The Gaussian is *infinitely flat* at infinity. Equivalently, the
Maclaurin expansion (Taylor expansion about
) of

(D.3) |

is

(D.4) |

for all . This follows from the fact that exponential growth or decay is faster than polynomial growth or decay. An exponential can in fact be viewed as an infinite-order polynomial, since

(D.5) |

We may call

- Padé approximation is
*maximally flat*approximation, and seeks to use all degrees of freedom in the approximation to match the leading terms of the Taylor series expansion. - Butterworth filters (IIR) are maximally flat at dc [#!JOSFP!#].
- Lagrange interpolation (FIR) is maximally flat at dc [#!PASP!#].
- Thiran allpass interpolation has maximally flat group delay at dc [#!PASP!#].

Another interesting mathematical property of essential singularities is
that near an *essential singular point*
the
inequality

(D.6) |

is satisfied at some point in

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University