Can $\tanh(\arctan(x))$ be simplified or re-expressed in terms of algebraic functions (or maybe a log here and there), kind of like the forward-inverse identities? (perhaps using the myriad of trig and hyperbolic formulas)

The only thing I found is the Gudermannian function, which in some sense yields the opposite of what I need

$$gd(x) = 2 \arctan\left(\tanh\left(\frac{x}{2}\right)\right)$$

Also, when I let Mathematica spit out the MacLaurin series, it seems that $\tan(\arctan(x))$ and $\tanh(\arctan(x))$ have similar expansions (differing only by the usual alternating sign). Is there a place where these kind of relationships between the trigonometric and hyperbolic functions are studied? Anything known about the properties of the coefficients of such series?


How do you feel about $$ \tanh \arctan x = \frac{(1-\mathrm{i}x)^\mathrm{i} - (1+\mathrm{i}x)^\mathrm{i}}{(1-\mathrm{i}x)^\mathrm{i} + (1+\mathrm{i}x)^\mathrm{i}} \text{?} $$

This is obtained from $\tanh y = \frac{\mathrm{e}^{2 y}-1}{\mathrm{e}^{2 y}+1}$ and $\arctan x = \frac{\mathrm{i}}{2} \log \frac{1-\mathrm{i} x}{1+\mathrm{i} x}$, followed by some simplifying.

  • $\begingroup$ How do you get that $i$ symbol like that? Using a \text block seems clumsy. $\endgroup$ – Oscar Lanzi Oct 21 '19 at 23:36
  • 2
    $\begingroup$ @OscarLanzi : Even worse: "\mathrm{e}" Similarly for the "$\mathrm{i}$". It's so common on MSE, I type it automatically now. (I even forget in my own writing that I long ago created macros in my personal set for these.) $\endgroup$ – Eric Towers Oct 21 '19 at 23:47
  • $\begingroup$ @Eric Towers: I would prefer something not involving "i" (kind of like the log formulas for the inverse hyperbolic functions), but it seems that such thing may not exist... $\endgroup$ – Tamref Relue Oct 22 '19 at 0:05
  • $\begingroup$ @EricTowers Bless you for putting $\rm e$ and $\rm i$ upright like the true numbers they are $\endgroup$ – gen-ℤ ready to perish Oct 22 '19 at 0:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.