Discussion
The value of the integral
\[ \frac{1}{2\pi i}\int_{|z| = 1} z^{3}\cos(\tfrac{1}{z})\,dz \]
is :
Answer: D
The value is 1/24.
The integral divided by \(2\pi i\) is the residue at 0, the coefficient of \(1/z\) in the Laurent series:
\[ \begin{aligned} z^{3}\cos\tfrac{1}{z} &= z^{3}\left(1 - \tfrac{1}{2z^{2}} + \tfrac{1}{24z^{4}} - \cdots\right) \\ &= z^{3} - \tfrac{z}{2} + \tfrac{1}{24z} - \cdots \end{aligned} \]
The coefficient of \(1/z\) is 1/24. The value 0 would hold if the function were analytic inside the circle, but it has an essential singularity at 0. -1/6 is the coefficient that \(\sin(1/z)\) would give.
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