The Inverse Hyperbolic Cosine  arccosh(z) .  Verify that the formula  

(v)              

is correct, we can verify this graphically.   But it is correct only in quadrants  I  and  IV.

Explore Formula (v) .

First, use Mathematica to determine the formula for ArcCosh[z].  Start with the identity  .  

And we can verify that    is the inverse.

We can use Mathematica to verify the formula graphically.   (But only for values of  z  in quadrants  I  and  IV.)

Remark. However, for other places it might not agree!  For Example in Quadrant II and III.  Here we must use the other branch of square root!  We use  .