Infinite Flatness at Infinity
The Gaussian is infinitely flat at infinity. Equivalently, the Maclaurin expansion (Taylor expansion about ) of
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
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
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 .
- Lagrange interpolation (FIR) is maximally flat at dc .
- Thiran allpass interpolation has maximally flat group delay at dc .
Another interesting mathematical property of essential singularities is that near an essential singular point the inequality
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].
Integral of a Complex Gaussian
Fitting a Gaussian to Data