Polar Coordinates to evaluate integrals

Fixxxer125
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Homework Statement


Use Polar coordinates to evaluate
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were C denotes the unit circle about a fixed point Z0 in the complex plane



The Attempt at a Solution


I've only used polar integrals to convert an integral in sin and cos into one in therms of z, find the residues and then use the residue theorum to evaluate the integral so I am not really sure where to go with this question? Any help would be greatly appreciated!
 
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Let z= z_0+ e^{i\theta}.
 
Have I done it correctly if I end up with a final answer of
2∏i(aZ02 + bZ0 + c)
Thanks!
 
Yes, in fact there is the "Cauchy integral formula" that says
\oint \frac{f(z)}{z-z_0} dz= 2\pi if(z_0)

Perhaps this problem was intended as an introduction to that.
 
Ah yes we have done that previously, I think I just need to do practise questions to bring all the theory together. Cheers
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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