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Re: [newbie] unexpected behaviour for x^x

From: Julien Bect
Subject: Re: [newbie] unexpected behaviour for x^x
Date: Fri, 12 Dec 2014 18:22:05 +0100
User-agent: Mozilla/5.0 (X11; Linux i686; rv:31.0) Gecko/20100101 Thunderbird/31.3.0

Le 12/12/2014 17:56, Jean Dubois a écrit :
However for x real: lim_{x-->0-} x^x is non-existing, even though numerically calculating lim_{x-->0-} x^x might suggest you get a complex number

What do you mean by "non-existing" ? "might suggest" ?

The logarithm of a complex number is perfectly well-defined, and it *is* a complex number.

Actually, the complex log is a multi-valued function, so the "well-defined" log I'm talking about is the principal value; see, e.g.,

To sum up:

1) x^x = exp (x * log (x)) is a perfectly well defined complex number, even for negative x, as soon as a branch of the complex log has been singled out

2) Octave computes the principal value of the log, i.e., log(z) is the only logarithm of z that has its imaginary part in (pi; pi].

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