09/03/2014, 01:04 PM
(This post was last modified: 09/03/2014, 01:31 PM by sheldonison.)

(09/01/2014, 10:24 PM)tommy1729 Wrote: It seems my friend ( our ? ) mick has given us ( myself and sheldon in particular ) credit for the use of " fake function theory "...

here is the link :

http://math.stackexchange.com/questions/...for-real-x

As I mentioned before z - ln(2sinh(z)/z) is asymptotic to ln(z) near the real line, but it fails to be entire ( sinh is periodic ! ).

Guess we could consider the fake ln(x^2 + 1) here too.

Unlike the half-exp we are considering more standard functions which might give intresting closed form results !

Thanks for the link Tommy! I've been overseas vacationing... I was able to use the basic recipe from this post to interpolate as an even function, just as you suggested! I will post more later, either here or at Mathstack.

I have also been working on an example that should allow us to make the "fake function" theory more rigorous. A very interesting asymptotic function is below. It is interesting because the corresponding function is simply , whose derivative is x, so , so all of the Taylor series coefficients of the interpolating function have an exact closed form, as do all of the error equations.

for this function, the Gaussian approximation is also the best approximation.

this is a converging Laurent series. We don't use the negative coefficients for the entire interpolant.

The Laurent series interpolant can also be expressed in a quickly converging closed form.

f(x) corresponds to n=0

- Sheldon