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I've cherrypicked the appropriate branch of arcosh to allow the method to 'work'.

Is this a valid integration method?

Using the substitution $x=\cosh\theta$ we get:

$$\int_{-1}^{1}\frac{dx}{\sqrt{1-x^{2}}}=\int_{\operatorname{arcosh}\left(-1\right)}^{\cosh^{-1}\left(1\right)}\frac{d\theta}{i}=-i\left(\cosh^{-1}\left(1\right)-\operatorname{arcosh}\left(-1\right)\right)=-i\left(0-\left(-i\pi\right)\right)=\pi$$

I would concede that it's rather convenient to simply choose $\operatorname{arcosh}\left(-1\right)=-i\pi$ so I'm wondering if in fact it's even a legal move.

In solving $$\frac{e^{w}+e^{-w}}{2}=-1$$ it can be observed that $w=i(2n+1)\pi$ where $n∈ℤ$, so I'm left wondering what exactly justifies choosing $$\operatorname{arcosh}\left(-1\right)=-i\pi$$ instead of choosing any of the infinitely many other options, such as $+i\pi$

Simon M
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    What is the difference between the functions $cosh^{-1}$ and $arccosh$? – MasB Oct 05 '23 at 20:33
  • If I understand correctly arcosh refers to the principal branch of the inverse hyperbolic cosine function when allowing complex numbers as arguments – Simon M Oct 05 '23 at 20:35
  • I was under the impression that $\cosh^{-1}\left(-1\right)$ is undefined over the real numbers – Simon M Oct 05 '23 at 20:37

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