- #1
Jamin2112
- 986
- 12
Homework Statement
∫C1(0) dz / (z * sin2(z))
Homework Equations
Residue Theorem material
The Attempt at a Solution
z * sin2(z)
= z * (1/2 - cos(2z)/2)
= z * [1/2 - (1/2)∑(-1)n(2z)n/(2n)! ]
= z3 + ...
---> z * sin2(z) has a zero of order 3 at z = 0
---> 1/(z * sin2(z)) has a pole of order 3 at z = 0
The circle C1(0) encloses z = 0 (its only singularity), so we have
2πi * Res[f, 0] = 2πi * 1/(3-1)! * limz-->0 d2/dz2 (z-0)3 1/(z * sin2(z)),
which I don't want to calculate if there's an easier way to do the problem. Is there an easier way? I'm too cool for calculating double derivatives.