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 Post subject: Contour Integration
PostPosted: Mon, 4 Apr 2011 15:39:51 UTC 
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C is the circle |z|=2, oriented counterclockwise in the complex plane.
\int_{C} \frac{dz}{(z-1)(z+3)^{2}}

What I did: There is a pole at z=1 (order 1) and a pole at z=-3 (order 2). Using the residue theorem, I calculated the residues as 1/16 and -1/16, respectively, so I was expecting the value of the integral to be zero, but that's incorrect. Any ideas what I did wrong?

Edit: I see the problem now. z=-3 is not in C. Now I need to figure out what to do in a case like this...


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 Post subject: Re: Contour Integration
PostPosted: Mon, 4 Apr 2011 15:46:50 UTC 
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StarKid wrote:
C is the circle |z|=2, oriented counterclockwise in the complex plane.
\int_{C} \frac{dz}{(z-1)(z+3)^{2}}

What I did: There is a pole at z=1 (order 1) and a pole at z=-3 (order 2). Using the residue theorem, I calculated the residues as 1/16 and -1/16, respectively, so I was expecting the value of the integral to be zero, but that's incorrect. Any ideas what I did wrong?

Edit: I see the problem now. z=-3 is not in C. Now I need to figure out what to do in a case like this...


Cauchy representation formula...

_________________
\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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 Post subject: Re: Contour Integration
PostPosted: Mon, 4 Apr 2011 15:47:25 UTC 
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StarKid wrote:
C is the circle |z|=2, oriented counterclockwise in the complex plane.
\int_{C} \frac{dz}{(z-1)(z+3)^{2}}

What I did: There is a pole at z=1 (order 1) and a pole at z=-3 (order 2). Using the residue theorem, I calculated the residues as 1/16 and -1/16, respectively, so I was expecting the value of the integral to be zero, but that's incorrect. Any ideas what I did wrong?

Edit: I see the problem now. z=-3 is not in C. Now I need to figure out what to do in a case like this...


You just use the residue theorem as usual. There's no problem. You sum the residues inside the curve, that's the rule.

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