Hm, this is strange. I just tried numerically the half iterate of \( e^x-1 \) and it looks quite convergent.
However I read in the article of Erdös and Jabotinsky [1]:
Now I am confused. If someone wants to compare, I computed the half iterate of \( f(x):=e^x-1 \) to be
Where is the mistake?
[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] M. Levin, MSc. Thesis, Israel Institute of Technology, 1960.
However I read in the article of Erdös and Jabotinsky [1]:
Quote:The function \( e^z-1 \) was shown by I. N. Baker [2] to have no real non-integer iterates. M. Levine [3] showed, using some results of the present paper, that this function and the functions \( z+z^2 \) and \( \frac{z}{1-z^2} \) have no analytic iterate.
Now I am confused. If someone wants to compare, I computed the half iterate of \( f(x):=e^x-1 \) to be
\( f^{\circ 1/2}(x)=x+{\frac {1}{4}}{x}^{2}+{\frac {1}{48}}{x}^{3}+{\frac {1}{3840}}{x}^{
5}-{\frac {7}{92160}}{x}^{6}+{\frac {1}{645120}}{x}^{7}+{\frac {53}{
3440640}}{x}^{8}-{\frac {281}{30965760}}{x}^{9}+O \left( {x}^{10}
\right) \)
I verified that indeed \( f^{1/2}\circ f^{1/2}=f \) and it converges for example for \( x=1.0 \)5}-{\frac {7}{92160}}{x}^{6}+{\frac {1}{645120}}{x}^{7}+{\frac {53}{
3440640}}{x}^{8}-{\frac {281}{30965760}}{x}^{9}+O \left( {x}^{10}
\right) \)
Code:
1.0,1.250000000,1.270833333,1.270833333,1.271093750,1.271017795,1.271019345,1.271034749,1.271025674,1.271025591,1.271029198,1.271027503Where is the mistake?
[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] M. Levin, MSc. Thesis, Israel Institute of Technology, 1960.