Which Method to Use for Evaluating Sums and Integrals?

[latentcorpse]

at the very end of this lecture
http://www.ph.ed.ac.uk/~pmonthou/Statistical-Mechanics/documents/SM10.pdf

and the very beginning of this lecture
http://www.ph.ed.ac.uk/~pmonthou/Statistical-Mechanics/documents/SM11.pdf

we look at evaluating this sum by making it into an integral by two different methods :

(i) the sum and integral are both in n-space
(ii) the sum in n-space is converted into either an integral in k-space or energy space.

How do you know which method to use? would it just depend on the information given to you in an exam, i.e. if you were given say $g(\epsilon)$ you would realize you have to convert to energy space - or am i completely missing the point here?

 

[turin]

It's just a change of variables. Both methods are basically equivalent, assuming the approximation at the very end of SM10 (10th wk. of Stat. Mech., I presume). However, the first one determines Z, whereas the second one gives the density of states, which is usually in terms of energy (momentum), not the mode number of energy (momentum).