Approximate solution of $x^a\tanh(x)=1$

Question

As the title says, I am concerned by approximations of the solution of $$x^a\tanh(x)=1\tag1$$ where $a$ is a positive real number. From a numerical point of view, there is no problem to obtain the exact solution.

When $a$ is large, the asymptotics is $$x \sim 1+\frac{\beta}{a}+\frac{\beta (\beta -4 \sinh (\beta))}{2 a^2}\qquad \text{where} \qquad \beta=\log (\coth (1))$$ For example, if $a=5$, the above formula leads to $1.04994$ while the exact solution is $1.05042$ (relative error of $0.05$%).

Now, what I wanted is to get for any value of $a$ a good estimate of the solution without any iterative calculations. So what I did is to use one single iteration of Newton and higher order methods as described here using, for conveniency $x_0=1$; this is equivalent to solving for zero the numerator of the $[1,k]$ Padé approximant ($k=0$ corresponding to Newton, $k=1$ to Halley, $k=3$ to Householder).

As a result, the approximate solution is given by $$x_{(k)}=1+\frac{P_k(a)}{P_{k+1}(a)}$$ where $P_n(a)$ is a polynomial of degree $n$ the coefficients of which being themself polynomials in $e$. For example, for Halley's method $(k=1)$ the expression is $$x_{(1)}=1+\frac{2 \left(e^2-1\right) a+4}{e^2(e^2-1) a^2+\left(5 e^2-1\right) a+2}$$

Below are given some results for small $k$'s (without any problem, I generated the expressions up to $k=10$) $$\left( \begin{array}{ccccccc} a & x_{(0)} & x_{(1)}& x_{(2)} & x_{(3)} & x_{(4)} & \text{exact} \\ 1 & 1.2384058 & 1.1970317 & 1.1997872 & 1.1996704 & 1.1996799 & 1.1996786 \\ 2 & 1.1353353 & 1.1124979 & 1.1142730 & 1.1141829 & 1.1141849 & 1.1141850 \\ 3 & 1.0944859 & 1.0791714 & 1.0802947 & 1.0802452 & 1.0802458 & 1.0802462 \\ 4 & 1.0725789 & 1.0611595 & 1.0619446 & 1.0619148 & 1.0619149 & 1.0619151 \\ 5 & 1.0589184 & 1.0498479 & 1.0504375 & 1.0504180 & 1.0504179 & 1.0504180 \end{array} \right)$$

Where came my surprise is when I tried for small values of $a$

$$\left( \begin{array}{ccccccc} a & x_{(0)} & x_{(1)}& x_{(2)} & x_{(3)} & x_{(4)} & \text{exact} \\ \frac{1}{2} & 1.38502 & 1.32696 & 1.32578 & 1.32618 & 1.32629 & 1.32626 \\ \frac{1}{4} & 1.55598 & 1.51622 & 1.49335 & 1.49428 & 1.49557 & 1.49535 \\ \frac{1}{8} & 1.71463 & 1.77409 & 1.70469 & 1.69606 & 1.70227 & 1.70214 \\ \frac{1}{16} & 1.83357 & 2.08297 & 1.97567 & 1.92313 & 1.93420 & 1.93904 \\ \frac{1}{32} & 1.90923 & 2.38806 & 2.32647 & 2.18582 & 2.17615 & 2.19868 \\ \frac{1}{64} & 1.95246 & 2.63209 & 2.74297 & 2.52022 & 2.42933 & 2.47520 \\ \frac{1}{128} & 1.97565 & 2.79525 & 3.15685 & 2.95906 & 2.72910 & 2.76427 \\ \frac{1}{256} & 1.98768 & 2.89148 & 3.49084 & 3.47976 & 3.13242 & 3.06280 \end{array} \right)$$ which are not ridiculously bad.

The problem is that when $a \to 0$, $x \to \infty$ and I have no idea for expanding $(1)$ as a series for so large values of $x$ mainly because of the $\tanh(x)$ function.

Any idea or suggestion would really be welcome. Thanks in advance.

