Proving the Equality of ArcTanh(ix) and iArcTan(x)

[Agnostic]

..just wanted to make sure I am reading this right..

 

[TD]

That is correct.

 

[dextercioby]

Here's the proof

$$\tanh ix=\frac{\sinh ix}{\cosh ix}=\frac{i\sin x}{\cos x}=i\tan x$$

Therefore

$$ix=\mbox{arc}\tanh \left(i\tan x\right)$$

Pick $x=\arctan y$

So

$$i\arctan y=\mbox{arc}\tanh \left(i\tan\left(\arctan y\right)\right)=\mbox{arc}\tanh iy$$

q.e.d.

Daniel.