hi Richard
In mathematics what you said,it should be -i*pi,is not so correct.
Raymond said
>Anyway, my reasoning is this:
>
>log(-1-%i*x) = 1/2*log(x^2+1) + %i*arg(-1-%i*x)
>
>As x -> 0 from above, the realpart has limit 0.
so far correct,I think.
But
> For the imaginary part
>arg(-1-%i*x) is -pi + eps for small x > 0,
> so the limit is -pi.
This is not so correct.For exsample,if we take principal value domain,arg is from 0 to 2pi,so arg(-1-%i*x)is pi + eps for
small x>0. (the branch cut is positive real axis).
in the sense of uniformization,
if in domain,arg is from -2pi to 0,arg(-1-%i*x) is -pi +eps
(B.
Thus Both -%i*%piand %i*%pi are correct answers in mathematics.
we shoud go with mathematics,I think.
gosei furuya
>limit(atan(x)/(1/exp(1)-exp(-(1+x)^2)),x,inf,plus) - wrong
>
> I think it should be -i*pi, and so does Mathematica.
> Commercial Macsyma thinks i*pi.
> Numerical evaluation as x->0 from real positive values confirms
> -i*pi is consistent with Maxima's numerics.
> RJF
>
>
> Raymond Toy wrote:
>
> >>>>>>"gosei" == go furuya writes:
> >>>>>>
> >>>>>>
> >
> > gosei> Hi Raimond
> > >> I've looked a bit more at this. A simpler example is
> > >>
> > >> limit(log(-1-%i*x),x,0,plus);
> > >>
> > >> Maxima returns log(-1) = %i*%pi. However, the correct answer is
> > >> -%i*%pi because we're always in the third quadrant.
> >
> > gosei> You may confused calculating limit in 2D with 1D.
> > gosei> In case of uniformization,both %i*%pi and -%i*%pi shoud be correct.
> > gosei> All that is required is to select domain of uniformization
> > gosei> ,such that principal value.
> >
> >What does uniformization mean? And why are they both correct? What
> >do you mean limit in 2D vs 1D?
> >
> >Perhaps I'm wrong in assuming log means principal log so the branch
> >cut is the negative real axis.
> >
> >Anyway, my reasoning is this:
> >
> >log(-1-%i*x) = 1/2*log(x^2+1) + %i*arg(-1-%i*x)
> >
> >As x -> 0 from above, the realpart has limit 0. For the imaginary part
> >arg(-1-%i*x) is -pi + eps for small x > 0, so the limit is -pi.
> >
> >Thus the correct answer should be -%i*%pi.
> >
> >Ray
> >
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>
>