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QUESTION

Find the principal part of the Laurent series of the function
$$\frac{e^{zt}}{z^2\left(z^2+4z+5\right)}$$ about the point $z=0$.

Show that this Laurent series converges for $0<|z|<\sqrt{5}$.

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ANSWER

Factorise: $z^2+4z+5=0\Rightarrow \left(z+2\right)^2=4-5
\Rightarrow z=-2 \pm i=z_{\pm}$ $$
\frac{e^{zt}}{z^2\left(z-z_+\right)\left(z-z_-\right)}
=z^{-2}e^{zt}\cdot\frac{1}{z+z_-}\cdot\frac{1}{1-\frac{z}{z_+}}\cdot\frac{1}{1-\frac{z}{z_-}}$$\\
We only need to expand to $O\left(z^{-1}\right)$, so
\begin{eqnarray*}
&=&z^{-2}\left(1+zt+\ldots\right)\frac{1}{z_+z_-}\left(1+\frac{z}{z_+}
+\ldots\right)\left(1+\frac{z}{z_-} +\ldots\right)\\
&=&\frac{1}{z_+z_-}z^{-2}\left[1+\left(t+\frac{1}{z_+}+\frac{1}{z_-}\right)z+\ldots\right]\\
&=&\frac{1}{z_+z_-}+\left(\frac{t}{z_+z_-}+\frac{z_++z_-}{\left(z_+z_-\right)^2}\right)z^{-1}+\ldots\\
&=&\frac{1}{3}z^{-2}+\left(\frac{t}{3}-\frac{4}{9}\right)z^{-1}+\ldots
\end{eqnarray*}

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