Laurent series (COMPLICATED, )

[libelec]

Laurent series (COMPLICATED, URGENT)

Homework Statement

Find the Laurent series for $$f(z) = \frac{{2{z^2}}}{{2{z^2} - 5z + 2}} + \sin (\frac{3}{{{z^2}}})$$ around z = 0, in which the series absolutely converges for z = -i.

The Attempt at a Solution

I have several problem with this. I tell you what I tried.

First: I separate the polinomial fraction in partial fractions. I don't know if the procedure is OK: $$\frac{{2{z^2}}}{{2{z^2} - 5z + 2}} = 2{z^2}\left( {\frac{2}{{3(z - 2)}} - \frac{2}{{3(z - 1/2)}}} \right)$$. I leave the 2z² there because it's its own Laurent Series around z=0, for all z.

Second: I compare the function with the geometrical function (considering that I have to find the Laurent series in a vecinity of infinite for the fraction with the (z - 1/2)) and I use a variable change for the sine:

a) $$\frac{2}{{3(z - \frac{1}{2})}} \to \frac{2}{{3z(1 - \frac{1}{{2z}})}}$$

$$\frac{2}{{3z(1 - \frac{1}{{2z}})}} = \frac{2}{{3z}}\sum\limits_{n = 0}^\infty {\frac{1}{{{2^n}{z^n}}}} {\rm{ }}\forall z/\left| z \right| > \frac{1}{2}$$


b)
$$\frac{{ -1}}{{3(z - \frac{1}{2})}} = \frac{2}{{3(\frac{1}{2} - z)}} = \frac{4}{{3(1 - 2z)}}$$

c) Since $$\sin (u) = \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}{u^{2n + 1}}}}{{(2n + 1)!}}}$$, then, using the change of variable u = 3/z², I get that:

$$\sin (\frac{3}{{{z^2}}}) = \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}{3^{2n + 1}}}}{{(2n + 1)!{z^{4n + 2}}}}} {\rm{ }}\forall z/\left| z \right| > 0$$


Then, I have that:

$$f(z) = 2{z^2}\left( {\frac{2}{{3z}}\sum\limits_{n = 0}^\infty {\frac{1}{{{2^n}{z^n}}}} - \frac{1}{3}\sum\limits_{n = 0}^\infty {\frac{{{z^n}}}{{{2^n}}}} } \right) + \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}{3^{2n + 1}}}}{{(2n + 1)!{z^{4n + 2}}}}} {\rm{, }}\forall z/\frac{1}{2}< \left| z \right|< 2$$

My question is, how can I come up with a sole expresion for the coefficients? I couldn't even find one for the series inside the parenthesis...

Thanks

 

[libelec]

I don't know why, but LaTeX doesn't let me put the expressions in order...