Talk:Zeta-function method for regularization

The text below has been removed from the page because it is based on a source which does not occur in either MathSciNet or Zentralblatt für Mathematik. --Ulf Rehmann 18:52, 3 September 2013 (CEST)

zeta regularization for integrals

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

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

this is the natural extension to integrals of the Zeta regularization algorithm , this recurrence equation is finite since for $$m-2r < -1 \qquad \int_{a}^{\infty}dxx^{m-2r}= -\frac{a^{m-2r+1}}{m-2r+1}$$ 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

How to Cite This Entry:
Zeta-function method for regularization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zeta-function_method_for_regularization&oldid=30323