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In this method, the Z-transform is expanded into a power series. This approach is useful when the Z-transform function is rational, allowing for the approximation of the inverse by expanding into a series and determining the signal coefficients term by term.
Conversely, every polynomial is a power series with only finitely many non-zero terms. Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform).
The probability generating function is an example of a generating function of a sequence: see also formal power series. It is equivalent to, and sometimes called, the z-transform of the probability mass function.
Faà di Bruno's formula gives coefficients of the composition of two formal power series in terms of the coefficients of those two series. Equivalently, it is a formula for the nth derivative of a composite function. Lagrange reversion theorem for another theorem sometimes called the inversion theorem; Formal power series#The Lagrange inversion ...
Z-transform – the special case where the Laurent series is taken about zero has much use in time-series analysis. Fourier series – the substitution z = e π i w {\displaystyle z=e^{\pi iw}} transforms a Laurent series into a Fourier series, or conversely.
Consider the formal power series in one complex variable z of the form = = where ,.. Then the radius of convergence of f at the point a is given by = (| | /) where lim sup denotes the limit superior, the limit as n approaches infinity of the supremum of the sequence values after the nth position.
Series multisection provides formulas for generating functions enumerating the sequence {+} given an ordinary generating function () where ,, , and <.In the first two cases where (,):= (,), (,), we can expand these arithmetic progression generating functions directly in terms of ():
The polylogarithm function is defined by a power series in z, ... (see e.g. discrete Fourier transform). Another important property, the inversion formula, ...