On Dec 5, 10:04 am, achrzesz <achrz...@wp.pl> wrote:
> On Dec 5, 5:31 am, Dan Drake <dr...@kaist.edu> wrote:
>
>
>
> > I keep wondering whether Sage is making a mistake, or I'm not
> > understanding complex analysis. I'm a little afraid to learn the answer.
> > :)
>
> > Take f(z) = (z-I)*(z-1)^2/(z-(-1/2-I/3)). It's analytic everywhere
> > except at -1/2-I/3, where it has a simple pole. So, if I integrate over
> > a circle centered at 0 of radius, say, 2, the answer should be
>
> > 2*pi*I*(residue of f at -1/2 - I/3),
>
> > which is pi*(181/27 + 19*I/36). However, when I try to do the contour
> > integral, I get:
>
> > sage: integrate(f(2*exp(I*t)) * 2*I*exp(I*t), (t, 0, 2*pi))
> > 0
>
> > even though the contour encloses the pole. It works if I center the
> > circle around the pole:
>
> > sage: integrate(f(-1/2-I/3 + exp(I*t)) * I*exp(I*t), (t, 0, 2*pi))
> > (19/36*I + 181/27)*pi
>
> > and also if I integrate over the square with vertices 1+i, 1-i, -1-i,
> > -1+i. What's wrong with the circle at the origin?
>
> > Note that Mathematica gets this right, although you need to ask for full
> > simplification: with f[z_] = (z-I)*(z-1)^2/(z-(-1/2-I/3)), you get
>
> > In[5]:= Integrate[f[2*Exp[I*t]] * 2*I*Exp[I*t], {t, 0, 2*Pi}]//FullSimplify
>
> > 181 19 I
> > Out[5]= (--- + ----) Pi
> > 27 36
>
> > Any ideas?
>
> > Dan
>
> > --
> > --- Dan Drake
> > ----- http://mathsci.kaist.ac.kr/~drake
> > -------
>
> > signature.asc
> > < 1KViewDownload
>
> sage: maxima('rectform(2*%pi*%i*residue((z-%i)*(z-1)^2/(z-(-1/2-%i/
> 3)),z,-1/2-%i/3))').sage()
> (19/36*I + 181/27)*pi
sage: f(z) = (z-I)*(z-1)^2/(z-(-1/2-I/3))
sage: 2*pi*I*f(z).maxima_methods().residue(z,-1/2 - I/3)
(19/36*I + 181/27)*pi
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