Find the Taylor series of argtanh(x) using sinh(x) and cosh(x)

Question

I just finished my exam a few hours a go, and there was 1 question I couldn't answer. I was asked to derive the Taylor series of $\arg\tanh(x)$ using the fact that $$\tanh(x)=\frac{\sinh(x)}{\cosh(x)},$$ $\cosh(x)=\frac{e^x+e^{-x}}{2}$ and $\sinh(x)=\frac{e^x-e^{-x}}{2}$.
Now I'm still scratching my head trying to figure out how I should have done this. What I did is differentiate $\cosh(x)$ and $\sinh(x)$, but then I didn't know what to do next.

I don't know if they will give solutions to this exam, but if they do, it's at least in 2 to 3 months, and that's a long time. Also I searched on the internet but didn't find something useful (maybe I didn't search well enough??),

Thanks for your help !

Answer 1

A convenient way to arrange the computation is by differentiating

$$tc=s,$$

(with hopefully obvious shorthands) giving

$$t'c+ts=c$$

then

$$t''c+2t's+tc=s, \\t'''c+3t''s+3t'c+ts=c, \\t''''c+4t'''s+6t''c+4t's+tc=s, \\t'''''c+5t''''s+10t'''c+10t''s+5t'c+ts=c, \\\cdots$$

If we set $x=0$, these simplify to

$$t_0=0, \\t'_0=1, \\t''_0+t_0=0, \\t'''_0+3t'_0=1, \\t''''_0+6t''_0+t_0=0, \\t'''''_0+10t'''_0+5t'_0=1, \\\cdots$$

More generally, the even coefficients are zero and

$$t_0^{(2n+1)}=1-\sum_{k=1}^{n} \binom {2n+1}{2k+1}t_0^{(2n+1-2k)}.$$