## Infinite Flatness at Infinity

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

(D.3) |

is

*zero*for all orders. Thus, even though is differentiable of all orders at , its series expansion fails to approach the function. This happens because has an

*essential singularity*at (also called a ``non-removable singularity''). One can think of an essential singularity as an infinite number of

*poles*piled up at the same point ( for ). Equivalently, above has an infinite number of zeros at , leading to the problem with Maclaurin series expansion. To prove this, one can show

(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

*infinitely flat*at in the ``Padé sense'':

- 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 [263].
- Lagrange interpolation (FIR) is maximally flat at dc [266].
- Thiran allpass interpolation has maximally flat group delay at dc [266].

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

(D.6) |

is satisfied at some point in

*every neighborhood*of , however small. In other words, comes arbitrarily close to

*every possible value*in

*any neighborhood*about an essential singular point. This was first proved by Weierstrass [42, p. 270].

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Integral of a Complex Gaussian

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