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I've come across a problem in my review that has me a little stuck. The question is as follows.

Determine the Laurent Series of the function $$\frac{e^z}{z^2-1}$$ in the domain $|z+1|>2$ centered at $z_0=-1$

Now with this problem, I've tried to rewriting this in numerous different ways, but I can't seem to make much progress on it. I've looked through my book for some key theorem or anything but cannot find one I can apply here.

How would I get started on doing this problem? Any help is much appreciated.

Surb
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myriagon
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1 Answers1

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$$ {e^{z} \over z^2 - 1} = {e^{z} \over (1 + z)(1 - z)}. $$

The Laurent series you want is: $$ {1 \over 1 + z} \; (\mbox{the Taylor series of $e^{z}$ at $z_{0} = -1$}) \; (\mbox{the Taylor series of $1/(1-z)$ at $z_{0} = -1$}) $$

Now, the Taylor series of $e^{z}$ at $z_{0} = -1$ is: $$ e^{z+1 - 1} = {1 \over e} e^{z + 1} = {1 \over e} \sum_{n \geq 0}{1 \over n!} (z + 1)^{n} $$ The Taylor series of $1 / (1 - z)$ at $z_{0} = -1$ is also not difficult to find.

avs
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