Integral of f(z) dz Around C1 & C2: Complex Math Solutions

[squenshl]

Homework Statement

We know sin(z) has zeros at integral multiples of pi. Let f(z) = z²/sin²(z)
How do I find the integral of f(z) dz around C1 (C1 is the circle |z| = 1 orientated anti-clockwise) and how do I find the integral of f(z) dz around C2 (C2 is the circle |z - pi| = 1 orientated anti-clockwise).

Homework Equations

The Attempt at a Solution

Do I use the Cauchy Integral formula for these integrals.
If not, how would I go about doing these.

 

[lanedance]

how about thinking residues?

 

[squenshl]

I got 1 as my integral.

 

[squenshl]

No. 1 is my value of f(z) (using L'Hopitals rule and the fact that f(z) has a removable singularity at z = 0 so this function is analytic).
My limits of integration are 0 and 2pi so my integral is 2pi.

 

[squenshl]

No again. We use the residue theorem
integral = 2 pi i (sum of the residues)
= 2 pi i (1)
= 2 pi i
and for the next integral I got 4 pi^2 i

 

[Office_Shredder]

If your function is analytic how can it have a residue at 0?