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Does ArcTanh(ix)=iArcTan(x)?

  1. Sep 19, 2005 #1

    ..just wanted to make sure im reading this right..
  2. jcsd
  3. Sep 20, 2005 #2


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    That is correct.
  4. Sep 20, 2005 #3


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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]


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

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


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


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