Talk:Complex logarithm

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[edit] History

I don't have a reference in front of me, but at some point it would be nice to include a History section in this article describing the original debate over logarithms of negative numbers. Before Euler figured it out, some intelligent men made some amusing mistakes, like forgetting about integration constants. Melchoir 21:33, 30 July 2006 (UTC)

Women too... —Preceding unsigned comment added by 132.206.33.67 (talk) 21:55, 3 October 2007 (UTC)

[edit] Cut and paste

I dug back through the history on this page, and discovered that somebody named TAB did a cut and paste job from the natural logarithm page to get this thing started. Here's the permanent link if you're interested.

Anyway, I guess that accounts for the rather abrupt introduction as the article stands right now. I'm aiming to regularize the language a little bit, and maybe split it up into sections, so it will look more like a regular Wikipedia article and less like a cut-and-paste job from another article. DavidCBryant 20:39, 16 January 2007 (UTC)

I've thought about it a little more, and have formed a game plan.

  • Explain the principal branch as the analytic continuation of ln(z) into the complex plane.
  • Explain the principal branch in terms of the exponential function.
  • Explain the multi-valued logarithm, and tie it in with the corkscrew Riemann surface.
  • Clean up the expressions for log(z) in terms of x + iy.
  • Add a history section, if I can get that figured out well enough. I like Melchoir's suggestion.
  • Add references and links as needed.

I'm probably talking to myself here, but this outline may eventually prove useful to somebody else. DavidCBryant 21:14, 16 January 2007 (UTC)

The first cut of this article is now completed. Thanks are due CMummert for his eagle eye, and for the lovely image of the "corkscrew" Riemann surface. After some reflection, I decided to omit the conversion formulae from Cartesian co-ordinates, since these are covered adequately in many articles.

I now intend to add one more section, the complex logarithm as a conformal mapping. I think this will round out the article nicely (except for the history section, which would also be nice – facts are needed to do a good job on that). DavidCBryant 13:46, 22 January 2007 (UTC)

[edit] Is log (0) defined?

It states that log (0) is not defined, but by just looking at the two graphs illustrating the real and imaginary parts of log and extrapolating the real part, would it not be fair to say that log (0) is equal to negative infinity real part and zero imaginary part? Its just a thought, maybe that means that it isnt defined?

Also if you think of this: xy = 0; where y = - infinity; 1/(x^infinity) = 0; therefore log (0) inverably = - infinity; —Preceding unsigned comment added by 81.129.146.7 (talk) 22:34, 16 February 2008 (UTC)

Can "negative infinity real part and zero imaginary part" really be taken as the definition of the value of a function? I don't think so. But even if it can, it's not the answer. Consider approaching zero from below; the answer would then be negative infinity real part and pi imaginary part. One thing you can be sure of is that 1/log(x) has a limit at 0 for x approaching zero. Dicklyon (talk) 05:16, 17 February 2008 (UTC)
On that note, is the principal branch of the log function defined for -1 (or any other with arg = pi) ? —Preceding unsigned comment added by 200.181.89.3 (talk) 00:59, 24 May 2008 (UTC)