Message: 6 Date: Thu, 19 Mar 2009 23:22:54 +0100 From: "wouter meeussen" <wouter.meeussen@pandora.be> To: "math-fun" <math-fun@mailman.xmission.com> Subject: [math-fun] error function: series development at infinity Message-ID: <000e01c9a8e1$3fb9f7f0$c887c554@pcwou> Content-Type: text/plain; charset="iso-8859-1"

dear math funners,

does anyone know of a series development of erf(x) round x-> infinity, or equivalently series of erf (1/x) around x=0? Alternatively, how to get approximations to log(1-erf(x)) for x around, say, 50 to 1000? Googled for it in vain,

Wouter.

One can try Mills ratio R(x) = cN(x)/D(N)(x) where cN(x) = 1 - N(x) is the complementary cumulative normal distribution and D(N) is the derivative. Then R(1/x) is a power series in x (!) with alternating coefficients a(n) = (-1)^n * (2*n)!/n!/(2^n) = (-1)^n * 2^n/Pi^(1/2)*GAMMA(n+1/2) (I have no proof at hand for this). They can be computed by recursion or approximately for large n and since they are alternating one has an error estimate for finite sums. May be that is just a re-formulation of the formula in Abramowitz.