Finding convergence region and various series expansion of $f(x)=\frac{1-\cos z}{z^2-3z+2}$

Question

Help me find convergence region and various series expansion of $f(x)=\frac{1-\cos z}{z^2-3z+2}$ .

I set $1>|z|$ , $1<|z|<2$, $2<|z|$, but I couldn't divide that.... help..

Moreover to solve $\int_{-\infty}^{\infty} \frac{1-\cos z}{z^2-3z+2}\,dz$ , am I right to evaluate the residue at $z=1$ and at $z=2$?

Answer 1

Hint. Note that for $0<|z|<1$, then $|z/2|<1$, $$\frac{1}{z^2-3z+2}=\frac{1}{2(1-z)(1-z/2)}.$$ For $1<|z|<2$, then $|1/z|<1$ and $|z/2|<1$, $$\frac{1}{z^3-3z+2}=-\frac{1}{2z(1-1/z)(1-z/2)}.$$ For $2<|z|$, then $|1/z|<1$ and $|2/z|<1$, $$\frac{1}{z^3-3z+2}=\frac{1}{z^2(1-1/z)(1-2/z)}.$$