FactSpewer

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Editing "Complex Logarithm"

Here is a link to a proposed draft of the complex logarithm page: User:FactSpewer/Draft of Complex Logarithm. Update Nov. 7, 2008: I have abandoned the plan to implement this as is, but intend to edit the existing article slowly, incorporating a more informal version of this material.

The draft is mostly based on parts IV.9 to IV.12 of the book Complex Function Theory by Donald Sarason, a well known researcher in the field.

I have kept some items from the old version, but for the most part it has been rewritten extensively.

Comments (and criticisms of the old version)

One of the main goals of the new version is to supply precise definitions with references, and to handle the multivalued nature with care. In particular, equations involving log z are not written unless a branch has already been specified.

The new version includes a precise definition of branch, and also a short discussion of the sense in which the Riemann surface includes the information from all branches.

The diagram with circles and rays was moved to the section discussing conformal mapping.

A branch of the complex logarithm function is not necessarily an extension of the natural logarithm on the positive real numbers; the branch might not even be defined on the positive real numbers!

Using the Taylor series for log(1+z) to define the complex logarithm defines the complex logarithm in only a limited region, and is less useful (and less elementary) than the direct approach of defining the principal value by using polar form. It is strange to begin the article (after the preamble) this way.

Regarding footnote 1 in the old version: it is not true for arbitrary power series that nonconvergence indicates a branch point, so the reasoning here is flawed (even though the conclusion is correct).

In calculating log(-1), the old version uses identities that are known to be false (or at least branch-dependent) for complex numbers.

Mentioning log z as a conformal map is worth doing, I think. So I kept this, but I tried to make it more concise.

Logarithms to the base i and other imaginary bases are unimportant, judging from the treatment they receive in complex analysis textbooks (many books do not even mention them). Devoting a section to this places detracts from more important topics, I feel.

Editing "Exponentiation"

The new version of the exponentiation has already been implemented, so the comments here regarding the draft have been removed.