I want to calculate the following integral: $\int_{|z-i| = 10} \left(z+\frac{1}{z}\right)^4$. I have been told to use the Residue Theorem, but I couldn't accomplish a correct calculation. Can anyone help?
2026-03-26 12:51:32.1774529492
Using the Residue Theorem for Complex Integrals
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Since $\left(z+\frac1z\right)^4$ is an even function, if you expand that expression, then the coefficient of $\frac1z$ will be $0$. Therefore, by the residue theorem, your integral is equal to $2\pi i\times0=0$.