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Answer to a math question 4+4

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I have a complex function I would like to integrate over. I can use two approaches and they should give the same solution. If I want to find the contour integral βˆ«π›Ύπ‘§Β―π‘‘π‘§ for where 𝛾 is the circle |π‘§βˆ’π‘–|=3 oriented counterclockwise I get the following: ∫2πœ‹0𝑖+3π‘’π‘–π‘‘βŽ―βŽ―βŽ―βŽ―βŽ―βŽ―βŽ―βŽ―βŽ―βŽ―βŽ―βŽ―βŽ―βŽ―βŽ―βŽ―βŽ―π‘‘(𝑖+3𝑒𝑖𝑑)=∫2πœ‹03𝑖(βˆ’π‘–+3π‘’βˆ’π‘–π‘‘)𝑒𝑖𝑑𝑑𝑑=18πœ‹π‘– If I directly apply the Residue Theorem, I would get βˆ«π›Ύπ‘§Β―π‘‘π‘§=2πœ‹π‘–Res(𝑓,𝑧=0)=2πœ‹π‘–

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