Edit

In his answer, Dr. Lutz Lehmann gave as an estimate $$x=\frac{1}{2} \left(\log \left(\frac{2}{a}\right)-\log \left(\log \left(\frac{1}{2} \log \left(\frac{2}{a}\right)\right)\right)\right)$$ and this looks to be an extremely good asymptotic formula for small values of $a$. For illustration using $a=10^{-p}$ $$\left( \begin{array}{ccc} p & \text{estimate} &\text{exact} \\ 1 & 1.9509850 & 1.7754966 \\ 2 & 2.6622064 & 2.6601002 \\ 3 & 3.6559407 & 3.6692855 \\ 4 & 4.7168233 & 4.7312764 \\ 5 & 5.8067083 & 5.8200164 \\ 6 & 6.9123770 & 6.9123770 \end{array} \right)$$

Answer 1

Taking the 1/1 Padé approximation $e^u=\frac{1+\frac12u}{1-\frac12u}$ of the exponential for $x^{-a}=e^{-a\ln x}$ results in comparing with $\tanh(x)=\frac{1-e^{-2x}}{1+e^{-2x}}$ in the approximation $$ \frac a2 \ln x=e^{-2x}\iff 2x=\ln\frac2a-\ln(\ln(x)) $$ which suggests $$ x_1=\frac12\left[\ln\frac2a-\ln\left(\ln\left(\frac12\ln\frac2a\right)\right)\right] $$ as a first approximation. To have a point of reference, use some method to get a more exact approximation of the solution, here as a quick-to-implement method use the fixed point iteration $x_{n+1}=\tanh^{-1}(x^{-a})$ and test the result on the original equation.

for n in range(2,20):
    a = 2**-n; x0 = log(2/a)/2;
    x = x1 = x0-log(log(x0))/2;
    for k in range(10): x=atanh(x**-a);
    print n, "& \\frac1{%d}&"%(2**n), x1, " & ", x, " & ", x**a*tanh(x)-1, "\\\\"

This gives on the sequence of $a$ in the last table in the question following table \begin{array}{rc|ll|l} n& a=2^{-n} & x_1 & x_{11} & x_{11}^a\cdot\tanh(x_{11})-1\\ \hline 2 & \frac1{4}& 2.66243094375 & 1.77633206765 & 0.0901611590769 \\ 3 & \frac1{8}& 1.94575146365 & 1.70270926716 & 0.000117344143408 \\ 4 & \frac1{16}& 2.03198848232 & 1.93904599147 & 8.19762613569e-07 \\ 5 & \frac1{32}& 2.23536105442 & 2.19868481845 & 9.19858500481e-09 \\ 6 & \frac1{64}& 2.4863932055 & 2.47520347264 & 1.16757048474e-10 \\ 7 & \frac1{128}& 2.76279455706 & 2.76427224535 & -8.78186412478e-13 \\ 8 & \frac1{256}& 3.05471753622 & 3.06279840543 & -3.53383988738e-13 \\ 9 & \frac1{512}& 3.35700216967 & 3.3685741973 & -4.59632332195e-14 \\ 10 & \frac1{1024}& 3.66663363816 & 3.68000759967 & -5.66213742559e-15 \\ 11 & \frac1{2048}& 3.98171067569 & 3.99593300799 & -7.77156117238e-16 \\ 12 & \frac1{4096}& 4.30096413533 & 4.31548248049 & 0.0 \\ 13 & \frac1{8192}& 4.62350862728 & 4.63799869386 & 0.0 \\ 14 & \frac1{16384}& 4.94870429689 & 4.96297571276 & -1.11022302463e-16 \\ 15 & \frac1{32768}& 5.27607516971 & 5.29001819157 & 0.0 \\ 16 & \frac1{65536}& 5.60525852668 & 5.61881285943 & -1.11022302463e-16 \\ 17 & \frac1{131072}& 5.93597223767 & 5.94910828218 & 0.0 \\ 18 & \frac1{262144}& 6.26799296135 & 6.28070027781 & 0.0 \\ 19 & \frac1{524288}& 6.60114117195 & 6.61342125003 & 0.0 \\ \end{array} This accuracy of the first digits holds also down to $a=10^{-16}$, after that the value of $x^a$ rounds to $1$ in floating point, which gives the unsolvable equation $1=\tanh(x)$.