Can the Residue Theorem be applied to these contour integrals?

[itsnogood]

Hi, first post here. I'm having some trouble with contour integration. Basically here's the question:

Contour Integral of

∫ 1+z dz
(z-1)(z²+9)

There are three cases:

l z l = 2
l z+1 l = 1
l z-\iota l = 3

Is each case a straightforward application of the residue theorem? Also, reading up in my textbook the poles should be contained within the contour so does on the boundary count?

Thanks in advance.

 

[vela]

Hi, first post here. I'm having some trouble with contour integration. Basically here's the question:

Contour Integral of

∫ 1+z dz
(z-1)(z²+9)

You should take a little time and learn LaTeX. It's not very difficult.
$$\int_C \frac{1+z}{(z-1)(z^2+9)}\,dz$$
There's a good tutorial here: https://www.physicsforums.com/showthread.php?p=3977517#post3977517

There are three cases:

l z l = 2
l z+1 l = 1
l z-\iota l = 3

Is each case a straightforward application of the residue theorem?

Yup.

Also, reading up in my textbook the poles should be contained within the contour so does on the boundary count?

No, the pole needs to be inside the contour. If it's on the contour, you have to treat it differently.