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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.,
`
https://en.wikipedia.org/wiki/Complex_logarithm
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].
`