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Find the residues of the following function + Cauchy Residue

  1. Apr 28, 2016 #1
    1. The problem statement, all variables and given/known data
    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/((z2 + 4)(z + 1))
    Cdz f(z)

    2. Relevant equations
    Cdz f(z) = 2πi ∑ Res(f, zi) (zi inside the contour C)

    3. 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.
     
  2. jcsd
  3. Apr 28, 2016 #2

    Samy_A

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    I for one don't see what's wrong.
    Second opinion would be appreciated, though.
     
  4. Apr 28, 2016 #3
    Yepp! I calculated the residues for a) and also got zero and for b) we indeed have that all poles are lying outside the contour. So it should be right, shouldn't it?
     
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