Difference between revisions of "Analytic continuation"
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| − | 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 <math>\sum_{n=1}^{\infty}{a_nr^{n}}</math> converges for <math>\mid | + | 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 <math>\sum_{n=1}^{\infty}{a_nr^{n}}</math> converges for <math>\mid r \mid <1</math>. 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 <math>\frac{a_n}{1-n}</math>. 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 <math>\sum_{n=1}^{\infty}{n^2} = \frac{1}{1-2} = -1</math>. Analytic continuations are used with the [[Riemann zeta function]], which allows us many interesting results, such as <math>\sum_{n=1}^{\infty}{n} = \frac{-1}{12} and \sum_{n=1}^{\infty}{n^2} = 0</math>. 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]]. |
Revision as of 19:10, 19 August 2015
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 
. 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.
