Talk:Zeta-function method for regularization

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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 \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 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

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