Saturday, 9 April 2016

complex analysis - Compute intGammafracefrac1zz1dz, where Gamma is the circle |z1|lefrac32, positively oriented.




Compute Γe1zz1dz, where Γ is the circle |z1|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:





  1. 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?

  2. After I find the Laurent series of e1z, how do I find the Laurent series for the e1zz1?


Answer



The residue in the essential singularity can be computed trough Taylor's expansions:
e1z=1+1z+12z2+,
1z1=1zz2z3,
from which (just multiply the two series):
Res(e1zz1,z=0)=k11k!=1e.



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