In this appendix, we evaluate certain sums and integrals that are useful in statistical mechanics. The sums are

Equation (A.4.1) is just the geometric series and can be proven by starting with the partial sums

Multiply this by x to get

Now subtract (A.4.8) from (A.4.7) and solve for S ₁( n). The result is

Since x < 1, taking the limit of (A.4.9) as n ? ? gives (A.4.1). Also, differentiating (A.4.9) gives

Equation (A.4.2) can be obtained from this by taking the limit for infinite n, but it is easier to differentiate (A.4.1) with respect to x to get

from which (A.4.2) follows immediately.

Equations (A.4.3) and (A.4.4) can be obtained by expanding the function f( x) = x ² in a Fourier series in the interval ?? to + ?. The result is

Letting x = 0 in this equation gives (A.4.3), and letting x = ? gives (A.4.4).

A similar procedure works for (A.4.5). Expand f( x) = x ⁴ in a Fourier series to get

Letting x = ?, (A.4.13) becomes

Replacing the first sum on the right by ?/6 according to (A.4.4) and solving for the second sum gives (A.4.5)

To evaluate (A.4.6), note that it has the form of a Taylor expansion, so a function F( x) exists such that

with the derivatives evaluated at x = 0 being...