Analytic continuation

Revision as of 19:59, 19 August 2015 by Pi3point14 (talk | contribs)

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. For example, the function $\sum_{n=1}^{\infty}{a_nr^{n}}$ converges for $\midn \mid <1$ (Error compiling LaTeX. Unknown error_msg). 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 $\frac{a_n}{1-n}$. 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 $\sum_{n=1}^{\infty}{n^2} = \frac{1}{1-2} = -1$. 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.