Use the residues theorem to calulate int_(|z-i|=3)(e^(z^2)-1)/(z^3-iz^2)dz

snaketao0g 2022-10-23 Answered
Use the residues theorem to calulate
| z i | = 3 e z 2 1 z 3 i z 2   d z
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Answers (1)

occuffick24
Answered 2022-10-24 Author has 13 answers
No, you miss the residue at z=0 since there are two poles z=0, z=i inside |z-i|=3. The correct answer is
| z i | = 3 e z 2 1 z 3 i z 2 d z = 2 π i ( Res ( f , i ) + Res ( f , 0 ) = 2 π i ( e z 2 1 z 2 | z = i + d d z e z 2 1 z i | z = 0 ) = 2 π i ( e 1 e + 1 ) = 2 π ( 2 e 1 ) i e .
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