fractional iteration of $xe^x$ has zero convergence radius? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T18:46:58Z http://mathoverflow.net/feeds/question/46731 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46731/fractional-iteration-of-xex-has-zero-convergence-radius fractional iteration of $xe^x$ has zero convergence radius? bo198214 2010-11-20T12:19:36Z 2010-11-24T22:51:02Z <p>The equation $f(f(x))=xe^x=x+x^2+\frac{x^3}{2}+\frac{x^4}{6}+\dots$ has a unique formal powerseries solution. Is its convergence radius 0 as was shown by Baker for the equation $f(f(x))=e^x-1$? Or more generally: Let $F(x)=xe^x$ does the unique formal powerseries solution of $F(f_t(x))=f_t(F(x))$ with $f_t(x)=x+t x^2 +\dots$ have convergence radius 0 for all non-integer numbers $t$?</p> http://mathoverflow.net/questions/46731/fractional-iteration-of-xex-has-zero-convergence-radius/47253#47253 Answer by Gottfried Helms for fractional iteration of $xe^x$ has zero convergence radius? Gottfried Helms 2010-11-24T18:29:54Z 2010-11-24T22:51:02Z <p>I don't have an analytically founded answer, but like in the thread on the iteration of sin(x) <a href="http://mathoverflow.net/questions/45608/formal-power-series-convergence" rel="nofollow">http://mathoverflow.net/questions/45608/formal-power-series-convergence</a> I've plotted the characteristics of the coefficients of the powerseries for the half-iterate of x*exp(x). In the first plot we see, that the log of the absolute values of the coefficients grow nearly linearly with their index, so this is a hypergeometric series. See <a href="http://go.helms-net.de/math/images/xexpxcoeff_c_256.png" rel="nofollow">growth</a>.<br> Also look at the <a href="http://go.helms-net.de/math/images/xexpxcoeff_d_256.png" rel="nofollow">differences</a> of that values; this is even more suggestive because of the typical pattern which suggests, that the differences of the logs grow asymptotic linearly with the logs of the index, which is just another expression for hypergeometric growthrate. So the convergence-radius for the half-iterate seems to be zero as is for the half-iterate of $exp(x)-1$ .</p>