Mittag-Leffler theorem

The Mittag-Leffler theorem guarantees a meromorphic function on the complex plane with prescribed principal parts at a discrete set of points. Its construction uses holomorphic correction terms to make an infinite sum converge.

Principal part: The sum of negative-power terms in a Laurent expansion around an isolated singularity. The theorem starts by prescribing these local data at each allowed pole.
Meromorphic function: A complex function holomorphic except at isolated poles. The theorem constructs a meromorphic function with exactly the requested principal parts.
Karl Weierstrass: A German mathematician who helped establish modern analysis and complex function theory. His work on entire functions parallels Mittag-Leffler's construction from prescribed local data.
Partial fraction decomposition: An algebraic expression of a rational function as a sum of simpler fractions. The theorem extends the pole-by-pole logic of partial fractions beyond finite rational sums.
Residue theorem: A theorem equating a contour integral to 2πi times the sum of enclosed residues. On compact settings, residue sums constrain pole data; the plane theorem allows arbitrary discrete prescriptions.
Laurent series: A series in integer powers of a complex variable that represents a function on an annulus. Laurent expansions encode each prescribed principal part near its pole.
Holomorphic function: A complex function that is complex differentiable throughout an open domain. The freedom to add holomorphic terms is central to assembling the prescribed poles.
Gösta Mittag-Leffler: A Swedish mathematician known for contributions to complex analysis and for founding Acta Mathematica. The theorem bears his name and reflects his work on meromorphic functions.
Hadamard factorization theorem: A theorem describing entire functions of finite order through their zeros and an exponential factor. Together, factorization and Mittag-Leffler constructions organize global functions from zeros or poles.
Riemann-Roch theorem: A theorem relating meromorphic functions with bounded poles on a compact Riemann surface to its genus. Unlike the plane theorem, it quantifies finite-dimensional restrictions on compact surfaces.

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Behnke–Stein theoremRelated: It exemplifies the global meromorphic constructions available on noncompact surfaces.