Not exactly my idea of fun, but here's how you do it: erf(x) = 1 - 2/sqrt(pi) int_x^inf exp(-t^2) dt Now integrate "by parts": int_x^inf exp(-t^2) dt = int_x^inf (d/dt exp(-t^2)) (- dt/(2t)) = exp(-t^2) (-1/(2t))|_x^inf - int_x^inf exp(-t^2) (dt/(2t^2)) = exp(-x^2) (1/(2x)) - int_x^inf exp(-t^2) (dt/(2t^2)) You can keep going, integrating by parts the resulting integral. Notice that each time you do this the integrand gets another factor of t^2 in the denominator. This allows you to bound the error term. Unfortunately, for fixed x you cannot get arbitrarily small errors by working out sufficiently many terms. That's because the numerical coefficients grow faster than a power (from repeated differentiation of 1/t^n with increasing n). That's no surprise because the function erf(x) has an essential singularity at x = inf. The successive integrations by parts give an asymptotic series, not a convergent series. As a result you are limited in what you can say about the error for x fixed. On the other hand, if you fix the number of terms, say keeping only those we calculated above, erf(x) = 1 - exp(-x^2)/(sqrt(pi) x) + ... then you can say a lot about how the error behaves as x --> inf. This should suffice for x ~ 50. Here's my idea of fun. Sum the series 1 - 1 + 2 - 6 + 24 - 120 + 720 - 5040 + ... Veit On Mar 19, 2009, at 11:22 PM, wouter meeussen wrote:
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. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun