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Hennie
 3 years ago
Compute the Laurent expansion of z/((z1)(2z)) for each of the following regions: abs(z) <1 and abs(z2)<1.
Hennie
 3 years ago
Compute the Laurent expansion of z/((z1)(2z)) for each of the following regions: abs(z) <1 and abs(z2)<1.

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Jemurray3
 3 years ago
Best ResponseYou've already chosen the best response.1Partial fraction decomposition would probably be a good start.

Hennie
 3 years ago
Best ResponseYou've already chosen the best response.0In the expression (A/(z1)) and B/(z2) is the value for A=1 and B=2 ?

Hennie
 3 years ago
Best ResponseYou've already chosen the best response.0For the laurent's expansion should I then use (1/(1z)) and ((1/(1z/2)) and expand each term seperately ?

Jemurray3
 3 years ago
Best ResponseYou've already chosen the best response.1It should be \[ \frac{1}{z1}  \frac{2}{z  2} \] so yes you're right... and now yes, you want to expand these two terms around those two points. For example, for z<1, we can rewrite that as \[ \frac{1}{1  z} + \frac{1}{1  z/2} = \sum_{n=0}^\infty z^n+ \sum_{k=0}^\infty (\frac{z}{2})^k \] you can simplify that if you'd like, bringing them both under the same summation sign and all that stuff, but that's the idea. Since z<1, these series converge. For z2<1, they don't necessarily converge, so you have to expand them about the point z = 2 for the second part, but the same deal applies.
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