# Find the residues of the following function + Cauchy Residue

• Poirot
In summary, in this conversation, the task at hand was to find the residues of a given function and then compute contour integrals for two different circles. The residues were found to be -1/10 - i/20, -1/10 + i/20, and 1/5 for poles at z = 2i, -2i, and -1, respectively. Using Cauchy's theorem, it was determined that the contour integral for the first circle (|z| = 3) was 0, as all poles lie within the contour. Similarly, for the second circle (|z| = 1/2), the poles lie outside the contour, resulting in a contour integral of 0 as

## 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.

Poirot said:

## 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.

Samy_A said:
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?