Meromorphic functions
Definition. A function is meromorphic on a domain if it is holomorphic except at isolated poles. Equivalently, locally it is a quotient of holomorphic functions with nonzero denominator not identically zero.
Prescribed principal parts
Theorem. The Mittag-Leffler theorem says that for a discrete set of points in a domain and prescribed finite principal parts at those points, there exists a meromorphic function having exactly those principal parts.
Convergence correction
Proof. One cannot simply sum all principal parts globally because the series may diverge. Subtract suitable Taylor polynomials from the local principal parts so the corrected series converges normally on compact subsets away from the poles.
Partial fractions
Example. Classical partial fraction expansions, such as expansions for \(\pi\cot\pi z\), are Mittag-Leffler constructions where the poles and principal parts are prescribed at the integers.
Relation to Weierstrass products
Remark. Weierstrass builds holomorphic functions with prescribed zeros; Mittag-Leffler builds meromorphic functions with prescribed poles. Together they describe the flexible global construction side of one-variable complex analysis.
Quotient representation
Corollary. On the complex plane, meromorphic functions can be represented as quotients of entire functions. Use Weierstrass products to create a denominator with the prescribed poles, then remove the poles from the product.
Prescribing principal parts
Theorem (Mittag--Leffler). Let \(\{a_n\}\subset\Omega\) be discrete, and at each \(a_n\) prescribe a finite principal part \[ P_n(z)=\sum_{k=1}^{m_n}c_{n,k}(z-a_n)^{-k}. \] There exists a meromorphic \(f\) on \(\Omega\) whose principal part at each \(a_n\) is \(P_n\) and which has no other poles.
Convergence corrections
Proof on the plane. Exhaust \(\mathbb C\) by discs and order \(a_n\to\infty\). On the \(n\)th compact disc, \(P_n\) is holomorphic for all sufficiently large \(n\). Approximate its Taylor expansion there by a polynomial \(q_n\) so accurately that \[ \sup_{|z|\le n}|P_n(z)-q_n(z)|\le2^{-n} \] whenever \(a_n\) lies outside the disc. Then \[ f(z)=\sum_n\{P_n(z)-q_n(z)\} \] converges normally away from the prescribed points. Near \(a_j\), every term except \(P_j\) is holomorphic, so the desired principal part remains. A domain version uses Runge approximation or a compact exhaustion with holomorphic corrections.
Partial fractions
Example. Prescribing simple poles of residue \(1\) at all integers leads, after symmetric correction, to \[ \pi\cot(\pi z) =\frac1z+\sum_{n=1}^{\infty} \left(\frac1{z-n}+\frac1{z+n}\right). \] The paired terms decay as \(O(n^{-2})\); the uncorrected sum \(\sum_{n\in\mathbb Z}(z-n)^{-1}\) does not converge absolutely.
Meromorphic functions as quotients
Theorem. Every meromorphic function on a domain in \(\mathbb C\) is locally, and globally on a planar domain, a quotient of holomorphic functions with no common zeros.
Proof. Use Weierstrass products or the zero-set theorem to construct a holomorphic \(g\) whose zeros match the poles of \(f\) with their orders. Then \(h=fg\) extends holomorphically across every pole, and \(f=h/g\). Remove common zero factors if necessary.
Additive and multiplicative existence theorems
Mittag--Leffler solves an additive problem: prescribe polar parts. Weierstrass solves a multiplicative problem: prescribe zeros. Together they construct meromorphic functions with specified divisors and local data. On a compact Riemann surface, global residue and divisor-degree constraints appear, so arbitrary planar prescriptions are no longer possible.