (- 1) + Log (- 1) = 2 π i \neq 0 = Log (1) = Log ((- 1) (- 1))$ .

Depends on

Dictionary for holomorphic logarithm branches, the principal logarithm, and principal powers

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 2 §3.4 The Logarithm (standard reference, not scraped)

The principal logarithm fails to turn multiplication into addition at (−1,−1)(−1,−1)