The Inverse Tangent  arctan(z) .  Verify that the formula  

(iii)              

is correct.  (At least for values of z in the upper half plane  .)

Explore Formula (iii).

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, for values of z in the upper half plane  .  

(c) 2006 John H. Mathews, Russell W. Howell