Method of calculating the vacuum energy divergence

[Haorong Wu]

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, 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.

 

[renormalize]

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?

Just after your quotation from pg. 16 of Birrell & Davies, Quantum field theory in curved space, it states:
"This method of temporarily making divergent quantities finite by continuing the dimension of spacetime away from integer values forms the basis of dimensional regularization (see chapter 6)."
What don't you understand about this explanation? Are you familiar with dimensional regularization?

 

[Haorong Wu]

Thanks, @renormalize. I mistakenly thought the integral value was somehow related to the integration in the equation.