08/10/2007, 10:04 PM
Oh damn, you were faster than me. I was just preparing a similar post!
So there remains for me the task of adding and correcting
First in the formula for the double binomial expansion the coefficient (-1)^{n-1-i} is missing. The correct formula is (I follow the usage of Ecalle and write \( f^{\circ t} \) for the \( t \)-th iteration instead of \( f^{[t]} \) as you write it) :
\( {f^{\circ t}}_n=\sum_{i=0}^{n-1}
(-1)^{n-1-i}\left(t\\i\right)\left(t-1-i\\n-1-i\right){f^{\circ i}}_{n} \)
A criterion for the convergence of this expression is given by Jabotinsky and Erdös in [1]:
Theorem 1: If the radius of convergence of the series \( L(z)=\frac{\partial f^{\circ s}(z)}{\partial s}|_{s=0} \), where \( \left({\frac{\partial f^{\circ s}}{\partial s}|_{s=0}}\right)_n = \sum_{i=1}^{n-1} \frac{(-1)^{i+1}}{i} {f^{\circ i}}_n \), is
\( \varrho>0 \) then there are radii \( \varrho(s)>0 \) such that \( f^{\circ s}(z) \) is analytic in s and in z for all finite complex s and for \( |z|<\varrho(s) \)
Theorem 2: If the radius of convergence of L(z) is 0 then the radius of convergence of \( f^{\circ s}(z) \) is 0 for almost all complex s and for almost all real s.
A famous example for this second case is \( f(x)=e^x-1 \), see [2]. It has a fixed point at 0, and is parabolic: \( (e^x-1)'(0)=e^0=1 \). But the parabolic iterates \( f^{\circ s} \) converge only for integer s.
However this looks more terrible than it is, because there are so called asymptotic expansions. That means that in the development point the series does not converge, but the function approximates in a certain way the (formal) power series in the point of development, see [5]. For this case Ecalle [3] showed, that there is a unique continuous iteration that has the formal continuous iteration as its asymptotic expansion (for series with \( f_0=0 \) and \( f_1=1 \)). The first however who treated this case was Szekeres in [4].
[1] P. Erdös and E. Jabotinksy, On analytic iteration, J. Analyse Math. 8, 1960/1961, 361-376.
[2] I. N. Baker, Zusammensetzungen ganzer Funktionen, Math. Z 69, 1958, 121-163.
[3] J. Ecalle, Théorie des invariants holomorphes, Publications mathématiques d'Orsay 67-74 09, 1974.
[4] G. Szekeres, Regular iteration of real and complex functions, Acta Math. 100, 1958, 203-258.
[5] W. Balser, From divergent power series to analytic functions, Lecture Notes in Mathematics, Springer, 1994.
So there remains for me the task of adding and correcting
First in the formula for the double binomial expansion the coefficient (-1)^{n-1-i} is missing. The correct formula is (I follow the usage of Ecalle and write \( f^{\circ t} \) for the \( t \)-th iteration instead of \( f^{[t]} \) as you write it) :
\( {f^{\circ t}}_n=\sum_{i=0}^{n-1}
(-1)^{n-1-i}\left(t\\i\right)\left(t-1-i\\n-1-i\right){f^{\circ i}}_{n} \)
A criterion for the convergence of this expression is given by Jabotinsky and Erdös in [1]:
Theorem 1: If the radius of convergence of the series \( L(z)=\frac{\partial f^{\circ s}(z)}{\partial s}|_{s=0} \), where \( \left({\frac{\partial f^{\circ s}}{\partial s}|_{s=0}}\right)_n = \sum_{i=1}^{n-1} \frac{(-1)^{i+1}}{i} {f^{\circ i}}_n \), is
\( \varrho>0 \) then there are radii \( \varrho(s)>0 \) such that \( f^{\circ s}(z) \) is analytic in s and in z for all finite complex s and for \( |z|<\varrho(s) \)
Theorem 2: If the radius of convergence of L(z) is 0 then the radius of convergence of \( f^{\circ s}(z) \) is 0 for almost all complex s and for almost all real s.
A famous example for this second case is \( f(x)=e^x-1 \), see [2]. It has a fixed point at 0, and is parabolic: \( (e^x-1)'(0)=e^0=1 \). But the parabolic iterates \( f^{\circ s} \) converge only for integer s.
However this looks more terrible than it is, because there are so called asymptotic expansions. That means that in the development point the series does not converge, but the function approximates in a certain way the (formal) power series in the point of development, see [5]. For this case Ecalle [3] showed, that there is a unique continuous iteration that has the formal continuous iteration as its asymptotic expansion (for series with \( f_0=0 \) and \( f_1=1 \)). The first however who treated this case was Szekeres in [4].
[1] P. Erdös and E. Jabotinksy, On analytic iteration, J. Analyse Math. 8, 1960/1961, 361-376.
[2] I. N. Baker, Zusammensetzungen ganzer Funktionen, Math. Z 69, 1958, 121-163.
[3] J. Ecalle, Théorie des invariants holomorphes, Publications mathématiques d'Orsay 67-74 09, 1974.
[4] G. Szekeres, Regular iteration of real and complex functions, Acta Math. 100, 1958, 203-258.
[5] W. Balser, From divergent power series to analytic functions, Lecture Notes in Mathematics, Springer, 1994.