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Let $f(z)=\frac{\sin z}{z^2}- \frac{\cos z}{z}$ then

  1. $f$ has a pole of order $2$ at $0$,

  2. $f$ has a simple pole at $0$,

  3. $\int_{|z|=1}f(z)=0$ anticlockwise,

  4. residue of $f$ at $0$ is $-2\pi i$,

I am not able to figure out which of the statement is correct or false. I think only 4 is correct by residue theorem.

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up vote 3 down vote accepted

It all becomes clear once you obtain the Laurent expansion $$\frac{\sin z}{z^2}=\frac{1}{z^2} \left(z-\frac{z^3}{3!}+O\left(z^5\right)\right)=\frac{1}{z}-\frac{z}{6}+O\left(z^3\right)$$ $$\frac{\cos z}{z}=\frac{1}{z} \left(1-\frac{z^2}{2!}+O\left(z^4\right)\right)=\frac{1}{z}-\frac{z}{2}+O\left(z^3\right)$$ $$f(z)=\frac{z}{3}+O\left(z^3\right)$$ Hence $f(z)$ is analytic and (3) is true.

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