I need to compute $$I=\int_C \dfrac{e^{\sqrt{1+u}}\cdot\sqrt[4]{1+u}}{\sqrt{u}} \,\mathrm {d}u$$ where $C$ is the unit circle. I am confused about whether I can use the residue theorem to compute it? Or do you have any suggestions? Thank you very much.
2026-03-28 15:18:40.1774711120
Can use residue theorem for this integral
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It's not so easy to use here residue theorem, because any root operation in complex analysis is multivalued. That means that this integral has more than one exact value, and before using the theorem you should choose for each root an exact branch which this root reaches, i.e.: this integral can reach up to $16$ different values.