- #1

Haorong Wu

- 415

- 90

- TL;DR Summary
- When calculating the vacuum energy divergence, an author said, "This divergence can be usefully analysed by performing the integral with n continued away from integral values". What does it mean?

In the book,

I could get the answer by letting ##k=m\tan t ## and using the properties of Beta functions and Gamma functions, but I still do not understand what it means by saying "with ##n## continued away from integral values".

Thanks ahead.

*quantum fields in curved space*, when calculating the vacuum energy divergence for scalar fields, it reads:$$\sum_{\mathbf k}\omega/2=(L^2/4\pi)^{(n-1)/2}\frac 1 {\Gamma ((n-1)/2)} \int_0^\infty (k^2+m^2)^{1/2}k^{n-2}dk .$$ This divergence can be usefully analyzed by performing the integral with ##n## continued away from integral values to obtain $$-L^{n-1}2^{-n-1}\pi^{-n/2}m^n \Gamma(-n/2).$$

I could get the answer by letting ##k=m\tan t ## and using the properties of Beta functions and Gamma functions, but I still do not understand what it means by saying "with ##n## continued away from integral values".

Thanks ahead.