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Renormalisation refers to replacing infinite results of a an infinite summation with a finite number through logically sort-of sensible means. A classical, intuitive example of Renormalisation is a Ramanujam Sum.

ExamplesEdit this section

Ramanujam SummmationEdit this section

Take, for example, the classical example, of the sum of all positive integers:

$ \sum_{k=1}^\infty k $

Clearly, thus sum diverges. However, notice that if one would use the (Wrongly, actually, to speak rigorously) definition Riemann Zeta Function that only applies to positive integers, that is,

$ \zeta(s)= \sum_{k=1}^\infty \frac{1}{k^{-s}} $

We see that this is just $ \zeta (-1) $. In reality, however, the zeta function is not defined this way for negative integers. If we use the actual definition of the Riemann Zeta Function to calculate this $ \zeta(-1) $, then we see that the result is $ \frac1{12} $. Therefore, we renormalise this sum to $ \frac1{12} $.

Of course, it is obvious that this is not rigorous, however, it is very useful from a Physics point of view.

Ramanujam summation

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The Ramanujam Summation is essentially a property of the partial sums, rather than a property of the entire sum, as that doesn't exist. If we take the Euler–Maclaurin summation formula together with the correction rule using Bernoulli numbers, we see that:

$ \begin{align} {} &\frac{1}{2}f\left( 0\right) + f\left( 1\right) + \cdots + f\left( n - 1\right) + \frac{1}{2}f\left( n\right) \\ = &\frac{1}{2}\left[f\left( 0\right) + f\left( n\right)\right] + \sum_{k=1}^{n-1}f\left(k\right)\\ = &\int_0^n f(x)\,dx + \sum_{k=1}^p\frac{B_{k + 1}}{(k + 1)!}\left[f^{(k)}(n) - f^{(k)}(0)\right] + R_p \end{align} $

Ramanujan[1] wrote it for the case $ p $going to infinity:

$ \sum_{k=1}^{x}f(k) = C + \int_0^x f(t)\,dt + \frac{1}{2}f(x) + \sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k - 1)}(x) $

where $ C $ is a constant specific to the series and its analytic continuation and the limits on the integral were not specified by Ramanujan, but presumably they were as given above. Comparing both formulae and assuming that $ R $ tends to 0 as $ x $ tends to infinity, we see that, in a general case, for functions $ f (x) $ with no divergence at $ x $ = 0:

$ C(a)=\int_0^a f(t)\,dt-\frac{1}{2}f(0)-\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(0) $

where Ramanujan assumed $ \scriptstyle a \,=\, 0 $. By taking $ \scriptstyle a \,=\, \infty $ we normally recover the usual summation for convergent series. For functions $ f ( x ) $ with no divergence at $ x =1 $ , we obtain:

$ C(a) = \int_1^a f(t)\,dt+ \frac{1}{2}f(1) - \sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(1) $

$ C(0) $ was then proposed to use as the sum of the divergent sequence.

In particular, the sum of all positive integers is

$ 1+2+3+\cdots = -\frac{1}{12}\ (\Re) $

where the notation $ \scriptstyle (\Re) $ indicates Ramanujan Summation.

External LinksEdit this section

ReferencesEdit this section

  1. Bruce C. Berndt, Ramanujan's Notebooks, Ramanujan's Theory of Divergent Series, Chapter 6, Springer-Verlag (ed.), (1939), pp. 133-149.