Mittag-Leffler Theorem

The Mittag-Leffler theorem is the additive counterpart of the Weierstrass factorization theorem: it constructs meromorphic functions with prescribed poles and principal parts.

Statement

Theorem9.10Mittag-Leffler theorem

Let $\{a_n\}_{n=1}^\infty$ be a sequence of distinct points in $\mathbb{C}$ with $|a_n| \to \infty$ , and let $p_n(z) = \sum_{k=1}^{m_n}\frac{c_{n,k}}{(z-a_n)^k}$ be prescribed principal parts. Then there exists a meromorphic function $f$ on $\mathbb{C}$ whose poles are exactly the $\{a_n\}$ with the specified principal parts.

Moreover, $f$ can be written as

$f(z) = \sum_{n=1}^{\infty}[p_n(z) - q_n(z)]$

where $q_n(z)$ are polynomials (subtracted to ensure convergence).

Proof

Step 1: Choice of convergence-ensuring polynomials.

For each $n$ , expand $p_n(z)$ as a power series in $z$ about the origin (valid for $|z| < |a_n|$ ):

$p_n(z) = \sum_{k=0}^{\infty}b_{n,k}z^k \quad \text{for } |z| < |a_n|.$

Let $q_n(z)$ be the partial sum $\sum_{k=0}^{N_n}b_{n,k}z^k$ . Choose $N_n$ large enough so that $|p_n(z) - q_n(z)| < 2^{-n}$ on $|z| \leq |a_n|/2$ .

Step 2: Convergence.

The series $\sum_{n=1}^\infty [p_n(z) - q_n(z)]$ converges uniformly on compact subsets of $\mathbb{C} \setminus \{a_n\}$ . For a compact set $K$ , choose $N$ so that $|a_n| > 2\max_{z\in K}|z|$ for $n > N$ . Then $|p_n(z) - q_n(z)| < 2^{-n}$ on $K$ for $n > N$ , giving uniform convergence of the tail.

Step 3: Meromorphicity.

Each partial sum is meromorphic (with poles at $a_1, \ldots, a_M$ ), and the tail converges to a holomorphic function. Therefore the sum is meromorphic with the prescribed poles and principal parts. $\blacksquare$

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Applications

ExamplePartial fraction of $\csc^2(\pi z)$

The function $\csc^2(\pi z) = 1/\sin^2(\pi z)$ has double poles at all integers with principal part $1/(z-n)^2$ . By the Mittag-Leffler theorem:

$\frac{1}{\sin^2(\pi z)} = \frac{1}{(\pi z)^2} + \sum_{n \neq 0}\left[\frac{1}{(z-n)^2} - \frac{1}{n^2}\right] + C.$

Simplifying with symmetry: $\pi^2\csc^2(\pi z) = \sum_{n=-\infty}^{\infty}\frac{1}{(z-n)^2}$ .

ExamplePartial fractions for rational functions

For a rational function $R(z) = P(z)/Q(z)$ with simple poles $a_1, \ldots, a_n$ and residues $r_1, \ldots, r_n$ :

$R(z) = P_\infty(z) + \sum_{k=1}^n \frac{r_k}{z - a_k}$

where $P_\infty$ is the polynomial part. This is the classical partial fraction decomposition, which the Mittag-Leffler theorem extends to infinitely many poles.

Relation to Weierstrass Factorization

RemarkDuality between products and sums

The Weierstrass theorem constructs entire functions with prescribed zeros via infinite products. The Mittag-Leffler theorem constructs meromorphic functions with prescribed poles via infinite sums. Together, they show:

| Property | Weierstrass | Mittag-Leffler | |----------|-------------|----------------| | Goal | Prescribed zeros | Prescribed poles | | Method | Infinite product | Infinite sum | | Correction terms | Exponential factors | Polynomial subtractions | | Non-uniqueness | Up to $e^{g(z)}$ | Up to entire function |

Both theorems are instances of solving the Cousin problems on $\mathbb{C}$ (which are always solvable because $\mathbb{C}$ is a Stein manifold).