## 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'':

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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