View source for Talk:Zeta-function method for regularization

<i>The text below has been removed from the [[Zeta-function method for regularization|page]] because it is based on a source which does not occur in either MathSciNet or Zentralblatt für Mathematik. 
</i> --[[User:Ulf Rehmann|Ulf Rehmann]] 18:52, 3 September 2013 (CEST)
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'' zeta regularization for integrals ''

The zeta function regularization may be extended in order to include divergent integrals \begin{equation} \int_{a}^{\infty}x^{m}dx  \qquad  m >0 \end{equation} by using the recurrence equation

\begin{equation} \begin{array}{l}
\int\nolimits_{a}^{\infty }x^{m-s} dx =\frac{m-s}{2} \int\nolimits_{a}^{\infty }x^{m-1-s} dx +\zeta (s-m)-\sum\limits_{i=1}^{a}i^{m-s}  +a^{m-s}  \\
-\sum\limits_{r=1}^{\infty }\frac{B_{2r} \Gamma (m-s+1)}{(2r)!\Gamma (m-2r+2-s)}  (m-2r+1-s)\int\nolimits_{a}^{\infty }x^{m-2r-s} dx \end{array} \end{equation}

this is the natural extension to integrals of the Zeta regularization algorithm , this recurrence equation is finite since for \begin{equation} m-2r < -1 \qquad \int_{a}^{\infty}dxx^{m-2r}= -\frac{a^{m-2r+1}}{m-2r+1} \end{equation}
the integrals inside the recurrence equation are convergent, this zeta regularization approach is used in physics (see
[[renormalization]]) in order to get finite amplitudes to the divergent integral.

[a7] Garcia , Jose Javier http://prespacetime.com/index.php/pst/article/view/498 The Application of Zeta Regularization Method to the Calculation of Certain Divergent Series and Integrals Refined Higgs, CMB from Planck, Departures in Logic, and GR Issues & Solutions vol 4 Nº 3 prespacetime journal http://prespacetime.com/index.php/pst/issue/view/41/showToc