Square root of x² − 1

How should we define √(z² − 1)? Well, you could square z, subtract 1, and take the square root. What else would you do?! The question turns out to be more subtle than it looks. When x is a non-negative real number, √x is defined to be the non-negative real number whose square is x. When x is a complex number √x is defined to be a function that extends √x from the real line to the complex plane by analytic continuation. But we can’t extend √x as an analytic function to the entire complex plane ℂ. We have to choose to make a “cut” somewhere, and the conventional choice is to make a cut along the negative real axis. Using the principle branch The “principle branch” of the square root function is defined to be the unique function that analytically extends √x from the positive reals to ℂ \ (−∞, 0]. Assume for now that by √x we mean the principle branch of the square root function. Now what does √(z² − 1) mean? It could mean just what we said at the top of the post: we square z, subtract 1,…