Does ArcTanh(ix)=iArcTan(x)?

  • Thread starter Agnostic
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  • #1
Agnostic
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..just wanted to make sure I am reading this right..
 

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  • #2
TD
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That is correct.
 
  • #3
dextercioby
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Here's the proof

[tex] \tanh ix=\frac{\sinh ix}{\cosh ix}=\frac{i\sin x}{\cos x}=i\tan x [/tex]

Therefore

[tex] ix=\mbox{arc}\tanh \left(i\tan x\right) [/tex]

Pick [itex] x=\arctan y [/itex]

So

[tex] i\arctan y=\mbox{arc}\tanh \left(i\tan\left(\arctan y\right)\right)=\mbox{arc}\tanh iy [/tex]

q.e.d.

Daniel.
 

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