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## Homework Statement

Find the residues of the function f(z), and compute the following contour integrals.

a) the anticlockwise circle, centred at z = 0, of radius three, |z| = 3

b) the anticlockwise circle, centred at z = 0, of radius 1/2, |z| = 1/2

f(z) = 1/((z

^{2}+ 4)(z + 1))

∫

_{C}dz f(z)

## Homework Equations

∫

_{C}dz f(z) = 2πi ∑ Res(f, z

_{i}) (z

_{i}inside the contour C)

## The Attempt at a Solution

I have found the poles and hence the residues:

z = 2i, -2i, -1

and the corresponding residues (respectively): -1/10 - i/20, -1/10 + i/20, 1/5

I'm fairly comfortable with these, however when computing the contour integrals using Cauchy's theorem I got that in a), all the poles lie in the contour and the sum of the residues = 0, so the contour integral = 0. But then with b), I found that because none of the poles lie in this circle of radius 1/2 (on the complex plane) then the contour integral also equals zero?

I think I must have done something wrong as I wouldn't expected an assignment to have 2 zero answers!

Thanks for any help.