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In James Brown's complex variables and applications, there is an exercise:

Let $f \left( z \right) = \frac{8 a^3 z^2}{\left( z^2 + a^2 \right)^3}$ with $a > 0$. Show that the principal part of $f \left( z \right)$ at $z = ai$ is

$- \frac{i / 2}{z - ai} - \frac{a / 2}{\left( z - ai \right)^2} - \frac{a^2 i}{\left( z - ai \right)^3}$

but the definition of principal part in many books is that: negative powers of $(z-z_0)$ in Laurent series.
We observe that $f(z)$ has two Laurent series( converge in $|z-ai|<2a$ and $|z-ai|>2a$), so the principal part of f(z) should also have two solutions. but this exercise just give the principal part of Laurent series converge in $|z-ai|<2a$, What wrong??!

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  • $\begingroup$ Which series converges in $|z-ai|>2a$ and which in $|z-ai|<2a$? $\endgroup$ – tomcuchta Nov 30 '12 at 7:16
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The principal part is the sum of the negative powers in the Laurent series that converges on $0 < |z-p| < r$, where $p$ is the singularity in question (and $r$ is the distance to the closest other singularity).

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