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Dealing with Repeated Poles Analytically

A pole of multiplicity $ m_i$ has $ m_i$ residues associated with it. For example,

$\displaystyle H(z)$ $\displaystyle \isdef$ $\displaystyle \frac{7 - 5z^{-1}+ z^{-2}}{\left(1-\frac{1}{2}z^{-1}\right)^3}$  
  $\displaystyle =$ $\displaystyle \frac{1}{\left(1-\frac{1}{2}z^{-1}\right)^3} + \frac{2}{\left(1-\frac{1}{2}z^{-1}\right)^2} + \frac{4}{\left(1-\frac{1}{2}z^{-1}\right)} \protect$ (7.12)

and the three residues associated with the pole $ z=1/2$ are 1, 2, and 4.

Let $ r_{ij}$ denote the $ j$th residue associated with the pole $ p_i$, $ j=1,\ldots,m_i$. Successively differentiating $ (1-p_iz^{-1})^{m_i}H(z)$ $ k-1$ times with respect to $ z^{-1}$ and setting $ z=p_i$ isolates the residue $ r_{ik}$:

\begin{eqnarray*} r_{i1} &=& \left.(1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i}\\... ...ac{d^3}{d(z^{-1})^3} (1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i} \end{eqnarray*}

or

$\displaystyle \zbox {r_{ik} = \left.\frac{1}{(k-1)!(-p_i)^{k-1}}\frac{d^{k-1}}{d(z^{-1})^{k-1}} (1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i}} $

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