Mittag-Leffler: Prescribing the Poles

The mirror image of Weierstrass

The previous guide built a function from scratch to have any zeros you asked for: the Weierstrass factorization theorem takes a prescribed list of zeros a_1, a_2, a_3, ... going off to infinity and produces an entire function vanishing exactly there. This guide asks the dual question. Instead of zeros, prescribe poles — and not just where the poles sit, but precisely how the function blows up at each one. The answer is the Mittag-Leffler theorem, the pole-side twin of Weierstrass, and the second pillar of constructive function theory.

To say what 'blows up like THIS' means precisely, recall the Laurent series. Near an isolated singularity z_k the function splits into a tame part (a power series, holomorphic at z_k) plus a principal part — the finite tail of negative powers c_(-1)/(z - z_k) + c_(-2)/(z - z_k)^2 + ... + c_(-m)/(z - z_k)^m. That principal part is the entire local description of a pole: it is what makes the function infinite there, and everything else is well-behaved. Mittag-Leffler's input is exactly a list of points together with one principal part you nail to each.

The naive sum, and why it diverges

Here is the obvious first attempt. You want a function whose principal part at z_k is some p_k(z) = c_(-1)/(z - z_k) + ... + c_(-m)/(z - z_k)^m. So just ADD them all up: f(z) = sum over k of p_k(z). Each term p_k contributes exactly the right blow-up at its own z_k and is holomorphic everywhere else, so if the sum converges it is meromorphic with precisely the poles and principal parts you ordered. Addition is the natural operation here, just as multiplication was for Weierstrass and zeros.

But the bare sum usually diverges. Picture poles at every nonzero integer, each a simple pole with principal part 1/(z - n). For a fixed z, the term 1/(z - n) behaves like -1/n for large n, and sum of 1/n is the harmonic series — divergent. The poles run off to infinity, yes, but not fast enough: their tails do not shrink quickly enough for the infinite sum to settle down. This is the exact same wall the infinite product hit in guide 1, and it is the same wall Weierstrass had to climb in guide 3. Prescribing is easy; making the prescription CONVERGE is the whole craft.

The fix: subtract a convergence-forcing polynomial

Weierstrass tamed his product by multiplying each factor by an exponential of a Taylor polynomial — the elementary factor. Mittag-Leffler does the additive analogue: he SUBTRACTS from each p_k a polynomial that cancels its slow-decaying tail, without touching the pole. Concretely, expand p_k(z) as a Taylor series around z = 0 (legitimate, since p_k is holomorphic near 0 when z_k is not 0), and let q_k(z) be the first few terms of that expansion. Then replace each term p_k by the corrected term p_k(z) - q_k(z).

Why does subtracting a polynomial leave the poles untouched? Because q_k is a polynomial — an entire function, with no poles anywhere. Subtracting it cannot create or destroy a singularity, and it does not alter the principal part of p_k at z_k one bit. What it DOES do is kill the leading terms of p_k's Taylor expansion, so the corrected term p_k - q_k is tiny on any fixed disk once |z_k| is large — small enough that the corrected sum sum over k of (p_k - q_k) converges uniformly on compact sets away from the poles. Uniform-on-compacta convergence is exactly what guarantees the limit is again holomorphic (off the poles), so f is genuinely meromorphic.

naive   :  f(z) = sum_k  p_k(z)                  -- right poles, but DIVERGES

fixed   :  f(z) = sum_k [ p_k(z) - q_k(z) ]      -- q_k = first terms of Taylor of p_k about 0

  * p_k(z)        principal part you prescribed at z_k  (the pole survives, untouched)
  * q_k(z)        a polynomial (entire, no poles) cancelling p_k's slow tail
  * the sum now converges uniformly on compacta away from the poles  ==>  f is meromorphic

Reading the recipe step by step

Let us walk the construction once, in order, so the moving parts are clear. Suppose you are given distinct points z_1, z_2, ... with |z_k| going to infinity, and for each a principal part p_k. (Points must not accumulate in any bounded region, or the poles would not be isolated and the result could not be meromorphic.) The procedure produces a meromorphic f with exactly those poles and principal parts.

1. Order the poles so |z_1| <= |z_2| <= ... and set aside any pole at the origin to add back at the end (its principal part needs no correction).
2. For each k, expand the principal part p_k(z) as a Taylor series about z = 0; this converges on the disk |z| < |z_k|, since p_k is holomorphic there.
3. Choose a degree n_k and let q_k be the Taylor polynomial of p_k up to degree n_k. Pick n_k large enough that the leftover p_k - q_k is, say, below 2^(-k) on the disk |z| <= |z_k|/2.
4. Form the corrected sum f(z) = sum over k of [ p_k(z) - q_k(z) ]. The 2^(-k) bounds make a geometric comparison, so the tail converges uniformly on every compact set that avoids the poles.
5. Conclude: f is meromorphic, with a pole exactly at each z_k whose principal part is the prescribed p_k, and no other poles.

Two honest caveats. First, f is NOT unique — adding any entire function g(z) to it changes nothing about the poles or principal parts, so 'the' Mittag-Leffler function is really a whole family differing by an entire function (just as a Weierstrass product is unique only up to multiplying by e raised to an entire function). Second, the choice of correction polynomials q_k is also not forced: any choice making the sum converge works, and different bookkeeping gives the same poles by different-looking formulas. The theorem guarantees existence and the poles; it does not hand you one canonical answer.

The headline example: the cotangent

The most beautiful payoff is a function you already know. The function pi cot(pi z) is meromorphic with a simple pole at every integer n, and a quick residue check (cosine over sine, sine vanishing simply) shows the residue at each pole is 1 — so its principal part at n is exactly 1/(z - n). That is a perfect Mittag-Leffler input: poles at all integers, each with principal part 1/(z - n). And as we saw, the naive sum of those diverges.

Apply the cure. For the pole at n, the correcting polynomial is the zeroth-order Taylor term of 1/(z - n) about 0, which is just the constant -1/n. Subtracting it, the corrected term becomes 1/(z - n) + 1/n, and pairing n with -n makes the harmonic-series problem cancel: 1/(z - n) + 1/(z + n) combines to 2z/(z^2 - n^2), which decays like 1/n^2 and sums beautifully. The result is the celebrated cotangent expansion: pi cot(pi z) = 1/z + sum over n>=1 of 2z/(z^2 - n^2). This is a genuine Mittag-Leffler series, and guide 5 will derive it and its cousins for csc, the partial fractions of sin, and friends in full.

Step back and see the whole arc of this rung. Guide 1 made infinite products converge; guide 2 read a function's zeros as the right data to factor on; guide 3 (Weierstrass) built a function with prescribed zeros multiplicatively. This guide is the additive mirror — Mittag-Leffler builds a function with prescribed poles, and the partial-fraction expansion is its everyday face. Together they say something sweeping: a meromorphic function is determined, up to an entire correction, by the LOCAL data at its zeros and poles. Guide 5 cashes the cotangent in for concrete identities you can use.