f(f(x))=exp(x)-1 and other functions "just in the middle" between linear and exponential
Let me see if I can summarize the conversation so far. If we want $f(f(z)) = e^z+z-1$, then there will be a solution, analytic in a neighborhood of the real axis. See either fedja's Banach space argument, or my sketchier iteration argument. The previous report of numerical counter-examples were in error; they came from computing $(k! f_k)^{1/k}$ instead of $f_k^{1/k}$. We do not know whether this function is entire. If it is, then there must be some place on the circle of radius $R$ where it is larger than $e^R$. (See fedja's comment here.)
If we want $f(f(z)) = e^z-1$, there is no solution, even in an $\epsilon$-ball around $0$. According to mathscinet, this is proved in a paper of Irvine Noel Baker, Zusammensetzungen ganzer Funktionen, Math. Z. 69 (1958), 121--163. However, there are two half-iterates (or associated Fatou coordinates $\alpha(e^z - 1) = \alpha(z) + 1$) that are holomorphic with very large domains. One is holomorphic on the complex numbers without the ray $\left[ 0,\infty \right)$ along the positive real axis, the other is holomorphic on the complex numbers without the ray $\left(- \infty,0\right]$ along the negative real axis. And both have the formal power series of the half-iterate $f(z)$ as asymptotic series at 0.
If we want $f(f(z))=e^z$, there are analytic solutions in a neighborhood of the real line, but they are known not to be entire.
I'll make this answer community wiki. What else have I left out of my summary?
Here is a related MO question. The answers to the new question contain further interesting information. Let me mention here a link with many references on "iterative roots and fractional iterations" one particular link on the iterative square root of exp (x) is here.
The following two links mentioned in the old blog discussion may be helpful
- http://www.math.niu.edu/~rusin/known-math/97/sqrt.exp (outdated link)
http://www.math.niu.edu/~rusin/known-math/99/sqrt_exp (outdated link)
http://web.archive.org/web/20140521065943/http://www.math.niu.edu/~rusin/known-math/97/sqrt.exp
- http://web.archive.org/web/20140521065943/http://www.math.niu.edu/~rusin/known-math/99/sqrt_exp
Allow me to leave alone the particular equation you mention and the issue of series, and focus instead on the general idea of finding functions "in the middle" between two families of functions. There is some extremely interesting mathematics in that idea.
The essence of this part of your question is that you have two families of functions, in your case the linear functions and the exponential functions, and the first family lies below the second in the sense that every function in the lower family is eventually dominated by every function in the upper family. Because of this, it is very natural to want to understand the functions that lie between the two classes. In what circumstances and for which types of families $L$ and $U$ can we always find a function $f$ filling the gap? That is, we seek a function $f$ that eventually dominates the functions in the lower family $L$ and is eventually dominated by the functions in the upper family $U$. It is natural to consider the cases where the families are maximal in some sense, and as a special case, one might consider what happens when they are linearly ordered by eventual domination.
Much of the content of this question is present already in the case of functions $f:\mathbb{N}\to \mathbb{N}$, and indeed, it turns out that much of the fundamental phenomenon occurs already for functions $g:\mathbb{N}\to 2$, which amounts to considering the quotient $P(\omega)/Fin$, as in this MO answer.
This way of thinking is intimately connected with the phenomenon of Hausdorff gaps.
First, if both families are countable (or are determined by a countable sub-family, which is true in your case), then it is an enjoyable exercise to show that one may always fill the gap (first proved by Hausdorff). That is, given two countable families of functions, members of the first always eventually dominated by members of the second, then there is a function filling the gap.
Second, Hausdorff constructed examples of families of functions that do not admit any function in the middle; these gaps cannot be filled. That is, he produced a lower family $L$ and and upper family $U$, such that every function in the lower family was eventually dominated by every function in the upper family, but there is no function just in the middle, filling the gap. His examples were unfilled gaps having uncountable order type $(\omega_1,\omega_1)$, in the sense that the both the lower and upper families are determined by an almost-increasing $\omega_1$-sequence of functions.
The unfillable nature of these gaps, however, admits extensive set-theoretic independence, in the sense that an unfilled gap can sometimes be filled by a function that is added by forcing, that is, by moving to a larger set-theoretic universe. At the same time, there are methods of sealing a gap, that prevent it from ever being filled in a cardinal-preserving forcing extension.
Kunen proved that it is consistent with Martin's axiom plus $\neg CH$ that there are unfilled gaps of type $(\omega_1,c)$ and $(c,c)$, where $c$ is the continuum, and also consistent that all such gaps are filled.
There is a unique formal power series solution with $f(0) = 0$ and $f'(0) = 1$. I had supposed that the coefficients would all be positive, which would imply that they are smaller than for $\exp(x)$ itself and thus that $f(x)$ is entire. No such luck. Maple gives me this:
$$f(x) = x + \frac{x^2}4 + \frac{x^3}{48} + \frac{x^5}{3840} - \frac{7x^6}{92160} + \frac{x^7}{645120} + \frac{53x^8}{3440640} + \cdots.$$
This doesn't say much about the possible radius of convergence of this series. On the other hand, expecting it to be entire may have been naive from the beginning, because it seems unlikely that $f(f(x))$ would be periodic in the imaginary direction.
Since Michael Lugo has found evidence that the Taylor series has zero radius of convergence, it's not a very good way to describe or even define $f(x)$. Is it clear that there is a unique $f(x)$ which is convex (at least for $x \ge 0$), and that that $f$ is smooth at 0 and real analytic away from $0$? There is a book on fractional iteration of functions that presumably addresses these issues.