An analytic continuation is when a function that normally converges in a disk of convergence or a half plane can be extended to a meromorphic function (or, in some cases, a holomorphic function). For example, the function converges for . In the complex plane, this makes a circle of radius 1 centered at (0,0). This is often referred to as a disk of convergence. Inside the disk, this particular function is equal to . We can now define it as the analytic continuation and treat it as an extension of the original function, so in this example, we might find that . Analytic continuations are used with the Riemann zeta function, which allows us many interesting results, such as and . Interestingly, even though these properties seem to be only in pure mathematics, they are often used in many areas of theoretical physics, especially string theory.