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Does ArcTanh(ix)=iArcTan(x)?

  1. Sep 19, 2005 #1

    Agnostic

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    ..just wanted to make sure im reading this right..
     
    Agnostic, Sep 19, 2005
  2. jcsd
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  3. Sep 20, 2005 #2

    TD

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    That is correct.
     
    TD, Sep 20, 2005
  4. Sep 20, 2005 #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.
     
    dextercioby, Sep 20, 2005
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