Dealing with Repeated Poles Analytically

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

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

$$= \frac{1}{\left(1-\frac{1}{2}z^{-1}\right)^3} + \frac{2}{\left(1-\frac{1}{2}z^{-1}\right)^2} + \frac{4}{\left(1-\frac{1}{2}z^{-1}\right)} \protect$$ (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 $p_i$, $i=1,\ldots,m_i$. Successively differentiating $(1-p_iz^{-1})^{m_i}H(z)$ $j$ 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

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