the cotangent partial-fraction expansion

The cotangent expansion is the single most celebrated partial-fraction identity, the model on which all the others are patterned. The function pi cot(pi z) is meromorphic with a simple pole at every integer n, and at each one its principal part is exactly 1/(z - n). Mittag-Leffler then assembles it as a convergent sum of those simple fractions: pi cot(pi z) = 1/z + sum over n >= 1 of [1/(z - n) + 1/(z + n)] = 1/z + sum over n >= 1 of 2z/(z^2 - n^2), valid for all non-integer z.

Why the pairing of +n with -n? Summing 1/(z - n) over all integers separately diverges (the terms behave like -1/n, the harmonic tail). Pairing the pole at n with the pole at -n produces 1/(z - n) + 1/(z + n) = 2z/(z^2 - n^2), whose terms now decay like 1/n^2, so the symmetric sum converges absolutely. This pairing is Mittag-Leffler's convergence correction made concrete — and it is forced by the structure of the poles, not a free choice. The lone term 1/z handles the pole at the origin.

This one identity is a workhorse. Differentiating it term by term gives the expansion pi^2 / sin^2(pi z) = sum 1/(z - n)^2. Expanding it as a power series about z = 0 and matching coefficients against the known Taylor series of cot produces the values of the Riemann zeta function at even integers — most famously sum 1/n^2 = pi^2/6 (the Basel problem) and sum 1/n^4 = pi^4/90. And integrating it recovers the sine product formula. The cotangent expansion is the hub from which the sine product, the sin^(-2) expansion, and the even-zeta values all radiate.