# Re: [isabelle] ln on negative numbers

For real r>0, the complex log (Ln) satisfies
Ln(-r) = ln(r) + pi*I
So I donât like extending the domain of ln using 0, as opposed to ln(|r|). This would extend the validity of
0 < ?z â Ln (complex_of_real ?z) = complex_of_real (ln ?z)
Thereâs still the question of ln(0). Note that Ln(z) is defined for all complex numbers except when z=0.
Larry Paulson
> On 29 Apr 2015, at 15:02, Manuel Eberl <eberlm at in.tum.de> wrote:
>
> Hallo,
>
> I am currently in the process of writing some automation for Landau symbols. One of the problems that I currently have is that I would like to rewrite things like
>
> O(Îx. (c*x) powr p)
>
> to
>
> O(Îx. c powr p * x powr p)
>
> However, for c < 0 this is simply not true (well, morally not true). If c < 0, and w.l.o.g. x > 0, we have
>
> c powr p = (c*x) powr p = exp (p * THE u. False)
>
>
> I therefore suggest to define the real logarithm as "ln x = (if x â 0 then 0 else THE u. exp u = x)". The practical implications of this should be small, and it would allow stating a lot of lemmas involving "powr" in a much simpler form and allow transformations such as the one above.
>
> Cheers,
>
> Manuel
>

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