Compute ∫Γe1zz−1dz, where Γ is the circle |z−1|≤32, positively oriented.
The numerator is not analytic in Γ so we can't use Cauchy integral formula. I'm thinking maybe I shold use residue theorem. But then I have these two questions:
- Should I look for the Laurent series of e1z around z=0? What if I look for the series around some other points in Γ?
- After I find the Laurent series of e1z, how do I find the Laurent series for the e1zz−1?
Answer
The residue in the essential singularity can be computed trough Taylor's expansions:
e1z=1+1z+12z2+…,
1z−1=−1−z−z2−z3−…,
from which (just multiply the two series):
Res(e1zz−1,z=0)=−∑k≥11k!=1−e.
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