Analytic continuation

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 $\sum_{n=1}^{\infty}{ar^{n}}$ converges for $\mid r  \mid <1$. 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 $\frac{a}{1-n}$. 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 $\sum_{n=1}^{\infty}{2^n} = \frac{1}{1-2} = -1$. Analytic continuations are used with the Riemann zeta function, which allows us many interesting results, such as $\sum_{n=1}^{\infty}{n} = \frac{-1}{12}$ and $\sum_{n=1}^{\infty}{n^2} = 0$. 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.

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