the Mittag-Leffler theorem
/ MIT-tahg LEFF-ler /
Weierstrass tells you that you may dictate the zeros of an entire function. Mittag-Leffler is the dual freedom for poles: you may dictate the poles of a meromorphic function. More than that — you may dictate not just where the poles sit but exactly how each one blows up, its full principal part. The theorem is the existence half of the design problem for meromorphic functions: there is always a meromorphic function on the plane with exactly the poles and exactly the singular behaviour you ask for.
Here is the statement. Choose any sequence of distinct points b_1, b_2, ... with |b_n| -> infinity, and to each attach a principal part P_n(z) — a finite tail of negative powers like c_(-1)/(z - b_n) + c_(-2)/(z - b_n)^2 + ... . Then there exists a meromorphic function f whose only poles are the b_n, with principal part exactly P_n at b_n. The naive guess, just adding up the principal parts as the sum of P_n(z), usually diverges; Mittag-Leffler's fix is to subtract from each P_n a correcting polynomial p_n(z) (a partial sum of its Taylor expansion about the origin) so that the corrected series sum (P_n(z) - p_n(z)) converges uniformly on compacta. Adding back a freely-chosen entire function gives the general such f.
This is the partial-fraction principle taken to the infinite case, and it is exactly parallel to Weierstrass: where Weierstrass forces convergence of a PRODUCT with convergence-supplying exponential factors, Mittag-Leffler forces convergence of a SUM with convergence-supplying polynomial subtractions. The two together give you total design freedom: prescribe zeros by a Weierstrass product, prescribe poles and principal parts by a Mittag-Leffler sum, and combine. The famous model expansions — pi cot(pi z), pi^2 / sin^2(pi z), and others — are concrete instances where the prescribed principal parts are 1/(z - n) (or its square) at every integer.
Prescribe a simple pole with principal part 1/(z - n) at every integer n. Naively summing 1/(z - n) over all n diverges. Mittag-Leffler subtracts the correction 1/(-n) = -1/n from each term (for n != 0), and the symmetrized series 1/z + sum over n != 0 of [1/(z - n) + 1/n] converges. It equals pi cot(pi z) — a meromorphic function with exactly the prescribed simple poles.
Prescribed principal parts 1/(z - n) at the integers, made convergent by Mittag-Leffler's subtractions — the result is pi cot(pi z).
Like Weierstrass, this is an existence theorem, not a uniqueness theorem: you can always add any entire function to f without changing a single pole or principal part, so the poles and principal parts never determine f on their own.