Cauchy theorem

eugene · Yang-Mills Theory Offline Joined: 16 May 2004 Posts: 970

Cauchy's Theorem for a disk. Let $f$ be analytic on a closed disk and let $\gamma$ be a path arund the boundary of the disk. Then $\int_{\gamma}f(z)dz=0.$
I must be misunderstanding something, but i'm just interested in question: does the theorem conclusion remains true if there,say,one singularity on the bopundary of the disk. For example does it follows from the theorem that $\int_{|z-1|=1}\frac{1}{z}dz=0$ ???

**Posted:** Sun Apr 17, 2005 9:21 pm

Myth

???
But this integral diverges! Confused

Anyway, Cauchy's theorem requires that the function is continuous in $\overline{D}$ and analytic inside of $D$ , $D$ is an open region with good boundary.

**Posted:** Sun Apr 17, 2005 9:27 pm

_________________
Myth is out of here

jmerry

If there are poles inside the region, the residue theorem applies, allowing you to calculate the value of the integral based on the residues at the poles.
If there is a simple pole on the boundary of the region (at a smooth point), the integral diverges but we can identify a "principal value" by taking a limit of integrals that omit a small length centered at the pole. This principal value is halfway between the value if the pole were inside and the value if the pole were outside.

Don't even ask about other kinds of singularities.

**Posted:** Sun Apr 17, 2005 9:54 pm

eugene · Yang-Mills Theory Offline Joined: 16 May 2004 Posts: 970

There was the reason i'was asking about this questioon regardless it's divegres,because at the calculus lesson we were calculating the integral $\displaystyle\int_{|z-1|=1}\frac{e^{z^2}}{z(z-6)}dz$ and my teacher said that it follows from Cauchy theorem that this integral equal to $0$ regardless it has singularity(simple pole $z=0$ ) on the integration boundary.
It seems to me that if even it were true it would be for example $\displastyle\int_{|z-1|=1}\frac{1}{z}dz=0$ in the same way, nevertheless it diverges.

I think that it is not true and you words:Myth,jmerry confirmed it(as far as i understand)

**Posted:** Sun Apr 17, 2005 11:30 pm

Myth

Are you sure your teacher was right? Confused

Thanks for PM Wink

**Posted:** Sun Apr 17, 2005 11:37 pm

_________________
Myth is out of here

eugene · Yang-Mills Theory Offline Joined: 16 May 2004 Posts: 970

it seems to me that he wasn't right. But like a proverb says the teachers shouldn't be argued with...

**Posted:** Sun Apr 17, 2005 11:48 pm