Why is this entire function a really good asymptotic for , where for large positive numbers, ?
As |x| gets larger, the error term is asymptotically , and the error term for . If we treat as an infinite Laurent series, than it does not converge.
I stumbled upon the result, using numerical approximations, so I can't really explain the equation for the coefficients, other than it appears to be the numerical limit of a pseudo Cauchy integral for the coefficients as the circle for the Cauchy integral path gets larger. I suspect the formula has been seen before, and can be generated by some other technique. By definition, for any entire function , we have for any value of real r:
The conjecture is that this is an equivalent definition for , where and