# transforming a sum to an integral : why does it work?

The problem at hand has been discussed in loads of previous questions (1, 2, 3)), and my version can be stated as follows.

Consider the sum $$\sum_{\mathbf k} \ln(1+e^{-(\alpha+\beta \varepsilon_k)}) \ .$$ We are summing over, say, two dimonsional k-space lattice $\mathbf k = \frac{2\pi}{L}(n_x,n_y),$ where the $n_x,n_y$ run through the positive integers. I want to understand how we can write this as an integral. Our energy is given by $\varepsilon_k = \frac{\hbar^2 k^2}{2m}$.

• -1. Not clear what you are asking. What is your question? How does it differ from those you have cited? "Is this a situation where hand-waving arguments are used in physics?" is a question teaching styles rather than the content of physics. Why this procedure works is a question about mathematics rather than physics. – sammy gerbil Feb 18 '17 at 15:25
• @sammygerbil My question is, indeed, why and how this procedure works. Since it is used extensively in statistical mechanics, I thought it would be on-topic at physics.stackexchange. – Mussé Redi Feb 18 '17 at 15:35
• @sammygerbil I removed the phrase about hand-waving, to avoid distraction from my main problem. – Mussé Redi Feb 18 '17 at 15:54
• Is your problem in the steps you show, or in the remaining steps until you get the energy integral? – anonymous Feb 18 '17 at 16:44
• @anonymous My problem is in the step where we subsitute a function $f(\mathbf k)$ in the sum $\sum_{\mathbf k} f(\mathbf k)$ and the assumptions made from here on to get the integral. I would be most satisfied if I could follow each (implicit) step made in this procedure. I've had (have been avoiding) this confusion for some time. – Mussé Redi Feb 18 '17 at 17:58

Say we are summing over discrete points $\mathbf k$ in k-space $$\sum_\mathbf{k}\ f(\mathbf k)\ .$$
If we want to rewrite this as an integral we have to assume that the separation between the points in k-space $\Delta k$ is negligible in comparison to the volume of the k-space $V$.