The Mittag-Leffler theorem on the decomposition of a meromorphic function is one of the main theorems of the theory of analytic functions, which gives an analogue of the decomposition of a rational function into simple fractions for meromorphic functions.
Let meromorphic function {\ displaystyle f (z)} has points {\ displaystyle z = a_ {k}, | a_ {1} | \ leqslant | a_ {2} | \ leqslant \ ldots \ leqslant | a_ {k} | \ leqslant \ ldots} poles with main parts {\ displaystyle g_ {k} ({\ frac {1} {z-a_ {k}}}) = G_ {k} (z)} let it go {\ displaystyle h_ {k} ^ {(p)} = G_ {k} (0) + G_ {k} ^ {1} (0) z + \ ldots + {\ frac {G_ {k} ^ {(p) } (0)} {p!}} Z ^ {p}} there will be segments of Taylor decompositions {\ displaystyle g_ {k} \ left ({\ frac {1} {z-a_ {k}}} \ right)} by degrees {\ displaystyle z} . Then there is such a sequence of integers {\ displaystyle p_ {k}} and such an entire function {\ displaystyle f_ {0} (z)} that for everyone {\ displaystyle z \ neq a_ {k}} decomposition takes place {\ displaystyle f (z) = f_ {0} (z) + \ sum _ {k = 1} ^ {\ infty} \ left \ {g_ {k} \ left ({\ frac {1} {z-a_ {k}}} \ right) -h_ {k} ^ {p_ {k}} (z) \ right \}} absolutely and uniformly converging in any finite circle {\ displaystyle | z | \ leqslant A} .
Any meromorphic function {\ displaystyle f (z)} representable as the sum of a series {\ displaystyle f (z) = h (z) + \ sum _ {n = 0} ^ {\ infty} \ left (g_ {n} (z) -P_ {n} (z) \ right)} where {\ displaystyle h} - whole function {\ displaystyle g_ {n}} - the main parts of the Laurent decompositions at the poles {\ displaystyle f (z)} numbered in ascending order of their modules, and {\ displaystyle P_ {n}} - some polynomials.