Re: [newbie] unexpected behaviour for x^x

[Jean Dubois]

2014-12-12 18:22 GMT+01:00 Julien Bect <address@hidden>:
> 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" ?
I added for x real, maybe I should have written "in R: lim_{x-->0-}
x^x is non-existing"

regards,
jean


>
> 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].
>
>
>
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