### Method

As can be seen from the code listing, this implementation of
`residuez` simply calls `residue`, which was written to
carry out the partial fraction expansions of -plane
(continuous-time) transfer functions :

where is the ``quotient'' and is the ``remainder'' in the PFE:

where is the order of the quotient polynomial in , and is the

*multiplicity*of the th pole. (When all poles are distinct, we have for all .) For , we define .

In the discrete-time case, we have the -plane transfer function

For compatibility with Matlab's

`residuez`, we need a PFE of the form such that

where .

We see that the -plane case formally does what we desire if we
treat -plane polynomials as polynomials in instead of
. From Eq.(J.2), we see that this requires reversing the
coefficient-order of `B` and `A` in the call to
`residue`. In the returned result, we obtain terms such as

**Next Section:**

Example with Repeated Poles

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The Padé-Prony Method