Inverting series with logs and W

Question

You've all heard it: what does a drowning analytic number theorist say? Log log log log....

I very frequently deal with the sorts of functions that one comes across and want to invert them. Generally there's no nice form for them, but a series (usually about $\infty$, but occasionally at other places) is just fine, and even just a few terms will suffice -- three, say.

Is there a good way to automate this tedious, tiresome process?

For my most recent example, in simplified form, I'm looking to solve for $a$ in $$ n = \exp\left(c\frac{\log a}{\log\log a}\right) $$ as a series about $n=\infty$ where I am pretending $c$ is a constant (it has a constant, nonzero limit). Of course $$ \frac1c\log n = \frac{\log a}{\log\log a} $$ but then you need to use Lambert's $W$, choose the right branches, and take series. There's got to be a way to automate this. As annoying as this example is, the last one had $$ \frac{\log\log x \log\log\log x}{\log\log\log\log x} $$ and that was a real pain, too. Yes, you can write $\ell=\log\log x$ and simplify it to $$ \frac{\ell\log\ell}{\log\log\ell} $$ but with three pieces trying to equate it to some $f(y)$ and invert was not fun.

Answer 1

There are some general and some specific methods for finding series representations. The peculiarity of your problem is finding an inverting series.

If you don't have the inverse relation, you can use Lagrange inversion in many cases.

see also, for example:

https://mathoverflow.net/questions/111400/good-computer-package-for-calculating-inverse-of-a-formal-power-series

Morse, Ph. M.; Feshbach, H.: Methods of theoretical physics. Volume 1. McGraw-Hill, Yew York/Toronto/London, 1953. p. 411: Inversion of Series

SFA, a package on symmetric functions considered as operators over the ring of polynomials for the computer algebra system Maple. J. Symbolic Comput. 29 (2000) 83-94

Dominici, D.: Nested derivatives: a simple method for computing series expansions of inverse functions. Int. J. Math. Math. Sci. (2003) (58) 3699-3715

Feinsilver, Ph.; Schott, R.: Operator Calculus Approach to Solving Analytic Systems. 2006

Feinsilver, Ph.; Schott, R.: Inversion of analytic functions via canonical polynomials: a matrix approach. 2007

Nachbagauer, K.: Power series solutions to holonomic differential equations and the general algebraic equation. Diploma thesis Johannes Kepler Universität Linz. 2009

Often series representations in terms of Bell polynomials are suitable.

Let $1<n\in\mathbb{N}$, $A$ an n-ary algebraic function and $f_1,...,f_n$ algebraically independent unary functions in the complex numbers. In general, we don't know how to invert $A(f_1(z),...,f_n(z))$ because the partial inverses of $A$ aren't closed-form functions because they are multi-valued.
If $A$ is unary, the partial inverses are given through the partial inverses of $f_1,...f_n$, if they are allowed.

There are some theorems for non-existence of elementary (partial) inverses of elementary functions and for non-existence of solutions of equations of elementary functions that are elementary numbers.

I assume you want to invert elementary functions or equations of elementary functions.

Wheeler, N.: Functional inversion strategies with emphasis on their application to inversion problems oosed by Napier, Lambert & Sommerfeld. 2017
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1.) Elementary functions, Lambert W, Generalized Lambert W, Hyper Lambert W, Generalized Hyper Lambert W

Inverse relations of some simpler $\mathbb{C}$-algebraic expressions of both $z$ and $e^z$ are presented in the literature:

How can we show that $A(z,e^z)$ and $A(\ln (z),z)$ have no elementary inverse?

What are the closed-form inverses of $x \sinh(x), x \cosh(x), x \tanh(x), x\ \text{sech}(x), x \coth(x), x\ \text{csch}(x)$?

What are the closed-form inverses of $x+\sinh(x)$, $x+\cosh(x)$?

see also:

Solve for $x$ in $^nx = i$

Consider also the posts for Inverse Beta Regularized at Math Stack Exchange, e.g. the answer of Tyma Gaidash at Surprising Generalizations.
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2.) Lambert W and Generalized Hyper-Lambert W for simple examples

I want to present the method of closed-form functions with the Special functions Lambert W and Generalized Hyper-Lambert W which offer themselves in your example.

$k,k_1,k_2,k_3\in\mathbb{N}$

$$^k\ln(x)= \left\{ \begin{array}{lr} x; & k=0 \\ \underbrace{\ln(\ln(...\ln(x)))}_{k\ \text{times}}; & \text{else} \end{array} \right\}$$

$$^k\exp(x)= \left\{ \begin{array}{lr} x; & k=0 \\ \underbrace{\exp(\exp(...\exp(x)))}_{k\ \text{times}}; & \text{else} \end{array} \right\}$$ $\ $

$$^{k_1}\exp\left(c\frac{^{k_2}\ln(x)}{^{k_2+1}\ln(x)}\right)=n:$$ $$x=\ ^{k_2}\exp\left(-\frac{^{k_1}\ln(n)}{c}W\left(-\frac{c}{^{k_1}\ln(n)}\right)\right)$$ $\ $

$$^{k_1}\exp\left(c\frac{^{k_2+1}\ln(x)}{^{k_2}\ln(x)}\right)=n:$$ $$x=\ ^{k_2}\exp\left(-\frac{c}{^{k_1}\ln(n)}W\left(-\frac{^{k_1}\ln(n)}{c}\right)\right)$$ $\ $

Your equations can be solved by Generalized Hyper-Lambert W in a systematic way. So the corresponding series representations are available. I want to show this for real $x$ and $n$.

$$F(n,m,x)= \left\{ \begin{array}{lr} \exp(x); & n=1 \\ \exp(c_{m-(n-1)}F(n-1,m,x)); & n>1 \end{array} \right\}$$

$$G(k,x)=xF(k+1,k+1,x)$$

Generalized Hyper-Lambert W $\text{HW}(c_1,...,c_k;x)$ is the inverse relation of $G(k,x)$.

$$^{k_1}\exp\left(c\frac{^{k_2}\ln(x)}{^{k_3}\ln(x)}\right)=n:$$

$$t= \left\{ \begin{array}{lr} -\text{HW}\left(-1,\underbrace{1,...,1}_{|k_2-k_3|-2\ \text{times}};-\left(\frac{\ ^{k_1}\ln(n)}{c}\right)^{\text{signum}(k_2-k_3)}\right); & |k_2-k_3|=1 \\ \text{HW}\left(-1,\underbrace{1,...,1}_{|k_2-k_3|-2\ \text{times}};\left(\frac{\ ^{k_1}\ln(n)}{c}\right)^{\text{signum}(k_2-k_3)}\right); & \text{else} \end{array} \right\}$$ $$x=\ ^{\max(k_2,k_3)}\exp(t)$$

Galidakis, I. N.: On solving the p-th complex auxiliary equation $f^{(p)}(z)=z$. Complex Variables 50 (2005) (13) 977-997

Galidakis, I. N.: On some applications of the generalized hyper-Lambert functions. Complex Variables and Elliptic Equations 52 (2007) (12) 1101-1119