5 Postcript added that summarizes present knowledge
4 Fixed the URL to Lipton's weblog.

On Dick Lipton's weblog, I posted a brief essay on demi-exponential functions, which I repeat here:

To expand upon Ken's remarks regarding demi-exponential functions (which is a fine name for them!), the analytic structure of these functions derives from the Lambert $W$ function, which is the subject of a classic article On the Lambert W Function (1996) by Corless, Gonnet, Hare, Jeffrey, and Knuth (yes, one somehow knew that Donald Knuth's name would arise in connection to such an interesting function ... to date this article has received more than 1600 references).

The connection arises via the following construction. Suppose that a demi-exponential function $d$ satisfies $d \circ d \circ \dots \circ d \circ z = \gamma \beta^z$, where $d$ is composed $k$ times. We say that $k$ is the order of the demi-function, $\gamma$ is the gain and $\beta$ is the base. It is easy to show that the fixed points of $d$ are given explicitly in terms of the $n$-th branch of the Lambert function as $z_f = -W_n(-\gamma \ln \beta)/\ln \beta$. Then by a series expansion about these fixed points (optionally augmented by a Pade resummation) it is straightforward to construct the demi-exponential functions both formally and numerically.

Provided the demi-exponential base and gain satisfy $\gamma \le 1/(e \ln \beta)$, such that the fixed points associated to the $n=-1$ branch of the $W$-function are real and positive, this construction yields smooth demi-exponential functions that pleasingly accord with our intuition of what demi-exponential functions should'' look like.

Counter-intuitively though, whenever the specified gain and base are sufficiently large that $\gamma > 1/(e \ln \beta)$, then the demi-exponential function has no real-valued fixed points, but rather develops jump-type singularities. In particular, the seemingly reasonable parameters $\beta=e$ and $\gamma=1$ have no smooth demi-exponential function associated to them (at least, that's the numerical evidence).

Perhaps this is one reason that demi-exponential functions have a reputation for being difficult to construct ... it is indeed very difficult to construct smooth functions for ranges of parameters such that no function has the desired smoothness!

It might be feasible (AFAICT) to write an article \emph{On On demi-exponential functions associated to the Lambert W Function}Function, and to include these functions in standard numerical packages (SciPy, MATLAB, Mathematica, etc.).

Some tough challenges would have to be met, however. Especially, there is at present no known integral representation of the demi-exponential functions (known to me, anyway), and yet such a representation would be very useful (perhaps even essential) in rigorously proving the analytical structures that the numerical Pade approximants show us so clearly.

Here's what these functions look like:

Mathematica script here (PDF).

On Scott Aaronson's Dick Lipton's weblog, I have posted numerical examples of half-exponential a brief essay on demi-exponential functions, which were numerically computed by high-order Padé approximant techniques.

The specific compositional equation solved I repeat here:

To expand upon Ken's remarks regarding demi-exponential functions (which is $f_\beta\circ f_\beta = \beta\ \text{exp}$, so that $\beta$ may be regarded as a half-exponential gain parameter.

The Padé approximants paint fine name for them!), the following picture analytic structure of these functions derives from the Lambert $f_\beta(z)$:

(1) YesW$function,$f_\beta(z)$which is analytic in$z$and$\beta$.the subject of a classic article On the Lambert W Function (2) No1996) by Corless,$f_\beta(z)$is not Gonnet, Hare, Jeffrey, and Knuth (yes, one somehow knew that Donald Knuth's name would arise in connection to such an entire interesting function ; it ... to date this article has a real-axis branch cut for$-\infty < z < -1/2$. (3) On received more than 1600 references). The connection arises via the following construction. Suppose that a demi-exponential function$-1/2 < z$real axis with d$ satisfies $\beta$ real, d \circ d \circ \dots \circ d \circ z = \gamma \beta^z$, where$f_\beta(z)$d$ is smooth iff composed $0 < \beta \le 1/e$k$times. For We say that$\beta > 1/e$k$ is the order of the demi-function, $\gamma$ is the function gain and $f_\beta(z)$ exhibits discrete jumps (see graph above) that are accompanied by \beta$is the well-known Gibbs phenomenon (that base. It is , numerical overshoot); these are associated easy to show that the well-known branch cut fixed points of$d$are given explicitly in terms of the Lambert function$W_{-1}(-\beta)$. Perhaps surprisingly, n$-th branch of the compositional relation Lambert function as $f_\beta\circ f_\beta z_f = -W_n(-\gamma \beta\ ln \text{exp}$ is satisfied even beta)/\ln \beta$. Then by a series expansion about these "jump" solutions. It thus appears that the conventional gain choice$\beta = 1$, which has often appeared in the mathematical literature fixed points (and also here on MathOverflowoptionally augmented by a Pade resummation) it is unfortunate from an analytic point of view; smaller values of straightforward to construct the demi-exponential functions both formally and numerically. Provided the demi-exponential base and gain satisfy$\beta \gamma \le 1/e$yield the smooth "half-exponential" behavior 1/(e \ln \beta)$, such that is most natural for information-theory purposes.

When I have a little more material (and time) available, I will post as a MathOverflow question/exercise the problems fixed points associated to the $n=-1$ branch of (1) proving the above assertions rigorously, and (2) providing numerical algorithms for computing $f_\beta(z)$ efficiently W$-function are real and to arbitrary accuracypositive, for all (complex) values this construction yields smooth demi-exponential functions that pleasingly accord with our intuition of$\beta$what demi-exponential functions should'' look like. Counter-intuitively though, whenever the specified gain and base are sufficiently large that$z$. In \gamma > 1/(e \ln \beta)$, then the interimdemi-exponential function has no real-valued fixed points, my algorithms for constructing high-order Padé approximants to $f_\beta(z)$ are herebut rather develops jump-type singularities. In summaryparticular, the strikingly simple (and numerically robust) analytic structure of the Padé approximants provides seemingly reasonable evidence that the above picture of the analytic structure of parameters $f_\beta(z)$ is likely \beta=e$and$\gamma=1$have no smooth demi-exponential function associated to confirmed by future more rigorous investigationsthem (at least, that's the numerical evidence). It therefore appears Perhaps this is one reason that —with considerable further mathematical work—$f_\beta(z)\$ may eventually take its place as a fully-qualified member of the pantheon of special demi-exponential functions have a reputation for being difficult to construct ... if it is indeed very difficult to construct smooth functions for no other reason, than ranges of parameters such that this no function has the desired smoothness!

It might be feasible (AFAICT) to write an article \emph{On demi-exponential functions associated to the Lambert W Function}, and to include these functions in standard numerical packages (SciPy, MATLAB, Mathematica, etc.). Some tough challenges would have to be met, however. Especially, there is at present no known integral representation of the demi-exponential functions (known to me, anyway), and yet such a distinctly pleasing appearancerepresentation would be very useful (perhaps even essential) in rigorously proving the analytical structures that the numerical Pade approximants show us so clearly.Good!

Here's what these functions look like:)

2 Added graph of half-exponential function
1