Answer to Question #231334 in Chemical Engineering for pavani

Question #231334

9)Find Laurent series expansion of the function 1/ z²+z-6 in the regions

1<|z-1 <4 and |z-1|> 4.


1
Expert's answer
2021-09-14T01:53:43-0400

From partial fraction

1(z2+z6)=1z1+1z2\frac{1}{(z^2+z-6)}=−\frac{1}{z−1}+\frac{1}{z−2}\\

so on either annulus, it is enough to find the Laurent series expansions of the functions 1z1\frac{1}{z−1} and 1z2\frac{1}{z−2} and then combine them term by term.

If z>2|z|>2  then note that

1z2=1z112z1\frac{1}{z−2}=\frac{1}{z} \frac{1}{1−2z^{−1}}

and the fact that |z|>2

|z|>2 implies that |2z

−1

|<1

|2z−1|<1. If you recall the geometric series formula,


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