On 7/9/14, Warren D Smith <warren.wds@gmail.com> wrote:
If x=erf(y), then say y=erfinv(x).
SMP jocks: What is the Maclaurin series of erfinv(x)?
Obviously the erfinv function is useful for statistics.
It also occurs to me: This series should have infinite radius of convergence, i.e. erfinv(x) should be a very well behaved "entire" function. Because: the derivative of erf is never zero anywhere in the complex plane, and the value of erf is always finite everywhere in the plane, so its inverse function should exist everywhere.
--Sorry, unfortunately, that reasoning was not valid. The same reasoning would lead to the "conclusion" that the inverse function of exp(y), i.e. ln(x), exists everywhere and is "entire" -- but actually, ln is a multivalued function and has a singularity at 0. The flaw in my wrong reasoning was that something infinitely far away in the y plane, may not be in the x plane. So... attempting to correct myself, I think the Maclaurin series for erfinv(x) ought to have radius of convergence equal to lim(y-->infinity) |erf(y)| = 1. Less nice. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)