Question

(Z+1) (z+5) as a Lauren series in the region 2<|z|<5

Answer 1

It is not indicated around which point the expansion into Lauren series has to be done. We assume that the point is $z_0=\frac{7}{2}=3.5$ . We have: $f(z)=z^2+6z+5=(z-z_0)^2+2zz_0-z_0^2+6z+5=(z-z_0)^2+(2z_0+6)(z-z_0)+2z_0^2+6z_0-z_0^2+5=(z-z_0)^2+(2z_0+6)(z-z_0)+z_0^2+6z_0+5=(z-3.5)^2+13(z-3.5)+12.25+26=(z-3.5)^2+13(z-3.5)+38.25$

The answer is: $f(z)=(z-3.5)^2+13(z-3.5)+38.25$