Closer look at an identity

The previous post derived the identity and said in a footnote that the identity holds at least for x > 1 and y > 1. That’s true, but let’s see why the footnote is necessary. Let’s have Mathematica plot The plot will be 0 where the identity above holds. The plot is indeed flat for x > 1 and y > 1, and more, but not everywhere. If we combine the two square roots and plot again we still get a valid identity for x > 1 and y > 1, but the plot changes. This is because √a √b does not necessarily equal √(ab) when the arguments may be negative. The square root function and the arccosh function are not naturally single-valued functions. They require branch cuts to force them to be single-valued, and the two functions require different branch cuts. I go into this in some detail here. Related posts Inverse cosine Branch cuts and Common Lisp Duplicating a plot from A & S The post Closer look at an identity first appeared on John D. Cook.