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

Thread starter Agnostic
Start date Sep 19, 2005

AI Thread Summary

The discussion confirms the equality of ArcTanh(ix) and iArcTan(x) through a mathematical proof. It begins with the identity tanh(ix) = i tan(x), leading to the expression ix = arcTanh(i tan(x)). By substituting x with arctan(y), it shows that i arctan(y) equals arcTanh(iy). The proof concludes with the statement "q.e.d." indicating the completion of the argument. This establishes a clear connection between the two functions in complex analysis.

Sep 19, 2005

Agnostic

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

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Sep 20, 2005

Homework Helper

That is correct.

Sep 20, 2005

dextercioby

Science Advisor

Insights Author

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.

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