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Homework Statement
Calculate the following integral along three different circular contours,
[tex]\int_{C_j}\frac{dz}{z(3z-1)^2(z+2)}[/tex]
where
[tex]C_1:0<r_1<1/3[/tex]
[tex]C_2:1/3<r_2<2[/tex]
[tex]C_3: r_3>2[/tex]
The Attempt at a Solution
The function has singularities at z=0, z=1/3 and z=-2. Thus all three contours enclose singularities and Cauchy's integral theorem doesn't hold (none of the integrals are immediately zero).
Along each circular contour,
[tex]z=re^{i \theta}\implies dz=ire^{i \theta}d \theta[/tex]
Am I going to need to use partial fractions for this? What is the best way to get started?
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