Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $t$?

share|improve this question
add comment

1 Answer

I don't have an analytically founded answer, but like in the thread on the iteration of sin(x) formal power series convergence 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 growth.
Also look at the differences 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 $ .

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.