The Inverse Hyperbolic Sine  arcsinh(z).  Verify that the formula  

(iv)              

is correct, we can verify this graphically.  (At least for values of z in the upper half plane first quadrant.)

Explore Formula (iv) .

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

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.)