- #1
zetafunction
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- 0
From the model used in the zeta regularization procedure to give a meaning to divergent series in the form [tex] 1+2+3+4+... [/tex] , we propose a similar method to give a finite meaning to divergent integrals in the form [tex] \int_{0}^{\infty}dx x^{m} [/tex] for positive 'm' in terms of the negative values of the Riemann Zeta function [tex] \zeta (s) [/tex].
for the case m=-1 due to the pole of the Riemann Zeta at s=1 we can use the expression of the functional determinant [tex] detA=exp(- \zeta '(0,a)) [/tex] where the comma means differentiation with respect to variable 'a' , in terms of the Hurwitz zeta
[tex] \prod (n+a) =exp(- \partial _{a} \zeta_{H}(0,a)) [/tex] this product must be understood in the sense of Zeta regularization
we give some examples for the one dimensional case and study how this method can be applied for multi-loop integral or multiple integrals by changing to polar coordinates and making the integral over the angular variables [tex] \int d\Omega [/tex] to be replaced by a discrete sum, so in the end we have one dimensional integrals in the form
[tex] \int_{0}^{\infty}dr r^{m-1}f(r) [/tex] using Laurent series expansion we can isolate the UV behavior and the regularize these integrals in terms of the negative values of the Riemann zeta, for the case of finite (convergent) integrals , the well-known result
[tex] \int_{0}^{n}dx x^{m}= \frac{n^{m+1}}{m+1} [/tex] is inmediatly obtained
FULL PAPER: avaliable at http://vixra.org/abs/1009.0047
for the case m=-1 due to the pole of the Riemann Zeta at s=1 we can use the expression of the functional determinant [tex] detA=exp(- \zeta '(0,a)) [/tex] where the comma means differentiation with respect to variable 'a' , in terms of the Hurwitz zeta
[tex] \prod (n+a) =exp(- \partial _{a} \zeta_{H}(0,a)) [/tex] this product must be understood in the sense of Zeta regularization
we give some examples for the one dimensional case and study how this method can be applied for multi-loop integral or multiple integrals by changing to polar coordinates and making the integral over the angular variables [tex] \int d\Omega [/tex] to be replaced by a discrete sum, so in the end we have one dimensional integrals in the form
[tex] \int_{0}^{\infty}dr r^{m-1}f(r) [/tex] using Laurent series expansion we can isolate the UV behavior and the regularize these integrals in terms of the negative values of the Riemann zeta, for the case of finite (convergent) integrals , the well-known result
[tex] \int_{0}^{n}dx x^{m}= \frac{n^{m+1}}{m+1} [/tex] is inmediatly obtained
FULL PAPER: avaliable at http://vixra.org/abs/1009.0047