Find the residues of the following function + Cauchy Residue

  • Thread starter Poirot
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  • #1
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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/((z2 + 4)(z + 1))
Cdz f(z)

Homework Equations


Cdz f(z) = 2πi ∑ Res(f, zi) (zi 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.
 

Answers and Replies

  • #2
Samy_A
Science Advisor
Homework Helper
1,241
510

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/((z2 + 4)(z + 1))
Cdz f(z)

Homework Equations


Cdz f(z) = 2πi ∑ Res(f, zi) (zi 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.
I for one don't see what's wrong.
Second opinion would be appreciated, though.
 
  • #3
58
5
I for one don't see what's wrong.
Second opinion would be appreciated, though.
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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