The Inverse Hyperbolic Tangent  arctanh(z) .  Verify that the formula  

(vi)               
            
is correct, we can verify this graphically.

Explore Formula (vi) .

First, use Mathematica to determine the formula for ArcTan[z].  Start with the identities  .  

The above formula looks different.  The following simplifications can be made and then the formulas will differ by the constant .  Since    both formulas are "right."  

   

And we can verify that    is the inverse.

We can use Mathematica to verify the formula graphically.   (At least for values of z in the upper half plane first quadrant .)