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How do I expand a function $f(z)$ in a particular region? For example, how would I expand $f(z)=(z^2-3z+2)^{-1}$ in the region $0<|z-1|<1.$? I believe this can be done by the binomial theorem. But how to arrange things to make the binomial expansion valid?

Please give me an idea for doing this.

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  • $\begingroup$ Is this a Mathematica question or a Math question? $\endgroup$ Commented Mar 21, 2014 at 13:31

1 Answer 1

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In Mathematica:

Series[1/(z^2 - 3 z + 2), {z, 1, 10}, Assumptions -> {0 < Abs[z - 1 ]< 1}]

(or whatever the top-order term is that you want).

Mathematical method:

Step (i): expand the function into partial fractions -- in Mathematica:

Apart[1/(z^2 - 3 z + 2)]

One of the fractions you get is the the Laurent series term with negative power of $z - 1$.

Step (ii): Expand the other term using a binomial series.

Step (iii): Add the results of Steps (i) and (ii).

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  • $\begingroup$ okay thanks.What to do if the region is |z|<1 ? $\endgroup$ Commented Mar 21, 2014 at 17:27
  • $\begingroup$ For $|z| < 1$, you'll need to expand around $z = 0$, not $z = 1$. And from the code I provided, I think you can see how to express that in Mathematica. $\endgroup$
    – murray
    Commented Mar 21, 2014 at 19:38
  • $\begingroup$ okay thanks.Can u explain the general case? $\endgroup$ Commented Mar 22, 2014 at 3:36

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